# 100 Austrian Journal of Statistics, Vol. 32 (2003), No. 1&2,

Save this PDF as:

Size: px
Start display at page:

Download "100 Austrian Journal of Statistics, Vol. 32 (2003), No. 1&2, 99-129"

## Transcription

2 100 Austrian Journal of Statistics, Vol , No. 1&, wic is just an approximation to te model 1, wit ɛ = σ. Here V is te noisy observation of an unknown regression function f, ɛ is te resolving noise and W x represents a standard Wiener process. Tere exists a uge literature on te equivalence between tese two models, cf. e.g. Brown and Low 1996 and Nussbaum 1996, but tis does not cover our main problem ere, namely adaptive non-parametric estimation. Our approac is greatly influenced by a recent paper, Lepski and Levit 1998, wic was a milestone in adaptive estimation of infinitely differentiable functions, in te wite noise model. Below we will explain main differences between our approac and tat of Lepski and Levit In non-parametric statistics, classes of functions are in general described by smootness parameters. In tis paper we sall study classes of functions defined in terms of positive parameters γ, β and r wose interpretation will be explained below. We will study estimation of f in 1, under te assumption tat f belongs to te functional class Aγ, β, r wic is te collection of all continuous functions suc tat f γ,β,r := γ e γt r F[f]t dt 1. 3 Here F[f] represents te Fourier transform of f. Te collection of all suc classes will be called functional scale. Note tat wen te parameters are assumed known, we are dealing wit te problem of non-parametric estimation muc studied recently, especially since te publications, Ibragimov and Has miskii 1981, 198, 1983, Stone 198. Te situation in wic neiter of tese parameters is known a priori is muc more realistic and complex. A real progress in tis problem wic is usually referred to as adaptive estimation, as been only acieved in te last decade, most notably since te publication of Lepski 1990, 1991, 199a, 199b. Furter progress was acieved in Lepski and Levit 1998, For all γ, β, r, te class Aγ, β, r is a class of infinitely differentiable functions, and eac of te parameters affects te smootness and te accuracy of te best nonparametric estimators in its own way. Te parameter γ is some kind of scale parameter: one can verify tat f A1, β, r if and only if 1 f Aγ, β, r. Terefore, γ γ of all parameters, it affects te smootness of f most dramatically. Te bigger is γ, te smooter are te functions of te class. Te parameter β can be interpreted as a size parameter and represents te radius of te corresponding L -ellipsoid defined by 3. Note tat f Aγ, 1, r if and only if βf Aγ, β, r. Terefore te bigger is β, te less smoot are te functions of te class. Finally, r can be best described as a parameter responsible for te type of smootness. It is well known tat for r = 1 all functions in te class Aγ, β, r admit bounded analytic continuation into te strip z = x + iy : y < γ} of te complex plane Paley- Wiener teorem, and terefore for all r > 1 te functions in Aγ, β, r are entire functions i.e. functions admitting analytic continuation into te wole complex plane. For r < 1 tese functions are only infinitely differentiable, and teir smootness increases togeter wit r. In te Gaussian wite noise model Lepski and Levit 1998 studied adaptive estimation for even broader classes of functions wit rapidly vanising Fourier transforms

5 L.M. Artiles and B.Y. Levit 103 Te Model Let us formalize our model. Definition 1 Let γ, β, r > 0 be given. We denote by Aγ, β, r te class of continuous functions f : R R, wose Fourier transforms F[f] satisfy γ f γ,β,r := e γt r F[f]t dt 1. 4 In tis study we use te following definition of te Fourier transform, F[f]t = e itx fx dx. 5 Note tat te Fourier inversion formula fx = 1 e itx F[f]t dt 6 certainly olds under assumption 4. It is easy to see tat for all γ, β, r > 0, functions in Aγ, β, r are infinitely differentiable. Now, let us consider te following observation model y l = fl + ξ l, l = 0, ±1, ±,..., 7 were ξ l are i.i.d. Gaussian random variables, N 0, σ, σ > 0. We assume tat te function f belongs to te family Aγ, β, r, for some γ, β, r > 0. Our purpose is to estimate te unknown function fx based on te vector of observations y =..., y, y 1, y 0, y 1, y,.... We will coose our optimal estimator from te family of kernel type estimators ˆf,s x, y = were k s, s 0, is te so-called sinc-function k s x l y l 8 k s x = and k s 0 = s. Tis kernel as te property sin sx x, 9 and terefore, according to te convolution teorem, F[k s ]t = [ s,s] t 10 F[f k s ]t = [ s,s] t F[f]t, 11 were * represents te convolution operator. Te kernel k s is just one of many possible, but its very tractable properties make it an attractive tool: it elps significantly in te searc of te most general possible results and

6 104 Austrian Journal of Statistics, Vol , No. 1&, clarifies te underlying ideas. For practical purposes some oter kernels, suc as de la Vallée Poussin kernel cf. Nikol skiĭ, 1975, p. 301, may be more relevant and typically would work better. Te parameter s is called te bandwidt. As we sall see in Section 4, for any fixed class tere exists an optimum bandwidt s. Te optimum bandwidt will depend on parameters γ, β, r, σ as well as te index of te model, called te bin-widt, wic in our asymptotic study will tend to zero. Denote by f x, y an arbitrary estimator of fx based on te observations y. To sorten te notation we will often write f x instead of f x, y. Let P f be te distribution of te vector y and let E f and Var f denote te expectation and te variance wit respect to tis measure. Wen tere is no possibility of confusion we will simply write P, E and Var respectively. Let W be te class of loss functions wx, x R, suc tat wx = w x, wx wy for x y, x, y R, and for some 0 < η < 1 e ηx wx dx <. Wit an appropriate normalizing factor σ to be defined sortly, and w W, we will consider te maximum risk, over a fixed functional class Aγ, β, r, given by sup E f w σ 1 f x, y fx f Aγ,β,r as a global measure of te error of te estimator f over te wole class Aγ, β, r. Wen te classes Aγ, β, r are considered fixed, our main goal is to find an estimator suc tat te corresponding maximum risk is as small as possible, i.e. acieves asymptotically te minimax risk inf f sup E f w f Aγ,β,r σ 1 f x, y fx were f is taken from te class of all possible estimators. In te adaptive setting, we sall allow γ, β, r to vary freely inside large scales K. Conditions under wic an adaptive study is suitable are presented and a notion of adaptive asymptotic optimality is introduced based on distinguising, among all possible functional scales, between te so-called non-parametric NP and pseudo-parametric PP scales. 3 Auxiliary Results In tis section we present, for te reader s convenience, two auxiliary results wic will be used in te subsequent sections. Te aim of te first lemma is to approximate summation formulas by integrals, wit a good approximation error in te case of very smoot integrands. Tis result is a version of te celebrated Poisson summation formula. It

7 L.M. Artiles and B.Y. Levit 105 as been used in a similar situation in Golubev, Levit and Tsybakov Below Aγ, β, r, γ, β, r > 0 are te functional classes of infinitely differentiable functions previously defined and k s x is te kernel 9. Lemma 1 Te following properties old: a Let f, g be continuous functions in L R suc tat F[f], F[g] L 1 R, ten gx lfl y = 1 e itx y F[g]t F[f]t dt + 1 l 0 l ei y e itx y F[g]t F[f] t + l dt = gx zfz y dz + 1 l 0 l ei y e itx y F[g]t F[f]t + l dt. b For arbitrary numbers s 1, s 0 s 1 s denote x = k s x k s1 x. Ten, uniformly in γ, β, r, s i 0, i = 1,, x R and f Aγ, β, r as 0 x lfl = 1 were c r = max1, r 1. e itx F[ ]t F[f]t dt + O e γ r s /c r 1/ dt s 1 γ eγtr, c Let s 1, s and x be as before. Ten, uniformly in s 1, s and x R, for 0, x l = s s O1 s s 1. Proof. a Te proof is based on te formula e i lx = δx l, 1 known in te teory of distributions cf. e.g. Antonsik et al., 1973, C Using te Fourier inversion formula, te distributional formula 1 and wit some algebra, one obtains Note tat x = k s x for s 1 = 0 and s = s.

8 106 Austrian Journal of Statistics, Vol , No. 1&, gx lfl y = = = = 1 = 1 = 1 e itx F[g]t e isy F[f]s e itx l F[g]t dt e is tl dt ds s t e itx F[g]t e isy F[f]s δ l dt ds e itx F[g]t e itx F[g]t e e isy F[f]s δ s t l ds dt l it+ e itx y F[g]t F[f]t dt + 1 l 0 e i l y y F[f] t + l dt e isl y F[f]s ds e itx y F[g]t F[f] t + l dt = gx zfz y dz + 1 l 0 e i l y e itx y F[g]t F[f] t + l dt. b If f Aγ, β, r ten f belongs to L R according to te Parseval s formula. Also, F[f] L 1 R according to 4 and te Caucy-Scwartz inequality. Tus we can apply te previous result in a, using g = and y = 0. Note tat F[ ]t = s1,s ] t. Applying te Fourier inversion formula, te Caucy-Scwartz inequality and te c r - inequality, we obtain after a few transformations x lfl 1 e itx F[ ]t F[f]t dt 1 e itx F[ ]tf[f]t + l dt l 0 1 1/ γ e γt r F[f]t dt l 0 F[ ]t γ e γ t+ 1/ l r dt

9 L.M. Artiles and B.Y. Levit l 0 1/ s 1,s ] t β γ e γt r e lγ r /c r dt 1 l 0 lγ e r /c r 1/ s 1,s ]t β γ eγtr dt = 1 1 s 1/ s 1 γ eγtr dt l=1 e l γ r /c r s 1/ s 1 γ eγtr dt e γ r /c r + e γ xr /c r dx 1 = O e s γ r /c r 1/ dt s 1 γ eγtr, 0, were te last asymptotic can be easily derived by partial integration. c Applying a and taking f = g = and x = y, we see tat x l = x l l x Terefore = 1 F[ ]t dt + 1 x l s j s i l l 0 l=1 e i l x F[ ]t F[ ] t + l dt. F[ ]t F[ ] t + l dt s 1,s ] t s 1,s ] t + l dt wic completes te proof of te lemma. 5s s 1 = O1 s s 1, Te following elementary properties will be used below. Tey will elp in bounding te bias and te approximation errors. Lemma For any positive γ and r te following inequality olds s e γtr dt s e γsr rγs r 13

10 108 Austrian Journal of Statistics, Vol , No. 1&, for all s > t 0 were t 0 satisfies rγt 0 r = 1 and s 0 e γtr dt = s eγsr 1 + o1 14 rγs r uniformly in r < r < r + for γs, were r, r + > 0 are arbitrary fixed numbers. For te first inequality see e.g. Lepski and Levit 1998, eqs..8,.10. Te second property can be easily proven by partial integration. 4 Minimax Regression in Aγ, β, r 4.1 Optimality in te Case of Fixed Classes Te first result we present in tis section is obtained in te classical framework, i.e. in a situation were te function fx altoug unknown belongs to a given class. In oter words, te parameter α = γ, β, r of te class is known and fixed. Denote for sortness Aα = Aγ, β, r. We will prove tat asymptotically minimax estimators can be found among kernel estimators using a specified bandwidt and we will also calculate to a constant teir maximal asymptotic risk, for a variety of loss functions. Teorem 1 Let α > 0 and ω W. Ten for any x R, te kernel estimator ˆf = ˆf,s, in 8 wit te bandwidt s = s α, σ = 1 γ 1 log 1/r, 15 γσ satisfies lim 0 sup E f w f Aα σ s ˆf x fx lim 0 inf f = sup E f w f Aα σ s f x fx = E wξ were f is taken from te class of all possible estimators of f and ξ N 0, 1. Proof. Upper bound for te risk. Let us first study te sample properties of te family of estimators we use. According to te model for te observations 7 and te formula for te estimator 8 one can split te error term as follows, ˆf,s x fx = k s x lfl fx + k s x lξ l := bf, x, s, + vσ, x, s,.

11 L.M. Artiles and B.Y. Levit 109 For simplicity we sall write below b s = bf, x, s,, v s = vσ, x, s,. Te mean square error can be decomposed as E ˆf,s x fx = b s + Var v s, 16 were b s is te bias and v s is a normally distributed zero mean stocastic term. First, let us consider te bias. In order to apply Lemma 1 we take s 1 = 0 and s = s. In tis case = k s. Now, applying Lemma 1b and te Fourier inversion formula for fx we see tat uniformly in f Aα b s = 1 s e itx F[k s ]t 1F[f]tdt + O e γ r /c r 1/ dt 0 γ eγtr, for 0. Furtermore, applying Caucy-Scwartz inequality, property 10, and definition of te class Aγ, β, r we get b s 1 e itx F[k s ]t 1F[f]t dt 1 t >s γ e γt r dt + O e s γ r /c r 0 γ eγtr dt 1 s + O e s γ r /c r 0 γ eγtr dt γ e γtr dt + O e s γ r /c r 0 γ eγtr dt. 17 Second, let us consider te variance term. From Lemma 1c, wit s 1 = 0 and s = s, we see tat Var v s = σ ksx l = σ s 1 + O1 s, 18 wen 0. For any s denote σ,s = σ s and for te cosen bandwidt s = s denote te resulting variance 19 σ = σ α, σ = σ s. 0 From equations we see tat te mean square error of te estimator ˆf,s satisfies E ˆf,s x fx σ,s σ,s Os + σ,s + σ,s O e γ r /c r s s 0 γ e γtr dt γ eγtr dt. 1 Now we sall verify tat, taking s = s as defined in 15, te term of te rigt and side of te previous equation is equal to σ o1. Before going into details, let us remark tat

12 110 Austrian Journal of Statistics, Vol , No. 1&, te bandwidt s is precisely te bandwidt tat balances te main terms of te bias and te variance in te mean square error, i.e. it minimizes σ s wit respect to s, since by 15 + Let us return to equation 1. Note first tat s γ e γtr dt e γs r = β γσ. s 0, wen 0. 3 Second, applying te identity and Lemma, we see tat σ s γ e γtr dt = β γσ s 1 rγs r = e γtr dt = s r log γ σ wen 0. Finally, applying te identity and trivial inequality s σ γ e r/c r 0 γ eγtr β dt γ σ e β = γ σ wen 0. Tus, from 1 and 3 5 we ave tat s e γtr dt s e γs r 1 = o1, 4 γ r /c r +γs r E ˆf x fx = σ 1 + o1, 0. e γ r /c r = o1, 5 Note tat wen we normalize te error of our estimator by σ, te normalized error term ˆf x fx/σ as a normal distribution, wit mean of order o1 and variance equal to 1 + o1 were te terms o1 are small uniformly in f Aα wen goes to zero. Because te loss function w as only countably many discontinuity points, applying te dominated convergence teorem lim sup 0 f Aα E f w σ 1 ˆf x fx = E w ξ. 6 Lower bound for te risk. Consider te parametric family of functions f θ z = θgz, gz = s k s z x. Tese functions satisfy f θ x = θ, and if we assume tat θ θ were θ = s s 0 γ eγtr dt 1 7

13 L.M. Artiles and B.Y. Levit 111 ten γ e γt r F[f θ ]t dt = θ s γ e γt r F[k s ]t dt θ s γ e γt r [ s,s ]t dt 1. Tus f θ Aα for all θ suc tat θ θ. Now, we can apply Kakutani s teorem using te fact tat g l < according to Lemma 1c, and see tat dp θ dp 0 1 y = exp σ θ y l gl θ g l were P θ = P fθ cf. e.g. Hui-Hsiung, 1975, Sect. II.. Te statistic }, 8 T = y l gl g l 9 is sufficient for te parameter θ of te family of distributions P θ. Obviously T is normally distributed. Given f θ l = θgl, we can easily verify tat T N σ θ,, 30 g l and applying Lemma 1c, wit s 1 = 0 and s = s, we see tat 1 σ g l = σ s ks x l = σ s 1 + O1 s, wen goes to zero. Tus, T can be represented as and, according to te previous arguments, T = θ + ϕ ξ were ξ N 0, 1 31 ϕ = σ g l = σ 1 + o1. 3 To derive te required lower bound, let us assume te unknown parameter θ as a prior density λθ; a convenient coice is λθ = 1 θ θ cos θ, θ θ.

14 11 Austrian Journal of Statistics, Vol , No. 1&, We obtain ten, due to te sufficiency of te statistic T, inf sup E f w f f Aα σ s f x fx inf f inf ˆθ inf ˆθ sup E f w θ <θ σ s f x f θ x sup E θ w ˆθ θ θ <θ σ s θ θ E θ w ˆθ θ λθdθ σ s θ = inf E θ w ˆθT θ λθdθ ˆθT θ σ s ϕ = E w ξ ϕ σ θ 1 x 1wxe x dx 1 + o1. Here te last equation follows from Levit According to 3, 0, wile applying identity and Lemma we see tat ϕ σ = 1 + o1, σ θ = γσ s 0 γe γtr dt = γs γe tr dt 0 γs γs e γs r 1 0, 33 rγs r wen 0. Tus we ave tat, according to te dominated convergence teorem, lim inf sup E f w 0 f Aα σ s ˆf x fx lim inf 0 inf f sup E f w f Aα σ s f x fx Togeter te relations 6 and 34 prove te teorem. 4. An Extension to Non-fixed Classes E wξ. 34 Up till now we assumed tat te classes Aα were fixed, i.e. not depending on te parameter, toug te function we wanted to estimate could vary freely witin te given class Aα and, in particular, could depend on. Te possible dependency of f on implies tat te estimated function could be as bad as our model allowed it to be wic justified te minimax approac of Teorem 1. To summarize, te assumption tat our functional class Aα is fixed implies tat te smootness properties of te elements of te class are fixed. However, we migt want to furter relax tis restriction by allowing te class itself depend on. Indeed, tere is neiter practical justification, nor a logical

15 L.M. Artiles and B.Y. Levit 113 requirement, tat te smootness of te underlying function remains te same wile te level of noise decreases and consequently te resolution of te available statistical procedures increases. Tis will become even more natural in te adaptive setting of Section 5 were te smootness of te underlying function is not known beforeand. Tus, as a first step towards introducing te adaptive framework, we let te parameters of te model γ, β and r depend on. Even so, tey still be assumed to be known to te statistician tis assumption will be abolised later in te adaptive framework of Section 5. Tis approac will allow us to explore te limits of te model were its parameters are allowed to cange freely. Let s be as defined in Teorem 1. Note tat now te optimum bandwidt s depends on also troug te parameters γ, β and r. Neverteless te statement of Teorem 1 still olds, as we sall see, under corresponding assumptions. Teorem Let w W, and let te parameters β = β, r = r, γ = γ and σ = σ be all positive and suc tat 0 < lim inf 0 r lim sup 0 r <, 35 Ten lim 0 sup f Aα lim sup 0 lim inf 0 γ E f w σ s ˆf x fx =, γ σ 36 1/r log β = 0. γ σ 37 lim 0 inf f sup f Aα E f w σ s f x fx were s, f and ˆf are te same as in Teorem 1. = = E wξ Remark 1 Note tat te conditions 35 and 37 imply s 0 wen 0. As a direct consequence of tis, we obtain consistency, provided σ is bounded, since ten σ s 0. However, our asymptotic optimality result doesn t require σ to be bounded; in oter words tey apply even wen tere is no consistency! Proof. We prove tis teorem following te same proof of Teorem 1. It is sufficient to see tat relations 3 5 and 33 still old for te class Aγ, β, r. Te limit 3 follows from 35 and 37, te limits 4 and 33 follow from 35 and 36. Finally 5 follows from te identity γ σ e γ r /c r = exp c 1 r γ r 1 c r log r γ β 1/r r } γ σ and conditions Note tat /γ 0, by 36 and 37. Te rest of te proof remains te same. 38

16 114 Austrian Journal of Statistics, Vol , No. 1&, Te important conclusion wic can be drawn from te last result is tat in order to prove asymptotic optimality of our estimation procedure, we do not ave to invoke te assumption not always realistic tat te smootness of te estimated function remains te same, even wen te level of noise decreases and, as a consequence, te resolution of available statistical metods increases. Note tat in tis more general situation te corresponding optimal rate of convergence σ α, σ = σ γ 1 log 1 r, 39 γ σ can be of any order, wit respect to any of te parameters, or σ, varying from extremely fast, parametric rates, to extremely slow, non-parametric ones, and even all te way down to no consistency at all. Te problem wic we will face in next section, is tat in practice we often do not know te real class at all. 5 Adaptive Minimax Regression 5.1 Adaptive Estimation in Functional Scales As a transition from te classical minimax setting, studied in te previous sections, to te adaptive setting we introduce functional scales A K = Aα α K }, 40 corresponding to a subset K R 3 + in te underlying parameter space. As our scales A K can be identified wit corresponding subsets K, we will speak sometimes about a scale K, instead of A K, wen tere is no risk tat could lead to a confusion. Sometimes we can tink of te scale A K as te collection of functions } f Aα α K. We will say tat some limit exists uniformly in A K to express tat it exists uniformly in f Aα for every α and tey converge uniformly in α K. Our goal is to estimate a function wic belongs to Aα for some α K. So, we must find an estimator, wic does not depend on α and suc tat it performs optimally well over te wole scale K. For tis new setting a new definition of optimality is necessary. We use te following definition wic was used in Lepski and Levit From now on we will restrict ourselves to te loss functions wx = x p, p > 0. Let A K be a functional scale and F a class of estimators f. Definition An estimator ˆf F is called p, K, F-adaptively minimax, at a point x R, if for any oter estimator f F lim sup 0 sup α K sup f Aα E f ˆf x fx p sup f Aα E f f x fx p 1.

17 L.M. Artiles and B.Y. Levit 115 Te simplest example of a scale A K can be obtained wen K is a fixed compact subset of R 3 +. Our results below cover a muc broader setting in wic te set K itself can depend on te parameter. In our approac, suc results serve two goals. First of all, tey allow a better understanding of te true scope of adaptivity of statistical procedures, since tey describe te extreme situation in wic an adaptation is still possible. In fact all wat is needed below is tat te assumptions of our non-adaptive Teorem old uniformly on te scale K; below we formulate tese assumptions more explicitly. Definition 3 A functional scale A K or te corresponding scale K is called a regular, or an R-scale if te following conditions are satisfied: 0 < lim inf 0 inf r lim sup sup r <, 41 α K α K 0 and for some 0 < δ < 1. lim sup 0 lim inf 0 sup α K inf α K γ σ 1 δ γ =, 4 1/r log β = 0 43 γ σ Te second goal tat can be acieved by considering more general scales K is to introduce te notion of optimality in adaptive estimation, by specifying a natural set of estimators F in te above Definition. Note tat witin a large scale A K, unknown functions f can vary from extremely smoot ones, allowing parametric rate σ O, to muc less smoot functions, allowing slower rates σ O δ, δ < 1, or even extremely slow rates σ Olog 1 1/. Te first possibility is not typical in non-parametric estimation and only can appen in some extreme cases. Tese ideas are made more precise by introducing te following terminology classifying functional scales A K into pseudoparametric PP and non-parametric NP scales depending of teir global rates of convergence. Definition 4 A functional scale A K or te corresponding parameter scale K is called a pseudo-parametric, or a PP scale if lim sup 0 b non-parametric, or an NP-scale if sup s α <, α K lim inf s α =. α K 0 We sall call regular pseudo-parametric and regular non-parametric scales respectively RPP and RNP scales. Since pseudo-parametric scales are not typical, in non-parametric estimation and can only appen in some extreme cases, we will only require our statistical procedure to

19 L.M. Artiles and B.Y. Levit 117 Te adaptive estimator. First, let us coose parameters, 1/ < l < 1, 1/ < δ < 1, p 1 > 0, l 1 = δl, and define te sequence of bandwidts s 0 = 0, s i = expi l for i = 1,.... For eac, we take a subsequence S = s 0, s 1,... s I } were I = arg max i s i log 1 1/}, 44 < 1. Our asymptotic study considers 0, tus, witout loss of generality, we define I just for < 1. Now, let us denote ˆf i x = ˆf,si x, σ i = Var ˆf i x, b i = E f ˆfi x fx, ˆσ i = σ s i, and define te tresolds Finally we define î = min σi,j = Var ˆf j x ˆf i x, ˆσ i,j = σ s j s i, λ j = p log s j + p 1 log δ s j. } 1 i I : ˆf j x ˆf i x λ j ˆσ i,j j i j I. 45 We will prove below tat te estimator ˆf x = ˆfîx satisfies bot te statements contained in Teorem 3. Let us get first some insigt into te algoritm. Te sequence S of bandwidts as several important properties. First, it is increasing, tus te variance of te corresponding estimators is also increasing. Second, according to te definition of R-scales te bandwidts s α, see eq. 43, are suc tat s α δ uniformly in K for some δ < 1, and small enoug. Tus, s I is large enoug for small enoug, so tat for eac α, te optimum bandwidt s α corresponding to Aα, can be sandwiced between two consecutive elements of te sequence S, i.e. tere exists iα = iα, suc tat Te sequence is also dense enoug so tat s iα 1 < s α s iα. s i+1 lim = 1. i s i Tis guarantees tat s α and s iα are asymptotically equivalent since s α for 0 in NP scales.

20 118 Austrian Journal of Statistics, Vol , No. 1&, Te sequence of tresolds λ j as been cosen in suc a way tat, for large i, j iα i j, te probability of te event ˆf j x ˆf i x > λ j Var 1/ ˆf j x ˆf i x, 46 is very small since, except for an event of a small probability, tis can only occur if te bias b j b i Var 1/ ˆf j x ˆf i x wic is not te case for bandwidts greater tan s α as we will see. Terefore, for any given i and j > i we reject s i in favor of te subsequent elements of te sequence S, if te event 46 occurs. Tis pairwise comparison is performed for every i, and from all te accepted s i we select te smallest, i.e. we coose te estimator wit te smallest variance. Note tat according to te previous argument no bandwidt s i, i iα will be rejected, wit ig probability. However it is possible tat a bandwidt s i, i < iα is cosen. In tat case our procedure warrants tat, cf. 45, ˆfîx ˆf iα x λ iα Var 1/ ˆfîx ˆf iα x1 + o1 Tus in te worst case te accuracy of ˆf decreases by a factor 1 + λ iα wic is of order log s α asymptotically as 0. In te next subsection we prove tat te accuracy of tis algoritm is asymptotically optimal in te adaptive setting, for all RNP scales subject to certain mild additional assumptions; see Teorems 1 and 6. Now, let us turn to te proof of te teorem. We start wit an auxiliary result needed in te proof were we use te same notations as tose used in describing te estimation procedure. Lemma 3 For 0, uniformly wit respect to i, j 1 i, j I and wit respect to α varying in a regular scale, a b j = o1ˆσ j for all j suc tat iα j I. b σ j = ˆσ j 1 + Olog 1 1/. c b j b i 1 + o1ˆσ i,j for all i, j suc tat iα i j I. d σ i,j = ˆσ i,j1 + Olog 1 1/. Proof. a Using te bound for te bias given in 17, equation, and Lemma we see, wit some algebra, tat b j 1 σ s j s j γ e γtr dt + O e s j γ r /c r βγe γtr dt e γs j r γ σ rγs j + O e γ r r /c r σ s j 0 jr γ σ eγs = ˆσ j e γs r γs j r rγs r + O e γ r /c r +γs r e γ r /c r +γs j r.

21 L.M. Artiles and B.Y. Levit 119 Now, given s j s α and using conditions 44 in te definition of te sequence of bandwidts S and conditions in te definition of R scales, we obtain b j = o1ˆσ j wen 0, uniformly wit respect to j iα j I and wit respect to α in K. b Tis is just a reformulation of te asymptotic relation 18 using te fact tat, according to 44, s j log 1 1/. c Applying Lemma 1b taking s 1 = s i and s = s j, and arguing as in 17 and in te proof a, we see tat b j b i 1 e itx F[ i,j ]tf[f]t dt + O e s j γ r /c r s i γ eγtr dt 1 sj s i γ e γtr dt + O σ s j s i = ˆσ i,j γ σ O e γs ir e γ r /c r s j + e γ r /c r σ s j s i s i γ eγtr dt γ σ eγs jr e γs r γs i r + O e γ r /c r+γs r e γ r /c r+γs j r = ˆσ i,j1 + o1, 0. d It follows directly from Lemma 1c, taking s 1 = s i and s = s j. Here, as in 18, we can verify tat σ i,j = σ k s j x l k si x l = σ s j s i 1 + O1 s j s i. 47 and tus, using 44, tis completes te proof of te lemma. We now proceed wit proving Teorem 3. For arbitrary f in any R-functional scale A K, R f := E ˆfîx fx p = R f + R+ f were and R f = E R + f = E î iα} ˆfîx fx p} î>iα} ˆfîx fx p}.

22 10 Austrian Journal of Statistics, Vol , No. 1&, Let us examine R f first. We ave } î iα ˆfîx ˆf } iα x λ iαˆσî,iα terefore R f E î iα} ˆfîx ˆf iα x λ iαˆσ iα }, ˆfîx ˆf iα x + ˆf p iα x fx E λ iαˆσ iα + ˆf p iα x fx p E λ iαˆσ iα + b iα + σ iα ξ were ξ N 0, 1. Now according to Lemma 3, a and b, uniformly wit respect to α in any regular scale σ iα = ˆσ iα 1 + o1 and b iα = o1ˆσ iα, 0. It follows tat for 0 uniformly wit respect to any RPP scale R f = Op/, 48 wile by te dominated convergence teorem, uniformly in any RNP scale R f ψp α1 + o1. 49 Now let us examine R + f. Consider te auxiliary events A i = ω : ˆf i x fx } λ iˆσ i. Applying Hölder s inequality we obtain R + f = E = I i=iα+1 î>iα} ˆfîx fx p = R +,1 f + R+, f, I i=iα+1 E E ˆf i x fx p î=i} A i + î=i} A c i î=i} ˆf i x fx p were R +,1 f = I i=iα+1 λ i ˆσ i p/ Pî = i

23 L.M. Artiles and B.Y. Levit 11 and We ave R +, f = Pî = i j=i+1 I i=iα+1 E 1/ ˆfi x fx p P 1/ A c i. P ˆf j 1 x ˆf i 1 x > ˆσ i 1,j 1 λ j By writing ˆf j x ˆf i x = σ i,j ξ + b j b i, were ξ N 0, 1, applying Lemma 3d, and using te well known bound on te tails of te normal distribution cf. Feller, 1968, Lemma, we find for some C > 0 and all small enoug P ˆf j x ˆf i x > λ j ˆσ i,j exp 1 ˆσ } i,j λ j C σ i,j Since by Lemma 3c and 44 ˆσ i,j P ξ > λ j b j b i σ i,j σ i,j exp 1 ˆσ λ i,j j σi j + Cλ j ˆσ i,j σ i,j } exp 1 ˆσ i,j λ j + Cλ j + 1 } σ i,j λ j 1 ˆσ i,j. σi j λ j σ i,j ˆσ i,j σ i,j = λ j O log 1 1/ = o1, 0, it follows from te last inequality tat for some C 1 > 0 P ˆf j x ˆf i x > λ j ˆσ i,j C 1 exp 1 } λj + Cλ j for all α, j i iα and all sufficiently small. Returning to 50 we obtain tat Pî = i C 1 exp 1 } λj 1 + Cλ j 1 j=i+1 = C 1 j=i exp pjl + p 1 j l 1 = C 1 j=i + C pj l + p 1 j l 1 } exp 1 } λj + Cλ j C 1 j=i exp pjl p } 1j l1 3 C 1 pl i1 l exp pil p } 1i l1 3 = C 1 pl i1 l s p/ i exp p 1i l 1 } 3 C s p/ i exp p 1i l 1 4 } 51

24 1 Austrian Journal of Statistics, Vol , No. 1&, for some C > 0 and all i iα, wen is sufficiently small. Terefore uniformly in A K R +,1 f = Op/ i=1 } i pl/ exp p 1 i l 1 /4 = O p/, 0. In order to obtain a bound on R +, f we write again ˆf i fx = b i + σ i ξ, ξ N 0, 1. Applying Lemma 3, a and b, in te same way as before, we ave PA c i P ξ > ˆσ i λ i b i P ξ > ˆσ i λ i σ i σ i σ i exp 1 ˆσ i λi } σ i } C 3 exp pi l p 1 i l 1 / } C 3 exp λ i + λ i } = C 3 s p i exp p 1 i l 1 /, for some C 3, all i iα and all α provided is small enoug. Tus, R +, f = I i=iα+1 I i=iα+1 E 1/ ˆf i x fx p P 1/ A c i ˆσ p i E1/ o o1ξ p P 1/ A c i σ p/ = O exp i=1 } p 1 i r 1 /4 = O p/, 0, 5 uniformly in A K. We can tus conclude tat, uniformly in any RPP scale K, our estimator satisfies E 1/ ˆf x fx p = O1, sup α K sup f Aα wile for any RNP scale K wen 0. sup α K sup f Aα E ψ 1 α ˆf x fx p 1 + o1, 5.3 Lower Bound: Optimality Results In Section 5. we ave establised an upper bound for te risk of adaptive procedures, by evaluating te quality of a proposed adaptive estimator. In tis section we will establis a lower bound for arbitrary suc estimator, wic will allow us to establis optimality of te proposed procedure in te sense of Definition.

25 L.M. Artiles and B.Y. Levit 13 Teorem 4 Let p > 0. Let A K be an arbitrary RNP scale suc tat quantities s = s α, s s α, and φ α can be defined in suc a way tat for all sufficiently small and α K and φ = φ α min p log s, rγ s r / 53 Denote lim inf φ =. 54 α K 0 Ten for any estimator f F 0 p x lim inf 0 Proof. Letting θ = φ inf ψ = ψ α = σ s sup α K f Aα E f ψ 1 φ. f x fx p 1. φ consider te following pair of functions: Note tat f 1 satisfies f 0 z 0, f 1 z = θ gz, gz = σ s k s x z. 55 σ s f 1 x = θ. Obviously f 1 is a continuous function and using 10, definition 15 of s, and Lemma, we get γ e γt r F[f 1 ]t dt = θ σ s γ e γt r F[k s ]t dt = θ γσ R s 0 γe γtr dt γ s = θ e γs rr s 0 γe γtr dt γ s = θ rγ s r e γ s r γs r 1 + o o1 1, φ rγ s r e γ s r γs r 1 + o1 56 uniformly in K for small enoug. Tus f 1 Aα for all sufficiently small and every α K.

26 14 Austrian Journal of Statistics, Vol , No. 1&, Let f Fp 0 x be an arbitrary estimator and denote f ten f x f 1 x = f ψ 1 f 1x = f wereas ψ 1 = ψ 1 σ f x f 0 x = σ f x = σ ψ fx = σ s σ f x and L = φ 1 θ; φ 1 θ = f L 57 φ f x = s 1/ φ f x log = f s exp + log φ }. 58 Denote q = exp φ } so tat by 54, q 0 uniformly wit respect to α for 0. Now, wit te tus defined f 1 Aα, for any f Fp 0 x, uniformly in α K as 0, we ave R := sup f Aα E ψ 1 f f x fx p E1 ψ 1 f x f 1 x p q E 0 σ f x f 0 x + 1 q E ψ 1 1 f p x f 1 x + Oq. According to 53 and 57 59, were R q exp φ + p log φ } E 0 f x p + 1 q E 1 f x L p + Oq 1 q E 1 Z f x p + f x L p + Oq 1 q E 1 inf x p 59 Z x p + x L p + Oq 60 φ Z = q exp + p log φ } dp 0 y. dp 1 For eac value of Z consider te optimization problem of minimizing te function: gx = Z x p + L x p. As was sown in Lepski and Levit 1998, minz, 1L p if p 1, min gx = x 1 + Z 1 p 1 p 1 L p if p > 1. 61

27 L.M. Artiles and B.Y. Levit 15 Tus for any p > 0 we can write min x gx = χl p, 6 were χ is defined by 61 and satisfies 0 < χ 1. Now, let us consider te likeliood corresponding to f 0 and f 1. Using te same arguments tat we used in 8 3 we can see tat dp 0 dp 1 dp 0 dp 1 y = exp = exp = exp 1 σ θξ θ s θ g l + θ y l gl θξ 1 } θ + O1 s } } k s x l were ξ N 0, 1 wit respect to P 1. Using te definition of θ, condition 53 and definition 55 we can see tat } y = 1 + o1 exp θ θξ, 0. Note tat by 54 Z = 1 + o1 exp φ + φ + p log φ φ φ ξ 1 } φ φ P 1 wen 0, ence χ P 1 1. Also L = 1 + o1, according to its definition. Terefore according to equations 60 6, uniformly in α K, R 1 ql p E 1 χ + Oq = 1 + o1, 0. Corollary 1 Let A K be an arbitrary RNP scale suc tat lim inf 0 rγs r inf = 63 α K log s were s is te optimum bandwidt defined in 15. Ten for any p > 0 and x R, te estimator ˆf of Teorem 3 is p, K, F p x-adaptively minimax at x. Proof. Tis is a consequence of Teorems 3 and 4. In order to prove te lower bound use te previous teorem taking s in place of s. Now, we prove a version of Teorem 4 under a weaker condition. It will be used below to provide an easily verifiable conditions for adaptive optimality of te estimator proposed in Section 5..,

28 16 Austrian Journal of Statistics, Vol , No. 1&, Teorem 5 Let A K be an arbitrary RNP scale suc tat lim inf 0 inf α K rγs r log log s = 64 were te optimum bandwidt s was defined in 15. Ten for any estimator f F 0 p x, lim inf 0 inf sup α K f Aα E f ψ 1 f x fx p 1, were ψ = ψ α = p log s σ s. Proof. We prove tis teorem in te same way as Teorem 4 by coosing φ = p log s and subsequently defining s in suc a way tat p log s rγ s r eγ s r γs r 1 65 for small enoug. Te point ere is tat condition 65 was only needed in proving 56, wic now becomes 65. We construct an appropriate s asymptotically equivalent to s tat satisfies te previous inequality for small enoug. Let us first, for fixed α, define te auxiliary bandwidt s as te solution of te equation γs r = γ s r + log rγ s r. 66 We know tat γs goes to infinity as goes to zero uniformly in regular scales. Tus from te previous equation, γ s goes to infinity too and we can see tat s r = 1 + log rγ s s γ s r = 1 + o1, uniformly in K according to 64. Tus te auxiliary bandwidt s is asymptotically equivalent to s. It also satisfies 64, see tat lim inf 0 inf α K rγ s r = lim inf log log s 0 inf α K rγs r 1+o1 lim inf log log s 0 inf α K rγs r log log s =. Now, let us define s equation = ϑ s were ϑ 0 < ϑ < 1 is te closest solution to 1 of te rγ s r log log s ϑ r log ϑ 1 = 1. We can see tat ϑ 1 as 0 tus implying tat s is asymptotically equivalent to s and s. Now, after few transformations, γ s r = γ s r + s s rγt r t 1 dt

29 L.M. Artiles and B.Y. Levit 17 s = γs r log rγ s r + rγt r t 1 dt s s γs r + log rγ s r + rγ s r t 1 dt = γs r + log rγ s r + rγ s r ϑ r log ϑ 1 s and we see tat = γs r + log rγ s r + log log s e γ s r e γs r rγ s r log s = e γs r p log s rγ s ϑr r γ s r /p e γs r p log s rγ s for small enoug. Te rest of te proof is te same as for Teorem 4. Finally, given σ ψ := p log s s is asymptotically equivalent to ψ we ave te proof of te lemma. Finally, we prove tat te estimator we constructed in Teorem 3 is adaptively minimax, for any RNP scale satisfying a condition just a little stronger tan condition 4 used in te definition of a regular scale. Teorem 6 Let K be a RNP scale suc tat lim inf 0 inf α K γσ C 1 δ for some δ 0 < δ < 1 and C > 0. Ten for any p > 0 and x R, te estimator ˆf of Teorem 3 is p, K, F p x-adaptively minimax at x. Proof. Te upper bound result was proved in Teorem 3. To prove te lower bound we notice tat rγs r = r log γσ r log C δ wile according to conditions for R scales log log s = log log 1 γ 1 log 1/r < log log 1 γσ tus rγs r log log s goes to infinity wen 0, uniformly wit respect to te scale K. Te desired lower bound follows now from Teorem 5.

30 18 Austrian Journal of Statistics, Vol , No. 1&, References P. Antonsik, J. Mikusiński, and R. Sikorski. Teory of Distribution. Te Sequential Approac. Elsevier, Amsterdam, L.D. Brown and M.G. Low. Asymptotic equivalence of nonparametric regression and wite noise. Ann. Statist., 4: , W. Feller. An Introduction to Probability Teory and its Applications, volume I. Wiley, New York, 3rd edition, G.K. Golubev and B.Y. Levit. Asymptotically efficient estimation for analytic distributions. Mat. Met. Statist., 5: , G.K. Golubev, B.Y. Levit, and A.B. Tsybakov. Asymptotically efficient estimation of analytic functions in Gaussian noise. Bernoulli, : , Kuo Hui-Hsiung. Gaussian Measures in Banac Spaces. Number 463 in Lect. Notes Mat. Springer-Verlag, Berlin-Heidelberg-New York, I.A. Ibragimov and R.I. Has minskii. Statistical Estimation, Asymptotic Teory. Springer, New York, I.A. Ibragimov and R.I. Has minskii. Bounds for te risks of non-parametric regression estimates. Teor. Probab. Appl., 7:84 99, 198. I.A. Ibragimov and R.I. Has minskii. Estimation of distribution density. Journ. Sov. Mat., 5:40 57, O.V. Lepski. On a problem of adaptive estimation in Gaussian noise. Teory Probab. Appl., 35: , O.V. Lepski. Asymptotically minimax adaptive estimation. I: Upper bounds. Optimally adaptive estimates. Teory Probab. Appl., 36:68 697, O.V. Lepski. Asymptotically minimax adaptive estimation. II: Scemes witout optimal adaptation. Adaptive estimators. Teory Probab. Appl., 7: , 199a. O.V. Lepski. On problems of adaptive estimation in wite Gaussian noise. Adv. Soc. Mat., 1:87 106, 199b. O.V. Lepski and B.Y. Levit. Adaptive minimax estimation of infinitely differentiable functions. Mat. Met. Statist., 7:13 156, O.V. Lepski and B.Y. Levit. Adaptive non-parametric estimation of smoot multivariate functions. Mat. Met. Statist., 8: , B.Y. Levit. On te asymptotic minimax estimates of te second order. Teory Prob. Appl., 5:55 568, 1980.

31 L.M. Artiles and B.Y. Levit 19 S. Nikol skiĭ. Approximation of Functions of Several Variables and Imbedding Teorems. Springer-Verlag, Berlin Heidelberg New York, M. Nussbaum. Asymptotic equivalence of density estimation and Gaussian wite noise. Ann. Statist., 4: , C.J. Stone. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10: , 198. Autors addresses: L.M. Artiles B.Y. Levit Eurandom Department of Matematics & Statistics P.O. Box 513 Queen s University 5600 MB Eindoven Kingston, ON, K7L 3N6 Te Neterlands Canada

### Verifying Numerical Convergence Rates

1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

### ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park

### FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

### The EOQ Inventory Formula

Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

### Geometric Stratification of Accounting Data

Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

### Distances in random graphs with infinite mean degrees

Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree

Computer Science and Engineering, UCSD October 7, 1999 Goldreic-Levin Teorem Autor: Bellare Te Goldreic-Levin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an n-bit

### Tangent Lines and Rates of Change

Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

### Instantaneous Rate of Change:

Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

### Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

### Optimized Data Indexing Algorithms for OLAP Systems

Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest

### CHAPTER 7. Di erentiation

CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.

### 2.28 EDGE Program. Introduction

Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.

### This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

### SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY

ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,

### Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

### Trapezoid Rule. y 2. y L

Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

### Differentiable Functions

Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.

### Schedulability Analysis under Graph Routing in WirelessHART Networks

Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,

### 7.6 Complex Fractions

Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

### Understanding the Derivative Backward and Forward by Dave Slomer

Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.

Improved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands

### Optimal Pricing Strategy for Second Degree Price Discrimination

Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases

### 2 Limits and Derivatives

2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

### 1 Derivatives of Piecewise Defined Functions

MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

### In other words the graph of the polynomial should pass through the points

Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

### Finite Difference Approximations

Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip

### OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS

OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous

### Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction

Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.

### Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

### College Planning Using Cash Value Life Insurance

College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded

### - 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

### Multivariate time series analysis: Some essential notions

Capter 2 Multivariate time series analysis: Some essential notions An overview of a modeling and learning framework for multivariate time series was presented in Capter 1. In tis capter, some notions on

### M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut

### Finite Volume Discretization of the Heat Equation

Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te one-dimensional variable coefficient eat equation, wit Neumann boundary conditions u t x

### Bonferroni-Based Size-Correction for Nonstandard Testing Problems

Bonferroni-Based Size-Correction for Nonstandard Testing Problems Adam McCloskey Brown University October 2011; Tis Version: October 2012 Abstract We develop powerful new size-correction procedures for

### Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)

Lecture 10 Limits (cont d) One-sided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define one-sided its, were

### Math 113 HW #5 Solutions

Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

### Equilibria in sequential bargaining games as solutions to systems of equations

Economics Letters 84 (2004) 407 411 www.elsevier.com/locate/econbase Equilibria in sequential bargaining games as solutions to systems of equations Tasos Kalandrakis* Department of Political Science, Yale

### TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS. Swati Dhingra London School of Economics and CEP. Online Appendix

TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS Swati Dingra London Scool of Economics and CEP Online Appendix APPENDIX A. THEORETICAL & EMPIRICAL RESULTS A.1. CES and Logit Preferences: Invariance of Innovation

### ACT Math Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

### Strategic trading in a dynamic noisy market. Dimitri Vayanos

LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral

### Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

### Cyber Epidemic Models with Dependences

Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas

### An inquiry into the multiplier process in IS-LM model

An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn

### 1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion

Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctions-intro.tex October, 9 Note tat tis section of notes is limitied to te consideration

### We consider the problem of determining (for a short lifecycle) retail product initial and

Optimizing Inventory Replenisment of Retail Fasion Products Marsall Fiser Kumar Rajaram Anant Raman Te Warton Scool, University of Pennsylvania, 3620 Locust Walk, 3207 SH-DH, Piladelpia, Pennsylvania 19104-6366

### Comparison between two approaches to overload control in a Real Server: local or hybrid solutions?

Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor

### Training Robust Support Vector Regression via D. C. Program

Journal of Information & Computational Science 7: 12 (2010) 2385 2394 Available at ttp://www.joics.com Training Robust Support Vector Regression via D. C. Program Kuaini Wang, Ping Zong, Yaoong Zao College

### Research on the Anti-perspective Correction Algorithm of QR Barcode

Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

### Unemployment insurance/severance payments and informality in developing countries

Unemployment insurance/severance payments and informality in developing countries David Bardey y and Fernando Jaramillo z First version: September 2011. Tis version: November 2011. Abstract We analyze

### A system to monitor the quality of automated coding of textual answers to open questions

Researc in Official Statistics Number 2/2001 A system to monitor te quality of automated coding of textual answers to open questions Stefania Maccia * and Marcello D Orazio ** Italian National Statistical

### Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Area-Specific Recreation Use Estimation Using the National Visitor Use Monitoring Program Data

United States Department of Agriculture Forest Service Pacific Nortwest Researc Station Researc Note PNW-RN-557 July 2007 Area-Specific Recreation Use Estimation Using te National Visitor Use Monitoring

### Catalogue no. 12-001-XIE. Survey Methodology. December 2004

Catalogue no. 1-001-XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods

### Pre-trial Settlement with Imperfect Private Monitoring

Pre-trial Settlement wit Imperfect Private Monitoring Mostafa Beskar University of New Hampsire Jee-Hyeong Park y Seoul National University July 2011 Incomplete, Do Not Circulate Abstract We model pretrial

### 2.23 Gambling Rehabilitation Services. Introduction

2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from \$69.2 million in 1995 to \$108 million in 2004. Te majority

### The modelling of business rules for dashboard reporting using mutual information

8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,

### Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student

### To motivate the notion of a variogram for a covariance stationary process, { Ys ( ): s R}

4. Variograms Te covariogram and its normalized form, te correlogram, are by far te most intuitive metods for summarizing te structure of spatial dependencies in a covariance stationary process. However,

### MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

### Proof of the Power Rule for Positive Integer Powers

Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

### Welfare, financial innovation and self insurance in dynamic incomplete markets models

Welfare, financial innovation and self insurance in dynamic incomplete markets models Paul Willen Department of Economics Princeton University First version: April 998 Tis version: July 999 Abstract We

### ME422 Mechanical Control Systems Modeling Fluid Systems

Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Multigrid computational methods are

M ULTIGRID C OMPUTING Wy Multigrid Metods Are So Efficient Originally introduced as a way to numerically solve elliptic boundary-value problems, multigrid metods, and teir various multiscale descendants,

### Analyzing the Effects of Insuring Health Risks:

Analyzing te Effects of Insuring Healt Risks: On te Trade-off between Sort Run Insurance Benefits vs. Long Run Incentive Costs Harold L. Cole University of Pennsylvania and NBER Soojin Kim University of

### Solution Derivations for Capa #7

Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select I-increases, D-decreases, If te first is I and te rest D, enter IDDDD). A) If R1

### Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1

Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te

### f(a + h) f(a) f (a) = lim

Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

### Working Capital 2013 UK plc s unproductive 69 billion

2013 Executive summary 2. Te level of excess working capital increased 3. UK sectors acieve a mixed performance 4. Size matters in te supply cain 6. Not all companies are overflowing wit cas 8. Excess

### An Interest Rate Model

An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal

### PLUG-IN BANDWIDTH SELECTOR FOR THE KERNEL RELATIVE DENSITY ESTIMATOR

PLUG-IN BANDWIDTH SELECTOR FOR THE KERNEL RELATIVE DENSITY ESTIMATOR ELISA MARÍA MOLANES-LÓPEZ AND RICARDO CAO Departamento de Matemáticas, Facultade de Informática, Universidade da Coruña, Campus de Elviña

### Free Shipping and Repeat Buying on the Internet: Theory and Evidence

Free Sipping and Repeat Buying on te Internet: eory and Evidence Yingui Yang, Skander Essegaier and David R. Bell 1 June 13, 2005 1 Graduate Scool of Management, University of California at Davis (yiyang@ucdavis.edu)

### Characterization of researchers in condensed matter physics using simple quantity based on h -index. M. A. Pustovoit

Caracterization of researcers in condensed matter pysics using simple quantity based on -index M. A. Pustovoit Petersburg Nuclear Pysics Institute, Gatcina Leningrad district 188300 Russia Analysis of

### SAT Subject Math Level 1 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

### Staffing and routing in a two-tier call centre. Sameer Hasija*, Edieal J. Pinker and Robert A. Shumsky

8 Int. J. Operational Researc, Vol. 1, Nos. 1/, 005 Staffing and routing in a two-tier call centre Sameer Hasija*, Edieal J. Pinker and Robert A. Sumsky Simon Scool, University of Rocester, Rocester 1467,

### What is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.

Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in- and outflows

### Chapter 7 Numerical Differentiation and Integration

45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

### THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

### Fuzzy Probability Distributions in Bayesian Analysis

Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:

### ON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION

Statistica Sinica 17(27), 289-3 ON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION J. E. Chacón, J. Montanero, A. G. Nogales and P. Pérez Universidad de Extremadura

### The Derivative. Not for Sale

3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously

### A strong credit score can help you score a lower rate on a mortgage

NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing

### Pretrial Settlement with Imperfect Private Monitoring

Pretrial Settlement wit Imperfect Private Monitoring Mostafa Beskar Indiana University Jee-Hyeong Park y Seoul National University April, 2016 Extremely Preliminary; Please Do Not Circulate. Abstract We

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### Writing Mathematics Papers

Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

### Predicting the behavior of interacting humans by fusing data from multiple sources

Predicting te beavior of interacting umans by fusing data from multiple sources Erik J. Sclict 1, Ritcie Lee 2, David H. Wolpert 3,4, Mykel J. Kocenderfer 1, and Brendan Tracey 5 1 Lincoln Laboratory,

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### Referendum-led Immigration Policy in the Welfare State

Referendum-led Immigration Policy in te Welfare State YUJI TAMURA Department of Economics, University of Warwick, UK First version: 12 December 2003 Updated: 16 Marc 2004 Abstract Preferences of eterogeneous

### Modeling User Perception of Interaction Opportunities for Effective Teamwork

Modeling User Perception of Interaction Opportunities for Effective Teamwork Ece Kamar, Ya akov Gal and Barbara J. Grosz Scool of Engineering and Applied Sciences Harvard University, Cambridge, MA 02138

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### 1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

### A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case

A New Cement to Glue Nonconforming Grids wit Robin Interface Conditions: Te Finite Element Case Martin J. Gander, Caroline Japet 2, Yvon Maday 3, and Frédéric Nataf 4 McGill University, Dept. of Matematics

### Simultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning

Simultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning Soo-Haeng Co Hoon Jang Taesik Lee Jon Turner Tepper Scool of Business, Carnegie Mellon University, Pittsburg,

### Strategic trading and welfare in a dynamic market. Dimitri Vayanos

LSE Researc Online Article (refereed) Strategic trading and welfare in a dynamic market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt