Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
6.2. Rao–Cram ́er Lower Bound and Efficiency 365 where the last equality follows from the transformationz=x−θ. Hence, in the loc ...
366 Maximum Likelihood Methods Differentiating with respect toθ,weobtain k′(θ)= ∫∞ −∞ ··· ∫∞ −∞ u(x 1 ,x 2 ,...,xn) [n ∑ 1 1 f(x ...
6.2. Rao–Cram ́er Lower Bound and Efficiency 367 Definition 6.2.2(Efficiency).In cases in which we can differentiate with respec ...
368 Maximum Likelihood Methods that the distribution is Γ(1, 1 /θ). Because theXis are independent, Theorem 3.3.1 shows thatW= ∑ ...
6.2. Rao–Cram ́er Lower Bound and Efficiency 369 Theorem 6.2.2.AssumeX 1 ,...,Xnare iid with pdff(x;θ 0 )forθ 0 ∈Ωsuch that the ...
370 Maximum Likelihood Methods It follows from (6.2.23)–(6.2.25) that n≥max{N 1 ,N 2 }⇒P [∣∣ ∣ ∣− 1 n l′′′(θ∗n) ∣ ∣ ∣ ∣≤1+Eθ^0 [ ...
6.2. Rao–Cram ́er Lower Bound and Efficiency 371 the sample meanXis asymptotically normal with meanθand varianceσ^2 /n,where σ^2 ...
372 Maximum Likelihood Methods Corollary 6.2.3.Under the assumptions of Theorem 6.2.2, √ n(θ̂n−θ 0 )= 1 I(θ 0 ) 1 √ n ∑n i=1 ∂lo ...
6.2. Rao–Cram ́er Lower Bound and Efficiency 373 y dl( (1)) dl( (0)) (1) (0) Figure 6.2.1: Beginning with the starting valuêθ(0 ...
374 Maximum Likelihood Methods Compare this with the variance of (n+1)Yn/n,whereYnis the largest observation of a random sample ...
6.2. Rao–Cram ́er Lower Bound and Efficiency 375 (a)Find the Fisher informationI(θ). (b)Show that the mle ofθ, which was derived ...
376 Maximum Likelihood Methods (a)Obtain a histogram of the data using the argumentpr=T.Overlaythepdfof aβ(4,1) pdf. Comment. (b ...
6.3. Maximum Likelihood Tests 377 Recall that the likelihood function and its log are given by L(θ)= ∏n i=1 f(Xi;θ) l(θ)= ∑n i=1 ...
378 Maximum Likelihood Methods Note that under the null hypothesis,H 0 ,thestatistic(2/θ 0 ) ∑n i=1Xihas aχ 2 distribution with ...
6.3. Maximum Likelihood Tests 379 Then Λ≤cis equivalent to−2log Λ≥−2logc. However, −2log Λ= ( X−θ 0 σ/ √ n ) 2 , which has aχ^2 ...
380 Maximum Likelihood Methods whereR∗n→0, in probability. By Theorems 5.2.4 and 6.2.2, the first term on the right side of the ...
6.3. Maximum Likelihood Tests 381 whereR 0 nconverges to 0 in probability. Hence the following decision rule defines an asymptot ...
382 Maximum Likelihood Methods conditions (R0)–(R5), but the results below can be derived rigorously; see, for example, Hettmans ...
6.3. Maximum Likelihood Tests 383 of efficiency for tests; see Chapter 10 and more advanced books such as Hettman- sperger and M ...
384 Maximum Likelihood Methods 6.3.8.Using the results of Example 6.2.4, find an exact sizeαtest for the hypotheses (6.3.21). 6. ...
«
15
16
17
18
19
20
21
22
23
24
»
Free download pdf