Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
6.6. The EM Algorithm 405 mutually independent. The conditional notation will prove useful here. Letg(x|θ) denote the joint pdf ...
406 Maximum Likelihood Methods Maximization Step: Let ̂θ(m+1)=ArgmaxQ(θ|θ̂(m),x). (6.6.6) Under strong assumptions, it can be ...
6.6. The EM Algorithm 407 By expression (6.6.1), the conditional distributionZgivenXis the ratio of (6.6.12) to (6.6.11); that i ...
408 Maximum Likelihood Methods Solving∂Q/∂θ=0forθdetermines the EM step estimates. In particular, given thatθ̂(m)is the EM estim ...
6.6. The EM Algorithm 409 An estimate of this expectation is the likelihood ofxibeing drawn from distribution f 2 (x), which is ...
410 Maximum Likelihood Methods where the parameterθsatisfies 0≤θ≤1. In this exercise, we obtain the mle ofθ. (a)Show that likeli ...
6.6. The EM Algorithm 411 (b)Using the data from Rao, compute the EM estimate ofθwith your program. List the sequence of EM esti ...
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Chapter 7 Sufficiency 7.1 MeasuresofQualityofEstimators In Chapters 4 and 6 we presented procedures for finding point estimates, ...
414 Sufficiency X=(X 1 +X 2 +···+X 9 )/9isN(θ,σ 2 9 ),Xis an unbiased estimator ofθ.Thestatistic X 1 isN(θ, σ^2 ), soX 1 is also ...
7.1. Measures of Quality of Estimators 415 If, in fact,θ=0,thenδ 2 (y) = 0 is an excellent decision and we haveR(0,δ 2 )=0. Howe ...
416 Sufficiency δ(Y) is also a statistic, and we could have started directly with a decision rule based on the observations in a ...
7.1. Measures of Quality of Estimators 417 Let us observe, however, that one of the corresponding estimators,Y/nand 1/Z, is bias ...
418 Sufficiency 7.1.3.LetY 1 <Y 2 <Y 3 be the order statistics of a random sample of size 3 from the uniform distribution ...
7.2. A Sufficient Statistic for a Parameter 419 he knows they will probably not equal the originalx-values) as follows. He notes ...
420 Sufficiency the conditional probability ofX 1 ,X 2 ,...,Xndoes not depend uponθ.Intuitively, this means that once the set de ...
7.2. A Sufficient Statistic for a Parameter 421 provided thatx 1 ,x 2 ,...,xnare such that the fixedy 1 =u 1 (x 1 ,x 2 ,...,xn), ...
422 Sufficiency where 0<xi<∞,i=1, 2 ,...,n. Since this ratio does not depend uponθ,the sumY 1 is a sufficient statistic fo ...
7.2. A Sufficient Statistic for a Parameter 423 wherewi=wi(y 1 ,y 2 ,...,yn),i=1, 2 ,...,n.ThepdfofY 1 ,sayfY 1 (y 1 ;θ), is giv ...
424 Sufficiency Thus the joint pdf ofX 1 ,X 2 ,...,Xnmay be written ( 1 σ √ 2 π )n exp [ − ∑n i=1 (xi−θ)^2 / 2 σ^2 ] ={exp[−n(x− ...
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