Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.6. Functions of a Parameter 445

The bootstrap process described above is often called thenonparametric boot-
strapbecause it makes no assumptions about the pdff(x;θ). In this chapter,
though, strong assumptions are made about the model. For instance, in the last
example, we assume that the pdf is normal. What if we make use of this information
in the bootstrap? This is called theparametric bootstrap. For the last example,
instead of sampling from the empirical cdfF̂n, we sample randomly from the nor-
mal distribution, using as meanxand as standard deviations, the sample standard
deviation. The R functionbootse2.Rperforms this parametric bootstrap. For our
run on the data set in the example, it computed the standard error as 3.162918.
Notice how close the three estimated standard deviations are.
Which bootstrap, nonparametric or parametric, should we use? We recommend
the nonparametric bootstrap in general. The strong model assumptions are not
needed for its validity. See pages 55–56 of Efron and Tibshirani (1993) for discussion.

EXERCISES

7.6.1.LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that is
N(θ,1),−∞<θ<∞. Find the MVUE ofθ^2.

Hint: First determineE(X

2

).

7.6.2. LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that is
N(0,θ). ThenY =

∑

Xi^2 is a complete sufficient statistic forθ. Find the MVUE

ofθ^2.

7.6.3.Consider Example 7.6.3 where the parameter of interest isP(X<c)forX
distributedN(θ,1). Modify the R functionbootse1.Rso that for a specified value
ofcit returns the MVUE ofP(X<c) and the bootstrap standard error of the
estimate. Run your function on the data inex763data.rdawithc= 11 and 3,000
bootstraps. These data are generated from aN(10,1) distribution. Report (a) the
true parameter, (b) the MVUE, and (c) the bootstrap standard error.

7.6.4.For Example 7.6.4, modify the R functionbootse1.Rso that the estimate
is the median not the mean. Using 3,000 bootstraps, run your function on the data
set discussed in the example and report (a) the estimate and (b) the bootstrap
standard error.

7.6.5.LetX 1 ,X 2 ,...,Xnbe a random sample from a uniform (0,θ) distribution.
Continuing with Example 7.6.2, find the MVUEs for the following functions ofθ.

(a)g(θ)=θ

2

12 , i.e., the variance of the distribution.

(b)g(θ)=^1 θ, i.e., the pdf of the distribution.

(c)Fortreal,g(θ)=e

tθ− 1

tθ , i.e., the mgf of the distribution.

7.6.6.LetX 1 ,X 2 ,...,Xnbe a random sample from a Poisson distribution with

parameterθ>0.