Lecture 14 November 10

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1 STATS 300A: Theory of Statistics Fall 015 Lecture 14 November 10 Lecturer: Lester Mackey Scribe: Ju Ya, Matteo Sesia Warig: These otes may cotai factual ad/or typographic errors Overview Hypothesis Testig Optimality Goal Recall that the hypothesis testig problem ca be formulated as H 0 : θ Ω 0 vs. H 1 : θ Ω 1. Here our goal is to fid a uiformly most powerful UMP level-α test φ which maximizes the power fuctio E θ1 φx subject to E θ0 φx α, for every θ 0 Ω 0 ad θ 1 Ω 1. I other words, φ maximizes the power over the alterative space while keepig the size of φ less tha the required level α over the etire ull set Strategies for Fidig UMPs Although the existece of a UMP test is ot geerally guarateed, there are some geeral purpose strategies to fid a UMP test whe oe exists. Oe well-studied strategy cotais the followig three steps: 1. Reduce the composite alterative to a simple alterative: If H 1 is composite, fix θ 1 Ω 1, ad test the ull hypothesis agaist the simple alterative θ = θ 1. Hope that does t deped o θ 1.. Collapse the composite ull to a simple ull: If H 0 is composite, collapse the ull hypothesis to a simple oe by averagig over the ull space Ω 0. We will discuss this strategy i today s lecture. 3. Apply Neyma Pearso lemma: Fid the MP LRT for testig the resultig simple ull versus the resultig simple alterative usig the NP lemma. Note that if the resultig test does ot deped o θ 1, the it will be UMP for the H 0 vs H Optimal Tests for Composite Nulls I previous lectures, our focus was o hypothesis testig problems with a simple ull. Here we itroduce a ew strategy to deal with cases with a composite ull. 14-1

2 14..1 The Model Cosider the case with a simple alterative: H 0 : X f θ, θ Ω 0 H 1 : X g, where g is kow. We ow impose a prior distributio Λ o Ω 0. So we cosider the ew hypothesis H Λ : X h Λ x = f θ x dλθ, Ω 0 where h Λ x is the margial distributio of X iduced by Λ. I order to reduce the problem to a simple versus simple case, let us test H Λ agaist H 1. Notice that the MP test give by the NP lemma should be checked to work for the origial composite ull. This task ca be achieved by pickig Λ to be the least favorable distributio which will be defied later. I the more geeral case of a composite ull vs. a composite alterative, oce a MP test for the composite ull vs. simple alterative is foud, we ca check whether it works for every θ 1 i the alterative parameter space. If so, the resultig test is UMP for the composite vs. composite case Least Favourable Distributio Let β Λ be the power of the MP level-α test φ Λ for testig H Λ vs. g. Defiitio 1 Least favorable Distributio. Λ is a least favorable distributio if β Λ β Λ for ay prior Λ. Hece, Λ will be the least favorable distributio if the MP test uder Λ has smaller power tha the MP test uder ay other prior distributio. The followig theorem ca help us to deal with the case of composite ull by usig the otio of least favorable distributio, which tells that if we choose Λ i the right way, we ca get the MP. Theorem 1 TSH Suppose φ Λ is a MP level-α test for testig H Λ agaist g. If φ Λ is level-α for the origial hypothesis H 0 i.e., E θ0 φ Λ x α, θ 0 Ω 0, the 1. The test φ Λ is MP for origial H 0 : θ Ω 0 vs. g.. The distributio Λ is least favorable. Proof. 1. Let φ be ay other level-α test of H 0 : θ Ω 0 vs. g. The φ is also a level-α test for H Λ vs. g, because E θ φ X = φ xf θ x dµx α, θ Ω 0, which implies that φ xh Λ x dµx = φ xf θ x dµxdλθ αdλθ = α. 14-

3 Sice φ Λ is MP for H Λ vs. g, we have φ xgx dµx φ Λ xgx dµx, Hece φ Λ is a MP test for H 0 vs. g, because φ Λ is also level α.. Let Λ be ay distributio o Ω 0. Sice E θ φ Λ x α, θ Ω 0, we kow that φ Λ must be level-α for H Λ vs. g. Thus β Λ β Λ, so Λ is the least favorable distributio Examples Example 1 Testig i the presece of uisace parameters. Let X 1,..., X be i.i.d. N θ, σ, where both θ, σ are ukow. We cosider testig H 0 : σ σ 0 agaist H 1 : σ > σ 0. To fid a UMP test, we follow the previously metioed strategy: 1. First we fix a simple alterative θ 1, σ 1 for some arbitrary θ 1 ad σ 1 > σ 0.. Secod, we choose a prior distributio Λ to collapse our ull hypothesis over. Ituitively, the least favorable prior should make the alterative hypothesis hard to distiguish. Hece, a rule of thumb cosists i cocetratig Λ o the boudary betwee H 1 ad H 0 i.e. the lie {σ = σ 0 }. Thus Λ will be a probability distributio over θ R for the fixed σ = σ 0. Aother useful observatio is that, give ay test fuctio φx ad a sufficiet statistic T, there exists a test fuctio η that has the same power as φ but depeds o x oly through T : ηt x = E[φx T x]. Hece, we ca restrict our attetio to the sufficiet statistics Y, U, where Y = X ad U = i=1 X i X. We kow that Y N θ, σ /, U σ χ 1, ad Y is idepedet of U by Basu s theorem. Thus, for Λ supported o σ = σ 0, we obtai the joit desity of Y, U uder H Λ as c 0 u 3 exp u exp y θ dλθ σ0 σ0 ad the joit desity uder alterative hypothesis θ 1, σ 1 as c 1 u 3 exp u exp y θ σ1 σ1 1. From the above observatios, we see that the choice of Λ oly affects the distributio of Y. To achieve miimal maximum power agaist the alterative i.e., to be least favorable, we eed to choose Λ such that the two distributios become as close as 14-3

4 possible. Uder the alterative hypothesis, Y N θ 1, σ 1. Uder H Λ, the distributio of Y is i a covolutio form, i.e., Y = Z + Θ for Z N 0, σ 0, Θ Λ, where Z ad Θ are idepedet. Hece, if we choose Θ N θ 1, σ 1 σ 0, Y will have the same distributio uder the ull ad the alterative, which is N choice of prior, the LRT rejects for large values of exp u σ1 θ 1, σ 1. Uder this, i.e., it rejects + u σ 0 for large values of u sice σ 1 > σ 0. So the MP test rejects H Λ if i=1 X i X lies above some threshold determied by the size costrait. I particular, it rejects if i=1 X i X > σ 0C 1,1 α, where C 1,1 α is the 1 α th quatile of χ Next we check if the MP test is level-α for the composite ull. For ay θ, σ with σ σ 0, the probability of rejectio is: i=1 P X i X θ,σ > σ 0 C 1,1 α = P χ σ σ 1 > σ 0 σ C 1,1 α α, while equality holds iff σ = σ 0. Hece, it follows from Theorem 1 that our test is MP for testig the origial ull H 0 vs. N θ 1, σ Fially, the MP level-α test for testig the composite ull H 0 vs. a arbitrarily chose θ 1, σ 1 does ot deped o the choice of θ 1, σ 1. Hece it is UMP for testig the origial composite ull vs. the composite alterative. Example Noparametric Quality Checkig. Idetical light bulbs have lifetime X 1,..., X with a arbitrary distributio P over R. Let u be a fixed threshold for a satisfactory lifetime ad PX u be the probability of a give light bulb beig usatisfactory. Give the data of sample lifetimes we may be iterested i testig whether the probability of havig a usatisfactory light bulb is too large: Here p 0 is a fixed quality parameter. H 0 : PX u p 0 vs. H 1 : PX u < p Before we start our search for the UMP test, let us reparametrize the distributio P as follows. Let P ad P + be the coditioal distributios of X X u ad X X > u respectively, ad let p = PX u. The, P has a oe-to-oe correspodece with P +, P, p. For ay fixed P, let p ad p + be the coditioal desities of P ad P + with respect to some measure µ existece of the desities ad base measure ca be justified, e.g. by Rado-Nikodym theorem i measure theory. The joit desity of X 1,..., X at values x 1,..., x whe x i1,..., x im u < x j1,..., x j m is the give by m m p m p x ij 1 p m p + x jk. j=1 1. As before, we fix a simple alterative P, P +, p 1 where p 1 < p k=1

5 . We ext choose a proper prior. We guess that Λ mostly cocetrates o the boudary poit p +, p, p 0. If so, for testig H Λ vs. the simple alterative, the LRT rejects for large values of m p m 1 1 p 1 m m p m 0 1 p 0, m which is equivalet to testig Bi, p 0 vs. Bi, p 1. Thus, the MP test, which rejects for small values of m = #{i : X i u} 1, is give by 1, if m < k φ Λ x = γ, if m = k 0, if m > k, where k ad γ are both determied by the level costrait E p0 φ Λ x = α. 3. Now we check if φ Λ is level-α for our composite ull H 0. Note that the power fuctio of φ Λ depeds o P oly through p = PX u. Give that this family has MLR i m, the power fuctio would be mootoe. So for ay p > p 0, the rejectio probability uder the ull is still smaller tha α. Hece, φ Λ is the MP test for testig the composite ull H 0 agaist the simple alterative H 1 : P, P +, p Fially, φ Λ has o depedece o the choice of alterative hypothesis. Therefore, φ Λ is UMP for testig the composite ull H 0 agaist the composite alterative H 1. 1 This test is called sig test sice it oly depeds o sigx i u. 14-5

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout