Suppose that X_1,X_2,...,X_n1, Y_1,Y_2,...,Y_n2, and W_1,W_2,...,W_n3 are independent ranom samples from normal distributions with respective unknown means μ1, μ2, and μ3 and variances (σ1)^2, (σ2)^2, (σ3)^2.
Find the likelihood ratio test for H_o: (σ1)^2 = (σ2)^2 = (σ3)^2 against the alternative of at least one inequality.
[source: Wackerly #10.108a]
I am clueless...can someone please go through the approach/steps to solving this problem? In particular, I am not sure how to find the likelihood function. If we simply have the random variables X_1,X_2,...,X_n, then the likelihood function is just the joint density f_X1,X2,...,Xn(x1,x2,...,xn), but here we have three sets of different random variables (X,Y, and W), what would be the likelihood function, then?
Thanks for helping!