LRTTwoMean - LRT Derivation of the Pooled Variance T-Test...

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LRT Derivation of the Pooled Variance T-Test Given that we have two independent random samples from two normal populations with equal but unknown variances. Now we derive the likelihood ratio test for: : = : H0 μ1 μ2 vs Ha μ1 μ2 Let = = μ1 μ2 μ , then, ω ={ -∞< = = <+∞, <+∞ μ1 μ2 μ 0 σ2 }, ={-∞< , <+∞, < <+∞} Ω μ1 μ2 0 σ2 = , =( ) + [- = - + = - ] Lω Lμ σ2 12πσ2 n1 n22exp 12σ2i 1n1xi μ2 j 1n2yj μ2 , and there are two parameters . =- + - = - + = - lnLω n1 n22ln2πσ2 12σ2i 1n1xi μ2 j 1n2yj μ2 , for it contains two parameters, we do the partial derivatives with μ and σ2 respectively and let the partial derivatives equal to 0. Then we have: = = + = + = + + μ i 1n1xi j 1n2yjn1 n2 n1x n2yn1 n2 = + [ = - + = - ] σω2 1n1 n2 i 1n1xi μ2 j 1n2yj μ2 ( )= , , =( ) + [- = - + = - ] L Ω Lμ1 μ2 σ2 12πσ2 n1 n22exp 12σ2i 1n1xi μ12 j 1n2yj μ22 , and there are three parameters. =- + - = - + = - lnLΩ
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This note was uploaded on 01/31/2011 for the course AMS 572 taught by Professor Weizhu during the Fall '10 term at SUNY Stony Brook.

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