Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
7.2. A Sufficient Statistic for a Parameter 425 Example 7.2.6.In Example 7.2.3 withf(x;θ)=e−(x−θ)I(θ,∞)(x), it was found that th ...
426 Sufficiency 7.2.8.What is the sufficient statistic forθif the sample arises from a beta distri- bution in whichα=β=θ>0? 7 ...
7.3. Properties of a Sufficient Statistic 427 That is, through this conditioning, the functionφ(Y 1 ) of the sufficient statisti ...
428 Sufficiency Example 7.3.1.LetX 1 ,...,Xnbe iid with pdf f(x;θ)= { θe−θx 0 <x<∞,θ> 0 0elsewhere. Suppose we want an ...
7.3. Properties of a Sufficient Statistic 429 We illustrate this remark in the following example. Example 7.3.2.LetX 1 ,X 2 ,X 3 ...
430 Sufficiency Of course,E[Υ(Y 3 )] =θand var[Υ(Y 3 )]≤var(Y 1 /3), but Υ(Y 3 ) is not a statistic, as it involvesθand cannot b ...
7.4. Completeness and Uniqueness 431 Let us consider the family{g 1 (y 1 ;θ):θ> 0 }of probability mass functions. Suppose tha ...
432 Sufficiency Example 7.4.1.Consider the family of pdfs{h(z;θ):0<θ<∞}. SupposeZ has a pdf in this family given by h(z;θ) ...
7.4. Completeness and Uniqueness 433 The statement thatY 1 is a sufficient statistic for a parameterθ, θ∈Ω, and that the family{ ...
434 Sufficiency (b)N(0,θ), where 0<θ<∞. 7.4.3.LetX 1 ,X 2 ,...,Xnrepresent a random sample from the discrete distribution ...
7.5. The Exponential Class of Distributions 435 (b)LetY=|X|. Show thatYis a complete and sufficient statistic forθ. 7.4.9. LetX ...
436 Sufficiency (b) ifXis a discrete random variable, thenK(x)is a nontrivial function of x∈S. For example, each member of the f ...
7.5. The Exponential Class of Distributions 437 Var(Y 1 )=np′(^1 θ) 3 {p′′(θ)q′(θ)−q′′(θ)p′(θ)}. Example 7.5.1. LetXhave a Poi ...
438 Sufficiency This theorem has useful implications. In a regular case of form (7.5.1), we can see by inspection that the suffi ...
7.5. The Exponential Class of Distributions 439 7.5.3.LetX 1 ,X 2 ,...,Xndenote a random sample of sizenfrom a distribution with ...
440 Sufficiency (a)Show thatY 1 =X 1 +X 2 +···+Xnis a complete sufficient statistic forθ. (b)Find the functionφ(Y 1 )thatistheMV ...
7.6. Functions of a Parameter 441 NowE(Y)=nθandE(Y^2 )=nθ(1−θ)+n^2 θ^2. Hence E [ Y n ( 1 − Y n )] =(n−1) θ(1−θ) n . If we multi ...
442 Sufficiency Solving foru(θ), we obtain u(θ)=g(θ)+ θg′(θ) n . Therefore, the MVUE ofg(θ)is u(Yn)=g(Yn)+ Yn n g′(Yn). (7.6.2) ...
7.6. Functions of a Parameter 443 7.6.8, the joint distribution ofX 1 andXis bivariate normal with mean vector (θ, θ), variances ...
444 Sufficiency 7.6.1 BootstrapStandardErrors Section 6.3 presented the asymptotic theory of maximum likelihood estimators (mles ...
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