1.34 For counts {n,}, the power divergence statistic for testing goodness of fit (Cressie and Read 1984, Read and Cressie 1988) is 2

λ(λ + 1)

λ

Ση; [(π/μ₁) – 1]

a. For λ = 1, show that this equals X2.

for −00 < λ

b. As a → 0, show that it converges to G2. [Hint: log t = limno(th-1)/h.]

c. As a → −1, show that it converges to 2 Σû; log(û₁/n₁), the minimum discrim-

ination information statistic (Gokhale and Kullback 1978).

i 2

d. For λ = −2, show that it equals (n₁ – μ₁)²/ni, the Neyman modified chi-

squared statistic (Neyman 1949).

e. For a = -, show that it equals 4 (√n; - √μ₁)², the Freeman-Tukey statistic

(Freeman and Tukey 1950).

ni

[Under regularity conditions, their asymptotic distributions are identical (Drost et al.

1989). The chi-squared null approximation works best for a near