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STAT 818P, Spring 2023

Homework Assignments

Homework 3

Due Date:  Thursday, May 11

Problem 1. (20 points) Assume Xn    X and Yn    Y .

(a) Does Xn + Yn    X + Y? Prove it or give a counter example.

(b) Does ╱Y(X)亢(亢)、     ╱   、Y(X) ? Prove it or give a counter example.

Problem 2. (10 points) Assume ╱Y(X)亢(亢)、      ╱   、Y(X) . Prove Xn    X and Yn    Y .

Problem 3. (30 points) Let X1 ,．．．, Xn  be a random sample from a distribution with mean µ, variance σ2 , E{X4 } < &, and let n  and Sn(2)  be the sample mean and variance, respectively.

(a) Find the asymptotic distribution of log Sn(2) .

(b) Show that ^n ) ╱S(X¯)、  - ╱µ2σ、]  N2 (0, Σ), and identify matrix Σ .

(c) Find the asymptotic distribution of n /Sn .

Problem  4.    (10  points)  Let  X1 ,．．．  , Xn   be  a  random  sample  from  Poisson(θ),  and  let

Zn  = n-1 Y I{Xi  = 0}. Find the asymptotic distribution of ╱Z(X¯)亢(亢)、 .

Problem 5. (30 points) Let Z1 ,．．．, Zn  be a random sample from a continuous distribution, and let Xk   =  Y I{Zi   >  Zk }.   It is know that Xk ’s are independent r.v.’s and Xk   has discrete uniform distribution U{0, 1,．．．, k - 1}.  The statistic Tn   = Yk(n)=1 Xk  represents the total number of discrepancies in the ordering of Z1 ,．．．, Zn , and may be used in hypothesis test

on the trend of the observations increasing or decreasing. Find E{Tn } and Var{Tn }, and show

that    N (0, 1).

Problem 6. (10 points) Let X1 ,．．．, Xn  be a random sample from d.f. F , and let Fn  be the

empirical d.f. of this random sample. Find the computing formula explicitly for

Tn(*)  = n .-o(o)[Fn(*)(x) - Fn (x)]2 dF0 (x)

where F0  is a continuous and strictly increasing d.f., and Fn(*)  is the empirical d.f. based on the bootstrap sample X1(*) ,．．．, Xn(*) .