Quiz 4 in Stat 610: Hypothesis Testing and Decision Rules, Quizzes of Statistics

Download Quiz 4 in Stat 610: Hypothesis Testing and Decision Rules and more Quizzes Statistics in PDF only on Docsity! STAT 610 Quiz 4 Newton: 5–1–07. Record all answers in the blue books. Show your work. 1. Consider a discrete-valued random X with probability distribution p(x; θ), where θ takes one of only two values: θ0 or θ1. For testing the null hypotheses H0 : θ = θ0, consider the decision rule, with some k > 0, φk(x) = 1 [p(x; θ1) > kp(x; θ0)] and suppose that Eθ0 [φk(X)] = α where 0 < α < 1. Suppose that φ(x) is any other decision rule with Eθ0 [φ(X)] ≤ α. Show that Eθ1 [φk(X)] ≥ Eθ1 [φ(X)]. 2. The distribution of the random variable X is a mixture of normals. Specifically, f(x; θ) = (1 − θ)p0(x) + θp1(x) where pµ is a normal density with mean µ and variance 1. (I.e. pµ(x) = 1 √ 2π exp [−(1/2)(x − µ)2].) Sketch f(x; θ) for various values of θ in (0, 1). Consider testing the null hypothesis H0 : θ = 0 versus the simple alternative H1 : θ = θ1 for some 0 < θ1 ≤ 1. Show that the likelihood ratio test is to reject if X is sufficiently large, and thus derive the uniformly most powerful level α test of H0 : θ = 0 versus H1 : θ > 0. 3. Two count variables X and Y are independent and Poisson distributed, with mean values θ1 > 0 and θ2 > 0, respectively. [Recall that if U is Poisson with mean µ, then p(u) = exp(−µ)µu/u! for u = 0, 1, 2, . . .. ] We are interested in the null hypothesis H0 : θ1 = θ2 and the alternative H1 : θ1 6= θ2. Derive the generalized likelihood ratio test statistic λ. Evaluate λ in the event X = 1 and Y = 2.