Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
7.9. Sufficiency, Completeness, and Independence 465 the underlying distribution is highly skewed to the right. WhileShas a dist ...
466 Sufficiency 7.9.4.LetXandY be random variables such thatE(Xk)andE(Yk) =0exist fork=1, 2 , 3 ,....IftheratioX/Yand its denomi ...
7.9. Sufficiency, Completeness, and Independence 467 7.9.12.LetY 1 <Y 2 <···<Ynbe the order statistics of a random samp ...
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Chapter 8 Optimal Tests of Hypotheses 8.1 MostPowerfulTests In Section 4.5, we introduced the concept of hypotheses testing and ...
470 Optimal Tests of Hypotheses levelof the test is the probability of a Type I error; i.e., α=max θ∈ω 0 Pθ(X∈C). (8.1.3) Note t ...
8.1. Most Powerful Tests 471 x 0 1 2 f(x;1/2) 1 / 32 5 / 32 10 / 32 f(x;3/4) 1 / 1024 15 / 1024 90 / 1024 f(x;1/2)/f(x;3/4) 32 / ...
472 Optimal Tests of Hypotheses Theorem 8.1.1. Neyman–Pearson Theorem. LetX 1 ,X 2 ,...,Xn,wheren is a fixed positive integer, d ...
8.1. Most Powerful Tests 473 and, from Equation (8.1.5), we obtain ∫ C L(θ′′)− ∫ A L(θ′′)≥ 1 k [∫ C∩Ac L(θ′)− ∫ A∩Cc L(θ′) ] . ( ...
474 Optimal Tests of Hypotheses As the next corollary shows, the best test given in Theorem 8.1.1 is an unbiased test. Corollary ...
8.1. Most Powerful Tests 475 Example 8.1.2.LetX′=(X 1 ,...,Xn) denote a random sample from the distri- bution that has the pdf f ...
476 Optimal Tests of Hypotheses For example, ifn=25andαis 0.05,c 1 =qnorm(0.95,0,1/5) = 0.329,using R. Hence, the power of the t ...
8.1. Most Powerful Tests 477 is a best critical regionC. Consider the case ofk=1andn= 1. The preceding inequality may be written ...
478 Optimal Tests of Hypotheses EXERCISES 8.1.1.In Example 8.1.2 of this section, let the simple hypotheses readH 0 :θ= θ′=0andH ...
8.2. Uniformly Most Powerful Tests 479 8.1.10.LetX 1 ,X 2 ,...,X 10 denote a random sample of size 10 from a Poisson distributio ...
480 Optimal Tests of Hypotheses As will be seen presently, uniformly most powerful tests do not always exist. However, when they ...
8.2. Uniformly Most Powerful Tests 481 Example 8.2.3.LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that isN(θ,1) ...
482 Optimal Tests of Hypotheses H 1 :θ=0.5. This critical region is also a uniformly most powerful one for testing H 0 :θ=0.1 ag ...
8.2. Uniformly Most Powerful Tests 483 maxθ≤θ′γ(θ), whereγ(θ) is the power function of the test. That is, the significance level ...
484 Optimal Tests of Hypotheses Example 8.2.5.LetX 1 ,X 2 ,...,Xnbe a random sample from a Bernoulli distri- bution with paramet ...
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