Thread: Combined likelihood and max likelihood estimator of two normal distributions

1. Combined likelihood and max likelihood estimator of two normal distributions

Hi guys ive been attempting a bank of revision questions in preparation for exams and I cannot seem to work out this question. For part a am I correct in assuming you take the log likelihoods of each distribution and add them together to combine the likelihoods?
For part b I cannot seem where to start so any help is appreciated.

Thanks

2. Re: Combined likelihood and max likelihood estimator of two normal distributions

Part a) Yes, the log-likelihood of the data is the sum of log-likelihoods corresponding to separate observations... In particular, if sigma_1 is not constrained to equal sigma_2, we have two completely separate likelihood maximization problems. You can maximize the sum over X's and the sum over Y's separately.

Part b) The maximum likelihood ratio test has the following form. Calculate

LR = sup{ LogLikelihood(sigma_1,sigma_2) | sigma_1 = sigma_2, sigma_1 > 0, sigma_2 > 0 } / sup{ LogLikelihood(sigma_1,sigma_2) | ssigma_1 > 0, sigma_2 > 0 }.

For some critical value K, which is a constant depending on the significance level, do:

if LR >= K accept H0: sigma_1 = sigma_2,
if LR < K accept H1: sigma_1 <> sigma_2.

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