Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
5.3. Central Limit Theorem 345 1 0 1 2 3 4 5 6 7 8 9 10 11 0.25 0.20 0.15 0.10 0.05 Figure 5.3.1:Theb ( 10 ,^12 ) pmf overlaid ...
346 Consistency and Limiting Distributions This is readily established by using the CLT and the same reasoning as in Example 5.3 ...
5.3. Central Limit Theorem 347 variance is constant; in particular, it is free ofp. Hence, we seek a transformation g(p) such th ...
348 Consistency and Limiting Distributions 5.3.9.Letf(x)=1/x^2 , 1 <x<∞, zero elsewhere, be the pdf of a random variableX. ...
5.4.∗Extensions to Multivariate Distributions 349 v′=(v 1 ,...,vp)as v= ∑p i=1 viei. The following lemma will be useful: Lemma 5 ...
350 Consistency and Limiting Distributions which is the desired result. Conversely, ifXnj →P Xjfor allj=1,...,p, then by the sec ...
5.4.∗Extensions to Multivariate Distributions 351 second moments. If we define thep×pmatrixSto be the matrix with thejth diagona ...
352 Consistency and Limiting Distributions Σwhich is positive definite. Assume that the common moment generating function M(t)ex ...
5.4.∗Extensions to Multivariate Distributions 353 A result that will prove to be quite useful is the extension of the Δ-method; ...
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Chapter 6 Maximum Likelihood Methods 6.1 MaximumLikelihoodEstimation Recall in Chapter 4 that as a point estimation procedure, w ...
356 Maximum Likelihood Methods As in Chapter 4, our point estimator ofθisθ̂=θ̂(X 1 ,...,Xn), whereθ̂max- imizes the functionL(θ) ...
6.1. Maximum Likelihood Estimation 357 Definition 6.1.1(Maximum Likelihood Estimator).We say that̂θ=θ̂(X)is a maximum likelihood ...
358 Maximum Likelihood Methods The log of the likelihood simplifies to l(θ)= ∑n i=1 logf(xi;θ)=nθ−nx− 2 ∑n i=1 log(1 + exp{−(xi− ...
6.1. Maximum Likelihood Estimation 359 Proof:First supposegis a one-to-one function. The likelihood of interest isL(g(θ)), but b ...
360 Maximum Likelihood Methods see Remark 5.2.3 for discussion onlim. It follows that for the sequence of solutions θ̂n,P[|̂θn−θ ...
6.1. Maximum Likelihood Estimation 361 is a mle ofθ. In particular, (4Y 1 +2Yn+1)/ 6 ,(Y 1 +Yn)/2, and (2Y 1 +4Yn−1)/ 6 are thre ...
362 Maximum Likelihood Methods (b)Approximate the mle by plotting the function in Part (a). Make use of the following R code whi ...
6.2. Rao–Cram ́er Lower Bound and Efficiency 363 (R4)The integral ∫ f(x;θ)dxcan be differentiated twice under the integral sign ...
364 Maximum Likelihood Methods Remark 6.2.1.Note that the information is the weighted mean of either [ ∂logf(x;θ) ∂θ ] 2 or − ∂^ ...
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