STAT 461/561- 2021 Assignments 1 1. [3+3+3] Assume X1, X2, . . . is a sequence of i.i.d. standard Poisson random variables. (a) max{X1, X2, . . . , Xn} = Op(log n). (b) max{X1, X2, . . . , Xn} = op(n). (c) max{X1, X2, . . . , Xn} 6= Op( p log n). 2. [5] Let an be a sequence of real values and Xn be a sequence of random variables. Suppose an !1 and an(Xn µ) d ! Y . If g(x) is a function satisfying Lipschitz condition: |g(x) g(y)| C|x y| for some constant C. Show that an{g(Xn) g(µ)} = Op(1). 3. [8] Let Xn, Yn, be a pair of independent Poisson distributed random variables with mean n 1 and n 2. Define Tn = (Yn/Xn) (Xn > K) for some K > 0. Show that p n(Tn 2/ 1) d ! N(0, 2) and work out the expression of 2. 4. [2+2+2] Suppose we have two i.i.d. samples fromN(0, 21) andN(0, 2 2). Let the observations be denoted as x1, x2, . . . , xm and y1, y2, . . . , yn. Design a test for H0 : 21 = 2 2 versus H1 : 2 1 6= 22. The requirements of this problem include: (1) Present the (same) test in all three ways as in Section 12.7; (2) Does your test statistic have the two desirable properties? (3) No actual data analysis is required. 1 5. [2+2+2] Suppose we have two i.i.d. samples fromN(µ1, 2) andN(µ2, 2). Let the observations by denoted as x1, x2, . . . , xm and y1, y2, . . . , yn. Sample I: -0.2212993 0.4495635 -0.9740405 -0.7886435 1.7338719 0.8720555 0.2192809 -1.8462444 0.3230189 1.0007282 Sample II: -0.4507532 -2.1045739 -1.3628046 1.2556175 -0.5325500 0.3671854 -0.7482886 -1.9928208 (a) Carry out the classical two-sample t-test for H0 : µ2 µ1 0 versus H1 : µ2 µ1 > 0 based on the above data. (b) Plot the type I error of this test as function of (µ2 µ1)/ . (c) Plot the type II error of this test as function of (µ2 µ1)/ . Hints: Review the material about non-central t-distribution. Choose sensible regions in (b) and (c), show your codes. 6. [3+2+2] Let (Xi, Yi), i = 1, 2, . . . , n be a set of iid bivariate observa- tions with their joint probably density function given by f(x, y; ✓1, ✓2) = xy ✓21✓ 2 2 exp( x ✓1 y ✓2 ). Consider the test problem for H0 : ✓1 = ✓2 versus H1 : ✓1 > ✓2. Let X¯n and Y¯n be sample means and define Tn = log{X¯n} log{Y¯n}. (a) Illustrate that Tn has the desired properties for the purpose of statistical significance test. (b) Suppose the observed value of Tn = t0. What is the p-value of the test based on Tn in a probability expression? Remark: I look for an expression in the spirit of P (Tn 2 [1, 2]). (c) Show that X¯n Y¯n has an F-distribution with some degrees of freedom. 2 欢迎咨询51作业君
