4.9. Bootstrap Procedures 303(a)Use Theorem 4.8.1 to generate an observation from this pdf.(b)Use the accept–reject algorithm to generate an observation from this pdf.4.8.19.Proceeding similar to Example 4.8.7, use the accept–reject algorithm to
generate an observation from atdistribution withr>1 degrees of freedom when
g(x) is the Cauchy pdf.4.8.20.Forα>0andβ>0, consider the following accept–reject algorithm:1.GenerateU 1 andU 2 iid uniform(0,1) random variables. SetV 1 =U
1 /α
1 and
V 2 =U
1 /β
2.
2.SetW=V 1 +V 2 .IfW≤1, setX=V 1 /W;elsegotostep1.3.DeliverX.Show thatXhas a beta distribution with parametersαandβ, (3.3.9). See Kennedy
and Gentle (1980).
4.8.21.Consider the following algorithm:
1.GenerateUandVindependent uniform (− 1 ,1) random variables.2.SetW=U^2 +V^2.3.IfW>1gotostep1.4.SetZ=√
(−2logW)/Wand letX 1 =UZandX 2 =VZ.Show that the random variablesX 1 andX 2 are iid with a commonN(0,1) distri-
bution. This algorithm was proposed by Marsaglia and Bray (1964).
4.9 BootstrapProcedures
In the last section, we introduced the method of Monte Carlo and discussed several
of its applications. In the last few years, however, Monte Carlo procedures have
become increasingly used in statistical inference. In this section, we present the
bootstrap, one of these procedures. We concentrate on confidence intervals and
tests for one- and two-sample problems in this section.4.9.1 Percentile Bootstrap Confidence Intervals
LetX be a random variable of the continuous type with pdff(x;θ), forθ∈Ω.
SupposeX=(X 1 ,X 2 ,...,Xn) is a random sample onXandθ̂=θ̂(X)isapoint
estimator ofθ. The vector notation,X, proves useful in this section. In Sections 4.2
and 4.3, we discussed the problem of obtaining confidence intervals forθin certain
situations. In this section, we discuss a general method called thepercentile boot-
strapprocedure, which is aresamplingprocedure. It was proposed by Efron (1979).