Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
8.2. Uniformly Most Powerful Tests 485 By our discussion above concerning mlr, the UMP levelαdecision rule for testing H 0 versu ...
486 Optimal Tests of Hypotheses 8.2.2.LetXhave a pdf of the formf(x;θ)=1/θ, 0 <x<θ, zero elsewhere. Let Y 1 <Y 2 <Y ...
8.3. Likelihood Ratio Tests 487 8.2.12.LetXhave the pdff(x;θ)=θx(1−θ)^1 −x,x=0,1, zero elsewhere. We testH 0 :θ=^12 againstH 1 : ...
488 Optimal Tests of Hypotheses it was shown that no UMP test exists for this situation. If we restrict attention to the class o ...
8.3. Likelihood Ratio Tests 489 independent random variables having the likelihood functions L(ω)= ( 1 2 πθ 3 )(n+m)/ 2 exp { − ...
490 Optimal Tests of Hypotheses The solutions forθ 1 ,θ 2 ,andθ 3 are, respectively, u 1 = n−^1 ∑n 1 xi u 2 = m−^1 ∑m 1 yi w′ =( ...
8.3. Likelihood Ratio Tests 491 and m ( Y− nX+mY n+m ) 2 = n^2 m (n+m)^2 (X−Y)^2. Hence the random variable defined by Λ^2 /(n+m ...
492 Optimal Tests of Hypotheses Definition 8.3.1.Let the random variableWbeN(δ,1); let the random variable V beχ^2 (r),andletW a ...
8.3. Likelihood Ratio Tests 493 Example 8.3.3(Power of the Two Samplet-Test).In Example 8.3.1 we had T= W 2 √ V 2 /(n+m−2) , whe ...
494 Optimal Tests of Hypotheses whereZhas a standard normal distribution. Hence the asymptotic test would use the critical regio ...
8.3. Likelihood Ratio Tests 495 Table 8.3.1:EmpiricalαLevels for the Nominal 0.05t-Test of Example 8.3.4. Empiricalα μc 0 5 10 1 ...
496 Optimal Tests of Hypotheses Remark 8.3.2.We caution the reader on this last test for the equality of two variances. In Remar ...
8.3. Likelihood Ratio Tests 497 EXERCISES 8.3.1.Verzani (2014) discusses a data set on healthy individuals, including their temp ...
498 Optimal Tests of Hypotheses (d)Determine the smallest value ofnso the power exceeds 0.80 to detectμ= 53. Hint:Modify the R f ...
8.3. Likelihood Ratio Tests 499 (b)Show that the likelihood ratio for testingH 0 :θ 3 =θ 4 withθ 1 andθ 2 unspec- ified can be b ...
500 Optimal Tests of Hypotheses Sample − 0. 389 − 2. 177 0.813 − 0. 001 1 − 0. 110 − 0. 709 0. 456 0. 135 Sample 0. 763 − 0. 570 ...
8.4.∗The Sequential Probability Ratio Test 501 are known numbers. For this section, we denote the likelihood ofX 1 ,X 2 ,...,Xn ...
502 Optimal Tests of Hypotheses A test of this kind is called Wald’ssequential probability ratio test.Fre- quently, inequality ( ...
8.4.∗The Sequential Probability Ratio Test 503 What is the probability of the procedure continuing indefinitely? What is the va ...
504 Optimal Tests of Hypotheses provided thatβis not equal to 0 or 1. Now letαaandβabe preassigned proper fractions; some typica ...
«
21
22
23
24
25
26
27
28
29
30
»
Free download pdf