Deriving the first- and second-order corrections to Pearson’s X2 (see Rao and Scott, 1981).
a. Suppose the random vector Y is normally distributed with mean 0 and covariance matrix . Then, if C is symmetric and positive definite, show that YTCY has the same distribution as ƩλiWi, where the Wi’s are independent χ2 1 random variables and the λi’s are the eigen values of CƩ.
b. Let ˆθ = (ˆθ11, . . . ,ˆθ1,(c−1), . . . ,ˆθ(r−1),1, . . . ,ˆθ(r−1),(c−1))T , where ˆθij = ˆpij − ˆpi+ˆp+j . Let A be the covariance matrix of ˆθ if a multinomial sample of size n is taken and the null hypothesis is true. Using (a), argue that ˆθ TA−1ˆθ asymptotically has the same distribution asλiWi , where the Wi are independent χ2 1 random variables, and the λi’s are the eigen values of A−1V(ˆθ).
c. What are E [ˆθ TA−1ˆθ] and V [ˆθ TA−1ˆθ] in terms of the λi’s?
d. Find E [ˆθ TA−1ˆθ] and V [ˆθ TA−1ˆθ] for a 2 × 2 table. You may want to use your answer in Exercise 14.