Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
7.6. Functions of a Parameter 445 The bootstrap process described above is often called thenonparametric boot- strapbecause it m ...
446 Sufficiency (a)Find the MVUE ofP(X≤1) = (1 +θ)e−θ. Hint: Letu(x 1 )=1,x 1 ≤1, zero elsewhere, and findE[u(X 1 )|Y =y], where ...
7.7. The Case of Several Parameters 447 7.6.11.Consider the situation of the last exercise, but suppose we have the following tw ...
448 Sufficiency and equals zero elsewhere. Accordingly, the joint pdf ofX 1 ,X 2 ,...,Xncan be written, for all points in its su ...
7.7. The Case of Several Parameters 449 the support does not depend on the vector of parametersθ, the spaceΩcontains a nonempty ...
450 Sufficiency Therefore, we can takeK 1 (x)=x^2 andK 2 (x)=x. Consequently, the statistics Y 1 = ∑n 1 X^2 i and Y 2 = ∑n 1 Xi ...
7.7. The Case of Several Parameters 451 that outcomejoccurs ispj; hence, ∑k j=1pj=1. LetX=(X^1 ,...,Xk−^1 ) ′and p =(p 1 ,...,pk ...
452 Sufficiency hence,K 1 (x)=x. The first term is easily seen to be a linear combination of the productsxixj,i, j=1, 2 ,...,k, ...
7.7. The Case of Several Parameters 453 7.7.2.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution that has a pdf of the fo ...
454 Sufficiency 7.7.8.In the notation of Example 7.7.3, show that the mle ofpjplisn−^2 YjYl. 7.7.9.Refer to Example 7.7.4 on suf ...
7.8. Minimal Sufficiency and Ancillary Statistics 455 which we findk+k(k+1)/2 joint sufficient statistics fork+k(k+1)/2 paramete ...
456 Sufficiency From these examples we see that the minimal sufficient statistics do not need to be unique, for any one-to-one t ...
7.8. Minimal Sufficiency and Ancillary Statistics 457 (c)Supposef(w) is the logistic pdf. As discussed in Example 6.1.2, the mle ...
458 Sufficiency for all reald. Hence Z=u(W 1 +θ, W 2 +θ,...,Wn+θ)=u(W 1 ,W 2 ,...,Wn) is a function ofW 1 ,W 2 ,...,Wnalone (not ...
7.8. Minimal Sufficiency and Ancillary Statistics 459 Example 7.8.6(Location- and Scale-Invariant Statistics).Finally, consider ...
460 Sufficiency (a)b(1,θ), where 0≤θ≤1. (b)Poisson with meanθ>0. (c)Gamma withα=3andβ=θ>0. (d)N(θ,1), where−∞<θ<∞. ( ...
7.9. Sufficiency, Completeness, and Independence 461 (b)T 2 = ∑n− 1 i=1(Xi+1−Xi) (^2) /S (^2). (c)T 3 =(Xi−X)/S. 7.8.7. With ran ...
462 Sufficiency Theorem 7.9.1.LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution having a pdff(x;θ),θ∈Ω,whereΩis an i ...
7.9. Sufficiency, Completeness, and Independence 463 complete sufficient statistic forθ. HenceY 1 must be independent of each lo ...
464 Sufficiency The joint pdf ofY 1 andYnis g(y 1 ,yn;θ)=n(n−1)(yn−y 1 )n−^2 / 2 n,θ− 1 <y 1 <yn<θ+1, zero elsewhere. A ...
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