Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
4.9. Bootstrap Procedures 305 LikewiseV(x∗i)=n−^1 ∑n i=1(xi−x) (^2) ; see Exercise 4.9.2. At first glance, this re- sampling the ...
306 Some Elementary Statistical Inferences size of the sample and provides a vector to store theθˆ∗s. In theforloop, the ith boo ...
4.9. Bootstrap Procedures 307 40 60 80 100 120 140 160 0 100 200 300 400 500 600 Frequency x* Figure 4.9.1:Histogram of the 3000 ...
308 Some Elementary Statistical Inferences cdfĤ.Letφ̂=m(̂θ)andφ̂∗=m(̂θ∗). We have P [ θ̂∗≤m−^1 (φ̂−z(1−α/2)σc) ] = P [ φ̂∗≤φ̂−z ...
4.9. Bootstrap Procedures 309 Whilej≤B, do steps 3–6. Obtain a random sample with replacement of sizen 1 fromz.Callthesample x∗ ...
310 Some Elementary Statistical Inferences There are three outliers in the data sets. Our test statistic for these data isv=y−x= ...
4.9. Bootstrap Procedures 311 replacement; see Exercise 4.9.10. Usually, the permutation tests and the bootstrap tests give very ...
312 Some Elementary Statistical Inferences 60 70 80 90 100 110 0 200 400 600 800 1000 Frequency z* Figure 4.9.3: Histogram of th ...
4.9. Bootstrap Procedures 313 4.9.3.LetX 1 ,X 2 ,...,Xnbe a random sample from a Γ(1,β) distribution. (a)Show that the confidenc ...
314 Some Elementary Statistical Inferences (a)Rewrite the percentile bootstrap confidence interval algorithm using the mean and ...
4.10.∗Tolerance Limits for Distributions 315 For each such samplej: (a)Label the sample of sizen 1 byx∗and label the sample of ...
316 Some Elementary Statistical Inferences the random interval, can we be led to an additional distribution-free method of stati ...
4.10.∗Tolerance Limits for Distributions 317 Because the distribution function ofZ=F(X)isgivenbyz, 0 <z<1, it follows from ...
318 Some Elementary Statistical Inferences We also note that the Jacobian is equal to 1 and that the space of positive probabili ...
4.10.∗Tolerance Limits for Distributions 319 EXERCISES 4.10.1.LetY 1 andYnbe, respectively, the first and thenth order statistic ...
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Chapter 5 Consistency and Limiting Distributions In Chapter 4, we introduced some of the main concepts in statistical inference, ...
322 Consistency and Limiting Distributions Definition 5.1.1.Let{Xn}be a sequence of random variables and letXbe a ran- dom varia ...
5.1. Convergence in Probability 323 Proof:Let>0 be given. Using the triangle inequality, we can write |Xn−X|+|Yn−Y|≥|(Xn+Yn) ...
324 Consistency and Limiting Distributions Proof:Using the above results, we have XnYn = 1 2 Xn^2 + 1 2 Yn^2 − 1 2 (Xn−Yn)^2 P → ...
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