Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
8.4.∗The Sequential Probability Ratio Test 505 can be rewritten, by taking logarithms, as −log 9< 6 ∑ xi− 459 n 200 <log 9 ...
506 Optimal Tests of Hypotheses sayx 1 ,x 2 ,x 3 ,.... These are usually plotted; and if they are between the LCL and UCL, we sa ...
8.5.∗Minimax and Classification Procedures 507 8.4.2.LetXhave a Poisson distribution with meanθ. Find the sequential proba- bili ...
508 Optimal Tests of Hypotheses random variablesX 1 ,X 2 ,...,Xndepend upon the parameterθ.Herenis a fixed positive integer. Thi ...
8.5.∗Minimax and Classification Procedures 509 whereβ=1−γ(θ′′) is the probability of the type II error. Let us now see if we can ...
510 Optimal Tests of Hypotheses or, equivalently, R(θ′′,C)≤R(θ′′,A). That is, R(θ′,C)=R(θ′′,C)≤R(θ′′,A). This means that max[R(θ ...
8.5.∗Minimax and Classification Procedures 511 choose the distribution indexed byθ′′; that is, we classify (x, y)ascomingfromthe ...
512 Optimal Tests of Hypotheses That is, if this linear function ofxandyin the left-hand member of inequality (8.5.3) is less th ...
8.5.∗Minimax and Classification Procedures 513 EXERCISES 8.5.1.LetX 1 ,X 2 ,...,X 20 be a random sample of size 20 from a distri ...
514 Optimal Tests of Hypotheses 8.5.6.LetXandY have a joint bivariate normal distribution. An observation (x, y) arises from the ...
Chapter 9 Inferences About Normal Linear Models 9.1 Introduction................................ In this chapter, we consider an ...
516 Inferences About Normal Linear Models X 1 ,X 2 ,...,Xnfrom an arbitrary distribution. Thus (n−1)S^2 = ∑n i=1 (Xi−X)^2 = ∑n i ...
9.2. One-Way ANOVA 517 analysis of variance(ANOVA). In short, we say that this example is a one-way ANOVA problem. Here thefull ...
518 Inferences About Normal Linear Models Since this derivation did not depend onσ, to find the mle ofσ, we substitutex·j forμji ...
9.2. One-Way ANOVA 519 To complete the test, we need to determine the distribution ofF underH 0. First consider the sum of squar ...
520 Inferences About Normal Linear Models oneway.test(elasticmod~ind,var.equal=T) F = 12.565, num df = 2, denom df = 19, p-value ...
9.2. One-Way ANOVA 521 9.2.5.Using the notation of this section, assume that the means satisfy the con- dition thatμ=μ 1 +(b−1)d ...
522 Inferences About Normal Linear Models (d)Note that one assumption for theF-test is that the random errorseijin Model (9.2.1) ...
9.3. Noncentralχ^2 andF-Distributions 523 The integral exists ift<^12. To evaluate the integral, note that tx^2 i σ^2 − (xi−μ ...
524 Inferences About Normal Linear Models ∑n 1 μ 2 i/σ (^2) , may be computed by replacing eachX iin the quadratic form by its m ...
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