Dr. Hackney STA Solutions pg 99

Dr. Hackney STA Solutions pg 99 - 7-2Solutions Manual for...

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Unformatted text preview: 7-2Solutions Manual for Statistical InferenceFor the given data,n= 14 and∑ixi= 323.6. Many computer programs can be usedto maximize this function. From PROC NLIN in SAS we obtain ˆα= 514.219 and, hence,ˆβ=323.614(514.219)=.0450.7.3 The log function is a strictly monotone increasing function. Therefore,L(θ|x)> L(θ|x) if andonly if logL(θ|x)>logL(θ|x). So the valueˆθthat maximizes logL(θ|x) is the same as thevalue that maximizesL(θ|x).7.5 a. The value ˆzsolves the equation(1-p)n=Yi(1-xiz),where 0≤z≤(maxixi)-1. Letˆk= greatest integer less than or equal to 1/ˆz. Then fromExample 7.2.9,ˆkmust satisfy[k(1-p)]n≥Yi(k-xi)and[(k+ 1)(1-p)]n<Yi(k+ 1-xi).Because the right-hand side of the first equation is decreasing in ˆz, and becauseˆk≤1/ˆz(soˆz≤1/ˆk) andˆk+ 1>1/ˆz,ˆkmust satisfy the two inequalities. Thusˆkis the MLE.b. Forp= 1/2, we must solve(12)4= (1-20z)(1-z)(1-19z), which can be reduced to thecubic equation-380z3+ 419z2-40z+ 15/16 = 0. The roots are .9998, .0646, and .0381,16 = 0....
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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