Suppose Xi,j are independently distributed as N(i, 2 i ); i = 1,...,s; j = 1,...,ni. Let

Suppose Xi,j are independently distributed as N(µi, σ2 i ); i =

1,...,s; j = 1,...,ni. Let S2 n,i = 

j (Xi,j − X¯i)

2, where X¯i = n−1 i



j Xi,j . Let Zn,i = log[S2 n,i/(ni − 1)]. Show that, as ni → ∞,

√ni − 1[Zn,i − log(σ2 i )] d

→ N(0, 2) .

Thus, for large ni, the problem of testing equality of all the σi can be approximately viewed as testing equality of means of normally distributed variables with known (possibly different) variances. Use