Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
6.3. Maximum Likelihood Tests 385 i.e.,V={v:v=a 1 ,for somea∈R}. Then in vector notation we can write the model as X=μ+e, μ∈V. ( ...
386 Maximum Likelihood Methods (a)Find Λ and−2logΛ. (b)Determine the Wald-type test. (c)What is Rao’s score statistic? 6.3.17.Le ...
6.4. Multiparameter Case: Estimation 387 we have labeled these (R6)–(R9). In this section, when we say “under regularity conditi ...
388 Maximum Likelihood Methods where the second equality follows becauseb>0. Setting this partial to 0, we obtain the mle ofa ...
6.4. Multiparameter Case: Estimation 389 Using (6.4.6) and (6.4.7) together, we obtain Ijk=−E [ ∂^2 ∂θj∂θk logf(X;θ) ] . (6.4.8) ...
390 Maximum Likelihood Methods Upon taking the negative of the expectations of the second partial derivatives, the information m ...
6.4. Multiparameter Case: Estimation 391 Thus, the information matrix can be written as (1/b)^2 times a matrix whose entries are ...
392 Maximum Likelihood Methods Next, we obtain the mles for a random sampleX 1 ,X 2 ,...,Xn. The likelihood function is given by ...
6.4. Multiparameter Case: Estimation 393 For any sequence that satisfies (1), √ n(̂θn−θ)→DNp( 0 ,I−^1 (θ)). The proof of this ...
394 Maximum Likelihood Methods EXERCISES 6.4.1.A survey is taken of the citizens in a city as to whether or not they sup- port t ...
6.5. Multiparameter Case: Testing 395 (a)If the constantbis defined by the equationP(X≤b)=0.90, find the mle of b. (b)Ifcis give ...
396 Maximum Likelihood Methods We may be interested though in testing thatμ=μ 0 ,whereμ 0 is a specified value. Here we are not ...
6.5. Multiparameter Case: Testing 397 Following Example 6.4.1, it is easy to show that under the reduced parameter space ω,̂σ 02 ...
398 Maximum Likelihood Methods Theorem 6.5.1.LetX 1 ,...,Xnbe iid with pdff(x;θ)forθ∈Ω⊂Rp. Assume the regularity conditions hold ...
6.5. Multiparameter Case: Testing 399 where the parameter space isω={p:0<p< 1 / 2 }. The likelihood underωis L(p)=pt^1 +t^ ...
400 Maximum Likelihood Methods Returning to the polling situation discussed at the beginning of this example, we would say the r ...
6.5. Multiparameter Case: Testing 401 in its denominator with a consistent estimate. Recall that̂pi→pi,i=1,2, in probability. Th ...
402 Maximum Likelihood Methods (a)Show that the likelihood ratio for testingH 0 :θ 1 =θ 2 ,θ 3 =θ 4 against all alternatives is ...
6.5. Multiparameter Case: Testing 403 6.5.9.Consider the two uniform distributions with respective pdfs f(x;θi)= { 1 2 θi −θi< ...
404 Maximum Likelihood Methods 6.5.12.Finish the derivation of the LRT found in Example 6.5.3. Simplify as much as possible. 6.5 ...
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