Math 281A Homework 2
Due: Oct 17, in class
1. Let X1,...,Xnbe i.i.d. from N(0,1), show that ¯
Xand (X1−¯
X,...,Xn−¯
X)are independent.
2. Suppose that random vector (X, Y )has probability density function
1
πe−x2+y2
2I(xy >0).
Does (X, Y )possess a multivariate normal distribution? Find the marginal distributions.
3. Suppose Tnand Snare sequences of estimators such that
√n(Tn−θ)d
Ð→ Nk(0,Σ),and Sn
P
Ð→ Σ,
for a certain vector θand a nonsingular matrix Σ. Show that
(a) Snis nonsingular with probability tending to one;
(b) {θ∶n(Tn−θ)⊺S−1
n(Tn−θ)≤χ2
k,α}is a confidence ellipsoid of asymptotic confidence level 1 −α.
4. Suppose that Xm∼Binomial(m, p1), Yn∼Binomial(n, p2)and they are independent. To test H0∶p1=
p2=a, we consider the test statistic
C2
m,n =(Xm−ma)2
ma(1−a)+(Yn−na)2
na(1−a).
(a) Find the limit distribution of C2
m,n as m, n →∞;
(b) How would you modify the test statistic if awere unknown? What’s the limit distribution after
modification? You don’t need to rigorously prove this question.
