Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 9
Fall 2014

Problem 9.1
Suppose X ∼ Np(θ,I) and consider testing H0 : θ ∈ Ω0 versus H1 : θ /∈ Ω0.
(a) Show that the likelihood ratio test statistic λ is equivalent to the distance D between X and Ω0, defined as
D = inf{kX − θk : θ ∈ Ω0}.
(b) Using part (a), a generalized likelihood ratio test will reject H0 if D > c. What is the significance level α for this test if p = 2 and Ω0 = {θ : θ1 ≤ 0,θ2 ≤ 0}?
Problem 9.2
Suppose that Y1,…,Yn are i.i.d. samples from a normal N(µ,σ2) distribution where σ2 > 0 is known. We wish to test the hypothesis H0 : µ = µ0 versus the alternative H1 : µ = µ1 > µ0 using the sample mean .
(a) Suppose that we reject the null hypothesis if Y¯n ≥ tα. Specify the choice of tα that yields a test of level α.
(b) Show that the power of this test can be expressed as
β(µ1) = Φ(zα + δn)
√ √
where zα = n(µ0 − tα)/σ, and δn = n(µ1 − µ0)/σ. Explain what happens to the power as (µ1 − µ0) increases/decreases, σ increases/decreases, or n increases. Why is this reasonable?
(c) Now suppose that we instead observe the thresholded quantities Zi = I[Yi ≥ µ0]. (This might happen when data communication rates are limited, so that only a single bit can be transmitted.) Using the normal approximation to the binomial distribution, show that the power function for a level-α test using the mean Z¯n can be approximated as
.
(d) Using the approximation Φ( ), show that the power from (c) further simplifies to
.
Note: Comparing this power to that from part (b) provides an indication of what is lost by the thresholding operation.
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Problem 9.3
(A nonparametric hypothesis test) A set of i.i.d. samples Y1,…,Yn is drawn from an unknown distribution F. The null hypothesis H0 asserts that F has median µ = µ0, while the alternative H1 asserts that µ > µ0. (Note that both hypotheses are composite, but neither is based on a parametric model, since only µ0 is given while the form of F is unknown.)
(a) Consider the statistic ]. Compute the exact distribution of S under the null hypothesis. How could this be useful in performing a hypothesis test?
(b) Consider the test δs(Y ) that rejects H0 when S exceeds some threshold s. Use asymptotic theory to approximate the level α(s) = E0[δs(Y )] of this test under H0 as a function of s. (This test is known as a nonparametric sign test.)
Problem 9.4
Suppose that we use a prior λ = [λ0 (1−λ0)] for a binary hypothesis test H0 : θ = θ0 versus H1 : θ = θ1 where λ0 = P[θ = θ0].
(a) Show that the Bayes risk under 0 − 1 loss can be written as
r(λ,δ) = λ0E0[δ(X)] + (1 − λ0){1 −E1[δ(X)]}.
(b) Show that any test minimizing the Bayes risk can be expressed as a likelihood ratiotest where the threshold t is a function of λ0.
(c) Suppose that we are given i.i.d. samples X1,…,Xn from Pθ (where θ is either equal to θ0 or θ1). For any fixed λ0 ∈ (0,1), let δn,λ denote the Bayes test derived in part (b).
Prove that limn→+∞ r(λ,δn) = 0.
Problem 9.5
For each of the following problems, compute the generalized likelihood ratio test, and compute its asymptotic distribution under the specified hypothesis H0.
(a) Let X1,…,Xn be an i.i.d. sample of N(µx,σx2) variates, and let Y1,…Yn be an i.i.d. sample of N(µy,σy2) variates. Consider testing H0 : µx = µy and σx2 = σy2.
(b) For i = 1,…,k, let Xi1,…,Xin be independent samples from Poisson distributions with means θi, respectively. Consider testing H0 : θ1 = θ2 = …θk.
(c) Let X1,…,Xn be an i.i.d. sample from the exponential distribution with parameter θ, and Y1,…Yn an i.i.d. sample from the exponential distribution with parameter µ. Consider testing H0 : µ = 2θ.
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