19 Another Look at Differentiability in Quadratic Mean

Size: px
Start display at page:

Download "19 Another Look at Differentiability in Quadratic Mean"

Transcription

1 19 Aother Look at Differetiability i Quadratic Mea David Pollard 1 ABSTRACT This ote revisits the delightfully subtle itercoectios betwee three ideas: differetiability, i a L 2 sese, of the square-root of a probability desity; local asymptotic ormality; ad cotiguity A mystery The traditioal regularity coditios for maximum likelihood theory ivolve existece of two or three derivatives of the desity fuctios, together with domiatio assumptios to justify differetiatio uder itegral sigs. Le Cam (1970) oted that such coditios are uecessarily striget. He commeted: Eve if oe is ot iterested i the maximum ecoomy of assumptios oe caot escape practical statistical problems i which apparetly slight violatios of the assumptios occur. For istace the derivatives fail to exist at oe poit x which may deped o θ, or the distributios may ot be mutually absolutely cotiuous or a variety of other difficulties may occur. The existig literature is rather uclear about what may happe i these circumstaces. Note also that sice the coditios are imposed upo probability desities they may be satisfied for oe choice of such desities but ot for certai other choices. Probably Le Cam had i mid examples such as the double expoetial desity, 1 / 2 exp( x θ ), for which differetiability fails at the poit θ = x. He showed that the traditioal coditios ca be replaced by a simpler assumptio of differetiability i quadratic mea (DQM): differetiability i orm of the square root of the desity as a elemet of a L 2 space. Much asymptotic theory ca be made to work uder DQM. I particular, as Le Cam showed, it implies a quadratic approximatio property for the log-likelihoods kow as local asymptotic ormality (LAN). Le Cam s idea is simple but subtle. Whe I first ecoutered the LAN property I wrogly dismissed it as othig more tha a Taylor expasio to quadratic terms of the log-likelihood. Le Cam s DQM result showed otherwise: 1 Yale Uiversity

2 306 David Pollard oe appears to get the beefit of the quadratic expasio without payig the twice-differetiability price usually demaded by such a Taylor expasio. How ca that happe? My iitial puzzlemet was ot completely allayed by a study of several careful accouts of LAN, such as those of Le Cam (1970; 1986, Sectio 17.3), Ibragimov & Has miskii (1981, page 114), Millar (1983, page 105), Le Cam & Yag (1990, page 101), or Strasser (1985, Chapter 12). Noe of the proofs left me with the feelig that I really uderstood why secod derivatives are ot eeded. (No criticism of those authors iteded, of course.) Evetually it dawed o me that I had overlooked a vital igrediet i the proofs: the square root of a desity is ot just a elemet of a L 2 space: it is a elemet with orm 1. By rearragig some of the stadard argumets I hope to covice the getle reader of this ote that the fixed orm is the real reaso for why a assumptio of oe-times differetiability (i quadratic mea) ca covey the beefits usually associated with two-times differetiability. I claim that the Lemma i the ext Sectio is the key to uderstadig the role of DQM A lemma The cocept of differetiability makes sese for maps ito a arbitrary ormed space (L, ). For the purposes of my expositio, it suffices to cosider the case where the orm is geerated by a ier product,,. I fact, L will be L 2 (λ), the space of fuctios square-itegrable with respect to some measure λ, but that simplificatio will play o role for the momet. Amapξ from R k ito L is said to be differetiable at a poit θ 0 with derivative, ifξ(θ) = ξ(θ 0 ) + (θ θ 0 ) + r(θ) ear θ 0, where r(θ) = o( θ θ 0 ) as θ teds to θ 0. The derivative is liear; it may be idetified with a k-vector of elemets from L. For a differetiable map, the Cauchy-Schwarz iequality implies that ξ(θ 0 ), r(θ) =o( θ θ 0 ). It would usually be a bluder to assume aively that the boud must therefore be of order O( θ θ 0 2 ); typically, higher-order differetiability assumptios are eeded to derive approximatios with smaller errors. However, if ξ(θ) is costat that is, if the fuctio is costraied to take values lyig o the surface of a sphere the the aive assumptio turs out to be o bluder. Ideed, i that case, ξ(θ 0 ), r(θ) ca be writte as a quadratic i θ θ 0 plus a error of order o( θ θ 0 2 ). The sequetial form of the assertio is more coveiet for my purposes. (1) Lemma Let {δ } be a sequece of costats tedig to zero. Let ξ 0, ξ 1,...be elemets of orm oe for which ξ = ξ 0 +δ W +r, with W aæxed elemet of L ad r =o(δ ). The ξ 0, W =0 ad ξ 0, r = 1 2 δ2 W 2 + o(δ 2).

3 19. Differetiability i Quadratic Mea 307 Proof. Because both ξ ad ξ 0 have uit legth, 0 = ξ 2 ξ 0 2 = 2δ ξ 0, W order O(δ ) + 2 ξ 0, r order o(δ ) + δ 2 W 2 order O(δ 2) + 2δ W, r + r 2 order o(δ 2). O the right-had side I have idicated the order at which the various cotributios ted to zero. (The Cauchy-Schwarz iequality delivers the o(δ ) ad o(δ 2 ) terms.) The exact zero o the left-had side leaves the leadig 2δ ξ 0, W uhappily exposed as the oly O(δ ) term. It must be of smaller order, which ca happe oly if ξ 0, W =0, leavig 0 = 2 ξ 0, r +δ 2 W 2 + o(δ 2 ), as asserted. Without the fixed legth property, the ier product ξ 0, r, which iherits o(δ ) behaviour from r, might ot decrease at the O(δ 2) rate A theorem Let {P θ : θ } be a family of probability measures o a space (X, A), idexed by a subset of R k. Suppose P θ has desity f (x,θ) with respect to a sigma-fiite measure λ. Uder the classical regularity coditios twice cotiuous differetiability of log f (x,θ) with respect to θ, with a domiated secod derivative the likelihood ratio f (x i,θ) f (x i,θ 0 ) ejoys the LAN property. Write L (t) for the likelihood ratio evaluated at θ equal to θ 0 + t/. The property asserts that, if the {x i } are sampled idepedetly from P θ0, the (2) L (t) = exp ( t S 1 2 t Ɣt + o p (1) ) for each t, where Ɣ is a fixed matrix (depedig o θ 0 )ads has a cetered asymptotic ormal distributio with variace matrix Ɣ. Formally, the LAN approximatio results from the usual poitwise Taylor expasio of the log desity g(x,θ) = log f (x,θ), followig a style of argumet familiar to most graduate studets. For example, i oe dimesio, log L (θ 0 + t/ ) = ( g(xi,θ 0 + t/ ) g(x i,θ 0 ) ) = t g (x i,θ 0 ) + t 2 g (x i,θ 0 ) +..., 2

4 308 David Pollard which suggests that S be the stadardized score fuctio, 1 g (x i,θ 0 ) N ( 0, var θ0 g (x,θ 0 ) ), ad Ɣ should be the iformatio fuctio, P θ0 g (x,θ 0 ) = var θ0 g (x,θ 0 ). The dual represetatio for Ɣ allows oe to elimiate all metio of secod derivatives from the statemet of the LAN approximatio, which hits that two derivatives might ot really be eeded, as Le Cam (1970) showed. I geeral, the family of desities is said to be differetiable i quadratic mea at θ 0 if the square root ξ(x,θ)= f (x,θ) is differetiable i the L 2 (λ) sese: for some k-vector (x) of fuctios i L 2 (λ), (3) ξ(x,θ)= ξ(x,θ 0 ) + (θ θ 0 ) (x) + r(x,θ), where λ r(x,θ) 2 = o( θ θ 0 2 ) as θ θ 0. Let us abbreviate ξ(x,θ 0 ) to ξ 0 (x) ad (x)/ξ 0 (x) to D(x). From (3) oe almost gets the LAN property. (4) Theorem Assume the DQM property (3). For each Æxed t the likelihood ratio has the approximatio, uder {P,θ0 }, where L (t) = exp ( t S 1 2 t Ɣt + o p (1) ), S = 2 D(x i ) N(0, I 0 ) ad Ɣ = 1 2 I I, with I 0 = 4λ( {ξ 0 > 0}) ad I = 4λ( ). Notice the slight differece betwee Ɣ ad the limitig variace matrix for S. At least formally, 2D(x) equals the derivative of log f (x,θ): igorig problems related to divisio by zero ad distictios betwee poitwise ad L 2 (λ) differetiability, we have 2 2D(x) = f (x,θ0 ) = f (x,θ0 ) θ θ log f (x,θ 0). Also, Ɣ agai correspods to the iformatio matrix, expressed i its variace form, except for the itrusio of the idicator fuctio {ξ 0 > 0}. The extra idicator is ecessary if we wish to be careful about 0/0. Its presece is related to the property called cotiguity aother of Le Cam s great ideas as is explaied i Sectio 5.

5 19. Differetiability i Quadratic Mea 309 At first sight the derivatio of Theorem 4 from assumptio (3) agai appears to be a simple matter of a Taylor expasio to quadratic terms of the log likelihood ratio. Writig R (x) = r(x,θ 0 + t/ )/ξ 0 (x), wehave log L (t) = 2log ξ(x i,θ 0 + t/ ) ξ(x i,θ 0 ) = 2log (1 + t ) D(x i ) + R (x i ). From the Taylor expasio of log( ) about 1, the sum of logarithms ca be writte as a formal series, 2 ( ) t D(x i ) + R (x i ) ( t 2 D(x i ) + R (x i )) +... (5) = 2t D(x i ) + 2 R (x i ) 1 ( t D(x i ) ) The first sum o the right-had side gives the t S i Theorem 4. The law of large umbers gives covergece of the third term to t P θ0 DD t. Mere oe-times differetiability might ot seem eough to dispose of the secod sum. Each summad has stadard deviatio of order o(1/ ), by DQM. A sum of such terms could crudely be bouded via a triagle iequality, leavig a quatity of order o( ), which clearly would ot suffice. I fact the sum of the R (x i ) does ot go away i the limit; as a cosequece of Lemma 1, it cotributes a fixed quadratic i t. That cotributio is the surprise behid DQM A proof Let me write P to deote calculatios uder the assumptio that the observatios x 1,...,x are sampled idepedetly from P θ0. The ratio f (x i,θ 0 + t/ )/f (x i,θ 0 ) is ot well defied whe f (x i,θ 0 ) = 0, but uder P the problem ca be eglected because P { f (x i,θ 0 ) = 0 for at least oe i} =0. For other probability measures that are ot absolutely cotiuous with respect to P, oe should be more careful. It pays to be quite explicit about behaviour whe f (x i,θ 0 ) = 0 for some i, by icludig a explicit idicator fuctio {ξ 0 > 0} as a factor i ay expressios with a ξ 0 i the deomiator. Defie D i to be the radom vector (x i ){ξ 0 (x i )>0}/ξ 0 (x i ), ad, for a fixed t, defie R i, = r(ξ i,θ 0 + t/ ){ξ 0 (x i )>0}/ξ 0 (x i ). The ξ(x i,θ 0 + t/ ) {ξ 0 ( i )>0} =1 + t D i + R i,. ξ 0 (x i )

6 310 David Pollard (6) (8) The radom vector D i has expected value λ(ξ 0 ), which, by Lemma 1, is zero, eve without the traditioal regularity assumptios that justify differetiatio uder a itegral sig. It has variace 1 4 I 0. It follows by a cetral limit theorem that S = 2 D i N(0, I 0 ). Also, by a (weak) law of large umbers, 1 D i D i P (D 1 D 1 ) = 1 4 I 0 i probability. To establish rigorously the ear-lan assertio of Theorem 4, it is merely a matter of boudig the error terms i (5) ad the justifyig the treatmet of the sum of the R (x i ). Three facts are eeded. (7) Lemma Uder {P }, assumig DQM, (a) max D i =o p ( ), (b) max R i, =o p (1), (c) 2R i, 1 4 t It i probability. Let me first explai how Theorem 4 follows from Lemma 7. Together the two facts (a) ad (b) esure that with high probability log L (t) does ot ivolve ifiite values. For (t D i / ) + R i, > 1 we may the a appeal to the Taylor expasio log(1 + y) = y 1 2 y β(y), where β(y) = o(y 2 ) as y teds to zero, to deduce that log L (t) equals 2 t D i + 2 R i, ( t ) D 2 i + R i, + ( t ) D i β + R i,, which expads to t S + 2 R i, 1 (t D i ) 2 2 t D i R i, R i, 2 + o p(1) ( Di 2 ) + Ri, 2. Each of the last three sums is of order o p (1) because D i 2 / = O p (1) ad P R2 i, = λ( ξ0 2 r(x 1,θ 0 + t/ ){ξ 0 > 0}/ξ0 2 ) λ r(,θ 0 + t/ ) 2 = o(1). By virtue of (6) ad (c), the expasio simplifies to t S 1 4 t It 1 4 t I 0 t + o p (1), as asserted by Theorem 4.

7 19. Differetiability i Quadratic Mea 311 Proof of Lemma 7. Assertio (a) follows from the idetical distributios: P {max D i >ɛ } P { D i >ɛ } = P { 1 >ɛ } ɛ 2 λ 2 1 { 1 >ξ 0 ɛ } 0 by Domiated Covergece. Assertio (b) follows from (8): P {max R i, >ɛ} ɛ 2 P Ri, 2 0. Oly Assertio (c) ivolves ay subtlety. The variace of the sum is bouded by 4 P R (x i ) 2, which teds to zero. The sum of the remaiders must lie withi o p (1) of its expected value, which equals 2P θ0 R 1, = 2λ ( ξ 0 r(,θ 0 + t/ ) ), a ier product betwee two fuctios i L 2 (λ). Notice that the ξ 0 factor makes the idicator {ξ 0 > 0} redudat. It is here that the uit legth property becomes importat. Specializig Lemma 1 to the case δ = 1/, with ξ (x) = ξ(x,θ 0 + t/ ) ad W = t, we get the approximatio to the sum of expected values of the R i,, from which Assertio (c) follows. A slight geeralizatio of the LAN assertio is possible. It is ot ecessary that we cosider oly parameters of the form θ 0 + t/ for a fixed t. By arguig almost as above alog coverget subsequeces of {t } we could prove a aalog of Theorem 4 if t were replaced by a bouded sequece {t } such that θ 0 + t /. The extesio is sigificat because (Le Cam 1986, page 584) the slightly stroger result forces a form of differetiability i quadratic mea Cotiguity ad disappearace of mass For otatioal simplicity, cosider oly the oe-dimesioal case with the typical value t = 1. Let ξ 2 be the margial desity, ad Q be the joit distributio, for x 1,...,x sampled with parameter value θ 0 + 1/.As before, ξ0 2 ad P correspod to θ 0. The measure Q is absolutely cotiuous with respect to P if ad oly if it puts zero mass i the set A ={ξ 0 (x i ) = 0 for at least oe i }. Writig α for λξ 2{ξ 0 = 0}, wehave Q A = 1 ( 1 Q {ξ 0 (x i ) = 0} ) = 1 (1 α ).

8 312 David Pollard By direct calculatio, α = λ ( r + / ) 2 {ξ0 = 0} =λ 2 {ξ 0 = 0}/ + o(1/). The quatity τ = λ 2 {ξ 0 = 0} has the followig sigificace. Uder Q, the umber of observatios ladig i A has approximately a Poisso(τ) distributio; ad Q A 1 e τ. I some asymptotic sese, the measure Q becomes more early absolutely cotiuous with respect to P if ad oly if τ = 0. The precise sese is called cotiguity: the sequece of measures {Q } is said to be cotiguous with respect to {P } if Q B 0 for each sequece of sets {B } such that P B 0. Because P A = 0 for every, the coditio τ = 0 is clearly ecessary for cotiguity. It is also sufficiet. Cotiguity follows from the assertio that L, the limit i distributio uder {P } of the likelihood ratios {L (1)}, have expected value oe. ( Le Cam s first lemma see the theorem o page 20 of Le Cam ad Yag, 1990.) The argumet is simple: If PL = 1 the, to each ɛ>0 there exists a fiite costat C such that PL{L < C} > 1 ɛ. From the covergece i distributio, P L {L < C} > 1 ɛ evetually. If P B 0the Q B P B L {L < C}+Q {L C} CP B + 1 P L {L < C} < 2ɛ evetually. For the special case of the limitig exp(n(µ, σ 2 )) distributio, where µ = 1 4 I I ad σ 2 = I 0, the requiremet becomes 1 = P exp ( N(µ, σ 2 ) ) = exp ( µ σ 2). That is, cotiguity obtais whe I 0 = I (or equivaletly, λ( 2 {ξ 0 = 0}) = 0), i which case, the limitig variace of S equals Ɣ. This coclusio plays the same role as the traditioal dual represetatio for the iformatio fuctio. As Le Cam & Yag (1990, page 23) commeted, The equality... is the classical oe. Oe fids it for istace i the stadard treatmet of maximum likelihood estimatio uder Cramér s coditios. There it is derived from coditios of differetiability uder the itegral sig. The fortuitous equality is othig more tha cotiguity i disguise. From the literature oe sometimes gets the impressio that λ 2 {ξ 0 = 0} is always zero. It is ot. (9) Example Let λ be Lebesgue measure o the real lie. Defie f 0 (x) = x{0 x 1}+(2 x){1 < x 2}. For 0 θ 1 defie desities f (x,θ)= (1 θ 2 ) f 0 (x) + θ 2 f 0 (x 2). Notice that (10) λ f (x,θ) f (x, 0) θ f (x, 1) 2 = ( 1 θ 2 1) 2 = O(θ 4 ).

9 19. Differetiability i Quadratic Mea 313 The family of desities is differetiable i quadratic mea at θ = 0 with derivative (x) = f (x, 1). For this family, λ 2 {ξ 0 = 0} =1. The ear-lan assertio of Theorem 4 degeerates: I 0 = 0adI = 4, givig L (t) exp ( t 2) i probability, uder {P,θ0 }. Ideed, as Aad va der Vaart has poited out to me, the limitig experimet (i Le Cam s sese) for the models {P,t/ : 0 t } is ot the Gaussia traslatio model correspodig to the LAN coditio. Istead, the limit experimet is {Q t : t 0}, with Q t equal to the Poisso(t 2 ) distributio. That is, for each fiite set T ad each h, uder {P,h/ } the radom vectors ( dp,t/ ) : t T dp,h/ coverge i distributio to ( ) dqt : t T, dq h as a radom vector uder the Q h distributio. The couterexample would ot work if θ were allowed to take o egative values; oe would eed (x) = f (x, 1) to get the aalog of (10) for egative θ. The failure of cotiguity is directly related to the fact that θ = 0 lies o boudary of the parameter iterval. I geeral, λ {ξ 0 = 0} must be zero at all iterior poits of the parameter space where DQM holds. O the set {ξ 0 = 0} we have 0 ξ(x,θ 0 +t/ ) = t + r, where r 0. Alog a subsequece, r 0, leavig the coclusio that t 0 almost everywhere o the set {ξ 0 = 0}. At a iterior poit, t ca rage over all directios, which forces = 0 almost everywhere o {ξ = 0}; at a iterior poit, {ξ = 0} =0 almost everywhere. More geerally, oe eeds oly to be able to approach θ 0 from eough differet directios to force = 0o{ξ 0 = 0} as i the cocept of a cotiget i Le Cam & Yag (1990, Sectio 6.2). The assumptio that θ 0 lies i the iterior of the parameter space is ot always easy to spot i the literature. Some authors, such as Le Cam & Yag (1990, page 101), prefer to dispese with the domiatig measure λ, by recastig differetiability i quadratic mea as a property of the desities dp θ /dp θ0, whose square roots correspod to the ratios ξ(x,θ){ξ 0 > 0}/ξ 0 (x). With that approach, the behaviour of o the set {ξ 0 = 0} must be specified explicitly. The cotiguity requiremet that P θ puts, at worst, mass of order o( θ θ 0 2 ) i the set {ξ 0 = 0} is the made part of the defiitio of differetiability i quadratic mea Refereces Ibragimov, I. A. & Has miskii, R. Z. (1981), Statistical Estimatio: Asymptotic Theory, Spriger-Verlag, New York.

10 314 David Pollard Le Cam, L. (1970), O the assumptios used to prove asymptotic ormality of maximum likelihood estimators, Aals of Mathematical Statistics 41, Le Cam, L. (1986), Asymptotic Methods i Statistical Decisio Theory, Spriger-Verlag, New York. Le Cam, L. & Yag, G. L. (1990), Asymptotics i Statistics: Some Basic Cocepts, Spriger-Verlag. Millar, P. W. (1983), The miimax priciple i asymptotic statistical theory, Spriger Lecture Notes i Mathematics pp Strasser, H. (1985), Mathematical Theory of Statistics: Statistical Experimets ad Asymptotic Decisio Theory, De Gruyter, Berli.

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Present Values, Investment Returns and Discount Rates

Present Values, Investment Returns and Discount Rates Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com May 2, 203 Copyright 20, CDI Advisors LLC The cocept of preset value lies

More information

MAXIMUM LIKELIHOODESTIMATION OF DISCRETELY SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH. By Yacine Aït-Sahalia 1

MAXIMUM LIKELIHOODESTIMATION OF DISCRETELY SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH. By Yacine Aït-Sahalia 1 Ecoometrica, Vol. 7, No. 1 (Jauary, 22), 223 262 MAXIMUM LIKELIHOODESTIMATION OF DISCRETEL SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH By acie Aït-Sahalia 1 Whe a cotiuous-time diffusio is

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

A note on the boundary behavior for a modified Green function in the upper-half space

A note on the boundary behavior for a modified Green function in the upper-half space Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s13661-015-0363-z RESEARCH Ope Access A ote o the boudary behavior for a modified Gree fuctio i the upper-half space Yulia Zhag1 ad Valery

More information

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014 1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value

More information

SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES

SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES MATH 57A. Itroductio Our geometric ituitio is derived from three-dimesioal space. Three coordiates suffice. May objects of iterest i aalysis, however, require far

More information

Escola Federal de Engenharia de Itajubá

Escola Federal de Engenharia de Itajubá Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José

More information

GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4 GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5

More information

THE PROBABLE ERROR OF A MEAN. Introduction

THE PROBABLE ERROR OF A MEAN. Introduction THE PROBABLE ERROR OF A MEAN By STUDENT Itroductio Ay experimet may he regarded as formig a idividual of a populatio of experimets which might he performed uder the same coditios. A series of experimets

More information

Asymptotic normality of the Nadaraya-Watson estimator for non-stationary functional data and applications to telecommunications.

Asymptotic normality of the Nadaraya-Watson estimator for non-stationary functional data and applications to telecommunications. Asymptotic ormality of the Nadaraya-Watso estimator for o-statioary fuctioal data ad applicatios to telecommuicatios. L. ASPIROT, K. BERTIN, G. PERERA Departameto de Estadística, CIMFAV, Uiversidad de

More information

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology

More information

Eigenvalues of graphs are useful for controlling many graph

Eigenvalues of graphs are useful for controlling many graph Spectra of radom graphs with give expected degrees Fa Chug, Liyua Lu, ad Va Vu Departmet of Mathematics, Uiversity of Califoria at Sa Diego, La Jolla, CA 92093-02 Edited by Richard V. Kadiso, Uiversity

More information

3. If x and y are real numbers, what is the simplified radical form

3. If x and y are real numbers, what is the simplified radical form lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

More information

A Direct Approach to Inference in Nonparametric and Semiparametric Quantile Models

A Direct Approach to Inference in Nonparametric and Semiparametric Quantile Models A Direct Approach to Iferece i Noparametric ad Semiparametric Quatile Models Yaqi Fa ad Ruixua Liu Uiversity of Washigto, Seattle Workig Paper o. 40 Ceter for Statistics ad the Social Scieces Uiversity

More information

I. Why is there a time value to money (TVM)?

I. Why is there a time value to money (TVM)? Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place. PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.

More information

Some Microfoundations of Collective Wisdom

Some Microfoundations of Collective Wisdom Some Microfoudatios of Collective Wisdom Lu Hog ad Scott E Page May 12, 2008 Abstract Collective wisdom refers to the ability of a populatio or group of idividuals to make a accurate predictio of a future

More information

Optimal Lossless Data Compression: Non-Asymptotics and Asymptotics Ioannis Kontoyiannis, Fellow, IEEE, and Sergio Verdú, Fellow, IEEE

Optimal Lossless Data Compression: Non-Asymptotics and Asymptotics Ioannis Kontoyiannis, Fellow, IEEE, and Sergio Verdú, Fellow, IEEE IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 204 777 Optimal Lossless Data Compressio: No-Asymptotics ad Asymptotics Ioais Kotoyiais, Fellow, IEEE, ad Sergio Verdú, Fellow, IEEE Abstract

More information

Groups of diverse problem solvers can outperform groups of high-ability problem solvers

Groups of diverse problem solvers can outperform groups of high-ability problem solvers Groups of diverse problem solvers ca outperform groups of high-ability problem solvers Lu Hog ad Scott E. Page Michiga Busiess School ad Complex Systems, Uiversity of Michiga, A Arbor, MI 48109-1234; ad

More information

HOW MANY TIMES SHOULD YOU SHUFFLE A DECK OF CARDS? 1

HOW MANY TIMES SHOULD YOU SHUFFLE A DECK OF CARDS? 1 1 HOW MANY TIMES SHOULD YOU SHUFFLE A DECK OF CARDS? 1 Brad Ma Departmet of Mathematics Harvard Uiversity ABSTRACT I this paper a mathematical model of card shufflig is costructed, ad used to determie

More information

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau

More information

Statistical Learning Theory

Statistical Learning Theory 1 / 130 Statistical Learig Theory Machie Learig Summer School, Kyoto, Japa Alexader (Sasha) Rakhli Uiversity of Pesylvaia, The Wharto School Pe Research i Machie Learig (PRiML) August 27-28, 2012 2 / 130

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

MEP Pupil Text 9. The mean, median and mode are three different ways of describing the average.

MEP Pupil Text 9. The mean, median and mode are three different ways of describing the average. 9 Data Aalysis 9. Mea, Media, Mode ad Rage I Uit 8, you were lookig at ways of collectig ad represetig data. I this uit, you will go oe step further ad fid out how to calculate statistical quatities which

More information

Terminology for Bonds and Loans

Terminology for Bonds and Loans ³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixed-paymet loa: series of (ofte equal) repaymets Bod is issued at some

More information

Ramsey-type theorems with forbidden subgraphs

Ramsey-type theorems with forbidden subgraphs Ramsey-type theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called H-free if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal.

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

Periodic Load Balancing on the N-Cycle: Analytical and Experimental Evaluation

Periodic Load Balancing on the N-Cycle: Analytical and Experimental Evaluation Periodic Load Balacig o the N-Cycle: Aalytical ad Experimetal Evaluatio Christia Rieß ad Rolf Waka Computer Sciece Departmet, Uiversity of Erlage-Nuremberg, Germay sichries@iformatik.stud.ui-erlage.de,rwaka@cs.fau.de

More information

Real-Time Computing Without Stable States: A New Framework for Neural Computation Based on Perturbations

Real-Time Computing Without Stable States: A New Framework for Neural Computation Based on Perturbations Real-Time Computig Without Stable States: A New Framework for Neural Computatio Based o Perturbatios Wolfgag aass+, Thomas Natschläger+ & Hery arkram* + Istitute for Theoretical Computer Sciece, Techische

More information

Statistica Siica 6(1996), 311-39 EFFECT OF HIGH DIMENSION: BY AN EXAMPLE OF A TWO SAMPLE PROBLEM Zhidog Bai ad Hewa Saraadasa Natioal Su Yat-se Uiversity Abstract: With the rapid developmet of moder computig

More information

SUPPORT UNION RECOVERY IN HIGH-DIMENSIONAL MULTIVARIATE REGRESSION 1

SUPPORT UNION RECOVERY IN HIGH-DIMENSIONAL MULTIVARIATE REGRESSION 1 The Aals of Statistics 2011, Vol. 39, No. 1, 1 47 DOI: 10.1214/09-AOS776 Istitute of Mathematical Statistics, 2011 SUPPORT UNION RECOVERY IN HIGH-DIMENSIONAL MULTIVARIATE REGRESSION 1 BY GUILLAUME OBOZINSKI,

More information

Performance Evaluation of the MSMPS Algorithm under Different Distribution Traffic

Performance Evaluation of the MSMPS Algorithm under Different Distribution Traffic Paper Performace Evaluatio of the MSMPS Algorithm uder Differet Distributio Traffic Grzegorz Dailewicz ad Marci Dziuba Faculty of Electroics ad Telecommuicatios, Poza Uiversity of Techology, Poza, Polad

More information

On the Biases in the Estimation of Inequality Using Bracketed Quantile Contributions

On the Biases in the Estimation of Inequality Using Bracketed Quantile Contributions O the Biases i the Estimatio of Iequality Usig Bracketed Quatile Cotributios Nassim Nicholas Taleb, Raphael Douady Retired optio trader, NYU Egieerig Risk Data, Cetre d Ecoomie de la Sorboe Abstract I

More information

Statistical Modeling of Non-Metallic Inclusions in Steels and Extreme Value Analysis

Statistical Modeling of Non-Metallic Inclusions in Steels and Extreme Value Analysis Statistical Modelig of No-Metallic Iclusios i Steels ad Extreme Value Aalysis Vo der Fakultät für Mathematik, Iformatik ud Naturwisseschafte der RWTH Aache Uiversity zur Erlagug des akademische Grades

More information

For customers Income protection the facts

For customers Income protection the facts For customers Icome protectio the facts We ve desiged this documet to give you more iformatio about our icome protectio beefits. It does t form part of ay cotract betwee you ad/or us. This iformatio refers

More information

Some comments on rigorous quantum field path integrals in the analytical regularization scheme

Some comments on rigorous quantum field path integrals in the analytical regularization scheme CBPF-NF-03/08 Some commets o rigorous quatum field path itegrals i the aalytical regularizatio scheme Luiz C.L. Botelho Departameto de Matemática Aplicada, Istituto de Matemática, Uiversidade Federal Flumiese,

More information

The Arithmetic of Investment Expenses

The Arithmetic of Investment Expenses Fiacial Aalysts Joural Volume 69 Number 2 2013 CFA Istitute The Arithmetic of Ivestmet Expeses William F. Sharpe Recet regulatory chages have brought a reewed focus o the impact of ivestmet expeses o ivestors

More information

A proposal for: Functionality classes for random number generators 1

A proposal for: Functionality classes for random number generators 1 Wolfgag Killma T-Systems GEI GmbH, Bo Werer Schidler Budesamt für Sicherheit i der Iformatiostechik (BSI), Bo A proposal for: Fuctioality classes for radom umber geerators Versio.0 8 September 0 The authors

More information

For customers Key features of the Guaranteed Pension Annuity

For customers Key features of the Guaranteed Pension Annuity For customers Key features of the Guarateed Pesio Auity The Fiacial Coduct Authority is a fiacial services regulator. It requires us, Aego, to give you this importat iformatio to help you to decide whether

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

A Software Reliability Growth Model for Three-Tier Client Server System

A Software Reliability Growth Model for Three-Tier Client Server System 2 Iteratioal Joural of Computer Applicatios (975 8887) A Software Reliability Growth Model for Three-Tier Cliet Server System Pradeep Kumar Iformatio Techology Departmet ABES Egieerig College, Ghaziabad

More information

Savings and Retirement Benefits

Savings and Retirement Benefits 60 Baltimore Couty Public Schools offers you several ways to begi savig moey through payroll deductios. Defied Beefit Pesio Pla Tax Sheltered Auities ad Custodial Accouts Defied Beefit Pesio Pla Did you

More information

The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs

The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs Joural of Machie Learig Research 0 2009 2295-2328 Submitted 3/09; Revised 5/09; ublished 0/09 The Noparaormal: Semiparametric Estimatio of High Dimesioal Udirected Graphs Ha Liu Joh Lafferty Larry Wasserma

More information

The Sample Complexity of Exploration in the Multi-Armed Bandit Problem

The Sample Complexity of Exploration in the Multi-Armed Bandit Problem Joura of Machie Learig Research 5 004) 63-648 Submitted 1/04; Pubished 6/04 The Sampe Compexity of Exporatio i the Muti-Armed Badit Probem Shie Maor Joh N. Tsitsikis Laboratory for Iformatio ad Decisio

More information

No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices

No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices No igevalues Outside the Support of the Limitig Spectral Distributio of Large Dimesioal Sample Covariace Matrices By Z.D. Bai ad Jack W. Silverstei 2 Natioal Uiversity of Sigapore ad North Carolia State

More information

Mean-Semivariance Optimization: A Heuristic Approach

Mean-Semivariance Optimization: A Heuristic Approach 57 Mea-Semivariace Optimizatio: A Heuristic Approach Javier Estrada Academics ad practitioers optimize portfolios usig the mea-variace approach far more ofte tha the measemivariace approach, despite the

More information

Consistency of Random Forests and Other Averaging Classifiers

Consistency of Random Forests and Other Averaging Classifiers Joural of Machie Learig Research 9 (2008) 2015-2033 Submitted 1/08; Revised 5/08; Published 9/08 Cosistecy of Radom Forests ad Other Averagig Classifiers Gérard Biau LSTA & LPMA Uiversité Pierre et Marie

More information

ON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS and A. RÉNYI. Introduction

ON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS and A. RÉNYI. Introduction ON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS ad A. RÉNYI Itroductio Dedicated to Professor P. Turá at his 50th birthday. Our aim is to study the probable structure of a radom graph r N which has give

More information

How to set up your GMC Online account

How to set up your GMC Online account How to set up your GMC Olie accout Mai title Itroductio GMC Olie is a secure part of our website that allows you to maage your registratio with us. Over 100,000 doctors already use GMC Olie. We wat every

More information

A Dynamic Theory of Public Spending, Taxation and Debt

A Dynamic Theory of Public Spending, Taxation and Debt This revisio: March 26 A Dyamic Theory of Public Spedig, Taxatio ad Debt Abstract This paper presets a dyamic political ecoomy theory of public spedig, taxatio ad debt. Policy choices are made by a legislature

More information

How Has the Literature on Gini s Index Evolved in the Past 80 Years?

How Has the Literature on Gini s Index Evolved in the Past 80 Years? How Has the Literature o Gii s Idex Evolved i the Past 80 Years? Kua Xu Departmet of Ecoomics Dalhousie Uiversity Halifax, Nova Scotia Caada B3H 3J5 Jauary 2004 The author started this survey paper whe

More information

The Power of Both Choices: Practical Load Balancing for Distributed Stream Processing Engines

The Power of Both Choices: Practical Load Balancing for Distributed Stream Processing Engines The Power of Both Choices: Practical Load Balacig for Distributed Stream Processig Egies Muhammad Ais Uddi Nasir #1, Giamarco De Fracisci Morales 2, David García-Soriao 3 Nicolas Kourtellis 4, Marco Serafii

More information

Noisy mean field stochastic games with network applications

Noisy mean field stochastic games with network applications Noisy mea field stochastic games with etwork applicatios Hamidou Tembie LSS, CNRS-Supélec-Uiv. Paris Sud, Frace Email: tembie@ieee.org Pedro Vilaova AMCS, KAUST, Saudi Arabia E-mail:pedro.guerra@kaust.edu.sa

More information

4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.

4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph. 4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics

More information

A Dynamic Theory of Public Spending, Taxation, and Debt

A Dynamic Theory of Public Spending, Taxation, and Debt America Ecoomic Review 2008, 98:1, 201 236 http://www.aeaweb.org/articles.php?doi=10.1257/aer.98.1.201 A Dyamic Theory of Public Spedig, Taxatio, ad Debt By Marco Battaglii ad Stephe Coate* This paper

More information

Sequential Learning for Optimal Monitoring of Multi-channel Wireless Networks

Sequential Learning for Optimal Monitoring of Multi-channel Wireless Networks Sequetial Learig for Optimal Moitorig of Multi-chael Wireless Networs Pallavi Arora Csaba Szepesvári Rog Zheg Departmet of Computer Sciece Departmet of Computig Sciece Departmet of Computer Sciece Uiversity

More information

INCOME PROTECTION POLICY CONDITIONS GUARANTEED PREMIUMS

INCOME PROTECTION POLICY CONDITIONS GUARANTEED PREMIUMS INCOME PROTECTION POLICY CONDITIONS GUARANTEED PREMIUMS Documet referece: MIMIIP12G This is a imptat documet Please keep it i a safe place. Icome Protectio Guarateed Premiums Policy Coditios Welcome to

More information

Experience Studies on Determining Life Premium Insurance Ratings: Practical Approaches

Experience Studies on Determining Life Premium Insurance Ratings: Practical Approaches Eurasia Joural of Busiess ad Ecoomics 2008, 1 (1), 61-82. Eperiece Studies o etermiig Life Premium Isurace Ratigs: Practical Approaches Mirela CRISTEA* Narcis Eduard MITU** Abstract The focus of this article

More information

TruStore: The storage. system that grows with you. Machine Tools / Power Tools Laser Technology / Electronics Medical Technology

TruStore: The storage. system that grows with you. Machine Tools / Power Tools Laser Technology / Electronics Medical Technology TruStore: The storage system that grows with you Machie Tools / Power Tools Laser Techology / Electroics Medical Techology Everythig from a sigle source. Cotets Everythig from a sigle source. 2 TruStore

More information

EFFICIENCY OF BROADBAND INTERNET ADOPTION IN EUROPEAN UNION MEMBER STATES

EFFICIENCY OF BROADBAND INTERNET ADOPTION IN EUROPEAN UNION MEMBER STATES he 11 th Iteratioal Coferece RELIABILIY ad SAISICS i RANSPORAION ad COMMUNICAION -011 Proceedigs of the 11 th Iteratioal Coferece Reliability ad Statistics i rasportatio ad Commuicatio (RelStat 11), 19

More information

Alcohol and drugs: JSNA support pack

Alcohol and drugs: JSNA support pack Alcohol ad drugs: JSNA support pack Key data to support plaig for effective alcohol prevetio, treatmet ad recovery SUFFOLK SUFFOLK SUPPORTING INFORMATION The health harms associated with alcohol cosumptio

More information

WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS

WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS A Thesis Submitted to the Graduate Faculty of the Louisiaa State Uiversity ad Agricultural ad Mechaical College i partial fulfillmet of the requiremets for

More information

A Secure Implementation of Java Inner Classes

A Secure Implementation of Java Inner Classes A Secure Implemetatio of Java Ier Classes By Aasua Bhowmik ad William Pugh Departmet of Computer Sciece Uiversity of Marylad More ifo at: http://www.cs.umd.edu/~pugh/java Motivatio ad Overview Preset implemetatio

More information

About our services and costs

About our services and costs About our services ad costs Cotets Whose products do we offer 2 Ivestmet 2 Isurace 2 Mortgages 2 Which services will we provide you with? 3 Ivestmet 3 Isurace 3 Mortgages 3 What will you have to pay us

More information

How To Find FINANCING For Your Business

How To Find FINANCING For Your Business How To Fid FINANCING For Your Busiess Oe of the most difficult tasks faced by the maagemet team of small busiesses today is fidig adequate fiacig for curret operatios i order to support ew ad ogoig cotracts.

More information

MARKOV MODEL M/M/M/K IN CONTACT CENTER

MARKOV MODEL M/M/M/K IN CONTACT CENTER MARKOV MODEL M/M/M/K IN CONTACT CENTER Erik CHROMY 1, Ja DIEZKA 1, Matej KAVACKY 1 1 Istitute of Telecommuicatios, Faculty of Electrical Egieerig ad Iformatio Techology, Slovak Uiversity of Techology Bratislava,

More information

A New Method To Simulate Bipolar Transistors Combining Analytical Solution And Currend-Based MC Method

A New Method To Simulate Bipolar Transistors Combining Analytical Solution And Currend-Based MC Method A New Method To Simulate Bipolar Trasistors Combiig Aalytical Solutio Ad Curred-Based MC Method Semesterwor Vicet Peiert ETH Zurich, Switzerlad performed at ITET 3.4.2007-18.5.2007 uder supervisio of Dr.

More information

Counterfactual Reasoning and Learning Systems: The Example of Computational Advertising

Counterfactual Reasoning and Learning Systems: The Example of Computational Advertising Joural of Machie Learig Research 14 (2013) 3207-3260 Submitted 9/12; Revised 3/13; Published 11/13 Couterfactual Reasoig ad Learig Systems: The Example of Computatioal Advertisig Léo Bottou Microsoft 1

More information

For customers Key features of the Personal Protection policy

For customers Key features of the Personal Protection policy For customers Key features of the Persoal Protectio policy The Fiacial Coduct Authority is a fiacial services regulator. It requires us, Aego, to give you this importat iformatio to help you to decide

More information

Current Year Income Assessment Form

Current Year Income Assessment Form Curret Year Icome Assessmet Form Academic Year 2015/16 Persoal details Perso 1 Your Customer Referece Number Your Customer Referece Number Name Name Date of birth Address / / Date of birth / / Address

More information

Unemployment Insurance Savings Accounts and Collective Wage Determination

Unemployment Insurance Savings Accounts and Collective Wage Determination DISCUSSION PAPER SERIES IZA DP No. 34 Uemploymet Isurace Savigs Accouts ad Collective Wage Determiatio Laszlo Goerke November 007 Forschugsistitut zur Zukuft der Arbeit Istitute for the Study of Labor

More information

Personal Retirement Savings Accounts (PRSAs) A consumer and employers guide to PRSAs

Personal Retirement Savings Accounts (PRSAs) A consumer and employers guide to PRSAs Persoal Retiremet Savigs Accouts (PRSAs) A cosumer ad employers guide to PRSAs www.pesiosauthority.ie The Pesios Authority Verschoyle House 28/30 Lower Mout Street Dubli 2 Tel: (01) 613 1900 Locall: 1890

More information

Major Coefficients Recovery: a Compressed Data Gathering Scheme for Wireless Sensor Network

Major Coefficients Recovery: a Compressed Data Gathering Scheme for Wireless Sensor Network This full text paper was peer reviewed at the directio of IEEE Commuicatios Society subject matter experts for publicatio i the IEEE Globecom proceedigs. Major Coefficiets Recovery: a Compressed Data Gatherig

More information

Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask

Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask Everythig You Always Wated to Kow about Copula Modelig but Were Afraid to Ask Christia Geest ad Ae-Catherie Favre 2 Abstract: This paper presets a itroductio to iferece for copula models, based o rak methods.

More information

Hybrid Logics and NP Graph Properties

Hybrid Logics and NP Graph Properties Hybrid Logics ad NP Graph Properties Fracicleber Martis Ferreira 1, Cibele Matos Freire 1, Mario R. F. Beevides 2, L. Measché Schechter 3 ad Aa Teresa Martis 1 1 Uiversidade Federal do Ceará, Departameto

More information

CHANGING CLAIMS-MADE INSURERS: IT S MORE THAN THE RETROACTIVE DATE

CHANGING CLAIMS-MADE INSURERS: IT S MORE THAN THE RETROACTIVE DATE HEALTH CARE PRACTICE CLAIMS-MADE ISSUES May 2010 www.willis.com CHANGING CLAIMS-MADE INSURERS: IT S MORE THAN THE RETROACTIVE DATE By Pamela K Haughawout, CPCU, ARM, RPLU Sadra Berkowitz, RN, JD Our cliets

More information

Prescribing costs in primary care

Prescribing costs in primary care Prescribig costs i primary care LONDON: The Statioery Office 13.50 Ordered by the House of Commos to be prited o 14 May 2007 REPORT BY THE COMPTROLLER AND AUDITOR GENERAL HC 454 Sessio 2006-2007 18 May

More information

Stéphane Boucheron 1, Olivier Bousquet 2 and Gábor Lugosi 3

Stéphane Boucheron 1, Olivier Bousquet 2 and Gábor Lugosi 3 ESAIM: Probability ad Statistics URL: http://wwwemathfr/ps/ Will be set by the publisher THEORY OF CLASSIFICATION: A SURVEY OF SOME RECENT ADVANCES Stéphae Bouchero 1, Olivier Bousquet 2 ad Gábor Lugosi

More information

Which Extreme Values Are Really Extreme?

Which Extreme Values Are Really Extreme? Which Extreme Values Are Really Extreme? JESÚS GONZALO Uiversidad Carlos III de Madrid JOSÉ OLMO Uiversidad Carlos III de Madrid abstract We defie the extreme values of ay radom sample of size from a distributio

More information

Cahier technique no. 194

Cahier technique no. 194 Collectio Techique... Cahier techique o. 194 Curret trasformers: how to specify them P. Foti "Cahiers Techiques" is a collectio of documets iteded for egieers ad techicias, people i the idustry who are

More information

supply-chain management (scm)

supply-chain management (scm) 7 supply-chai maagemet (scm) 7. 1 Theory: SCM ad logistics Learig outcomes Lear about the theory of supply-chai maagemet. Use collocatios coected with supply-chai maagemet. Desig a supply chai. Itroductio

More information

G r a d e. 5 M a t h e M a t i c s. Number

G r a d e. 5 M a t h e M a t i c s. Number G r a d e 5 M a t h e M a t i c s Number Grade 5: Number (5.N.1) edurig uderstadigs: the positio of a digit i a umber determies its value. each place value positio is 10 times greater tha the place value

More information

client communication

client communication CCH Portal cliet commuicatio facig today s challeges Like most accoutacy practices, we ow use email for most cliet commuicatio. It s quick ad easy, but we do worry about the security of sesitive data.

More information

Settling Insurance Claims After a Disaster What you need to know about how to file a claim how the claim process works what s covered and what s not

Settling Insurance Claims After a Disaster What you need to know about how to file a claim how the claim process works what s covered and what s not Settlig Isurace Claims After a Disaster What you eed to kow about how to file a claim how the claim process works what s covered ad what s ot First Steps Cotact your aget or compay immediately. Fid out

More information

A model of Virtual Resource Scheduling in Cloud Computing and Its

A model of Virtual Resource Scheduling in Cloud Computing and Its A model of Virtual Resource Schedulig i Cloud Computig ad Its Solutio usig EDAs 1 Jiafeg Zhao, 2 Wehua Zeg, 3 Miu Liu, 4 Guagmig Li 1, First Author, 3 Cogitive Sciece Departmet, Xiame Uiversity, Xiame,

More information

A Parent s Guide To Understanding Student Discipline Policies and Practices In Virginia Schools

A Parent s Guide To Understanding Student Discipline Policies and Practices In Virginia Schools A Paret s Guide To Uderstadig Studet Disciplie Policies ad Practices I Virgiia Schools OH o, How could this happe? what should I do? Virgiia Departmet of Educatio The Paret Guide to Uderstadig Studet Disciplie

More information

NUMERICAL VERIFICATION OF THE RELATIONSHIP BETWEEN THE IN-PLANE GEOMETRIC CONSTRAINTS USED IN FRACTURE MECHANICS PROBLEMS.

NUMERICAL VERIFICATION OF THE RELATIONSHIP BETWEEN THE IN-PLANE GEOMETRIC CONSTRAINTS USED IN FRACTURE MECHANICS PROBLEMS. Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems NUMERICAL VERIFICAION OF HE RELAIONSHIP BEWEEN HE IN-PLANE GEOMERIC CONSRAINS

More information

Throughput and Delay Analysis of Hybrid Wireless Networks with Multi-Hop Uplinks

Throughput and Delay Analysis of Hybrid Wireless Networks with Multi-Hop Uplinks This paper was preseted as part of the ai techical progra at IEEE INFOCOM 0 Throughput ad Delay Aalysis of Hybrid Wireless Networks with Multi-Hop Upliks Devu Maikata Shila, Yu Cheg ad Tricha Ajali Dept.

More information

Making NTRU as Secure as Worst-Case Problems over Ideal Lattices

Making NTRU as Secure as Worst-Case Problems over Ideal Lattices Makig NTRU as Secure as Worst-Case Problems over Ideal Lattices Damie Stehlé 1 ad Ro Steifeld 2 1 CNRS, Laboratoire LIP (U. Lyo, CNRS, ENS Lyo, INRIA, UCBL), 46 Allée d'italie, 69364 Lyo Cedex 07, Frace.

More information

Flood Emergency Response Plan

Flood Emergency Response Plan Flood Emergecy Respose Pla This reprit is made available for iformatioal purposes oly i support of the isurace relatioship betwee FM Global ad its cliets. This iformatio does ot chage or supplemet policy

More information

Crowds: Anonymity for Web Transactions

Crowds: Anonymity for Web Transactions Crowds: Aoymity for Web Trasactios Michael K. Reiter ad Aviel D. Rubi AT&T Labs Research I this paper we itroduce a system called Crowds for protectig users aoymity o the worldwide-web. Crowds, amed for

More information

What is IT Governance?

What is IT Governance? 30 Caada What is IT Goverace? ad why is it importat for the IS auditor By Richard Brisebois, pricipal of IT Audit Services, Greg Boyd, Director ad Ziad Shadid, Auditor. from the Office of the Auditor Geeral

More information

Alcohol data: JSNA support pack. Key data to support planning for effective alcohol prevention, treatment and recovery in 2015-16

Alcohol data: JSNA support pack. Key data to support planning for effective alcohol prevention, treatment and recovery in 2015-16 Key data to support plaig for effective alcohol prevetio, treatmet ad recovery i 2015-16 Croydo (usig latest available data) ABOUT THIS JSNA SUPPORT PACK The health harms associated with alcohol cosumptio

More information

Optimal Backpressure Routing for Wireless Networks with Multi-Receiver Diversity

Optimal Backpressure Routing for Wireless Networks with Multi-Receiver Diversity CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS), INVITED PAPER ON OPTIMIZATION OF COMM. NETWORKS, MARCH 2006 1 Optimal Backpressure Routig for Wireless Networks with Multi-Receiver Diversity Michael

More information