5.2. Convergence in Distribution 3275.1.5.Consider the R functionconsistmeanwhich produces the plot shown in
Figure 5.1.1. Obtain a similar plot for the sample median when the distribution
sampled is theN(0,1) distribution. Compare the mean and median plots.
5.1.6.Write an R function that obtains a plot similar to Figure 5.1.1 for the situ-
ation described in Example 5.1.2. For the plot chooseθ= 10.
5.1.7.LetX 1 ,...,Xnbe iid random variables with common pdf
f(x)={
e−(x−θ) x>θ,−∞<θ<∞
0elsewhere.
(5.1.3)This pdf is called theshifted exponential.LetYn=min{X 1 ,...,Xn}.Prove
thatYn→θin probability by first obtaining the cdf ofYn.5.1.8.Using the assumptions behind the confidence interval given in expression
(4.2.9), show that √
S^21
n 1
+S^22
n 2
/√
σ^21
n 1
+σ 22
n 2P
→ 1.5.1.9.For Exercise 5.1.7, obtain the mean ofYn.IsYnan unbiased estimator of
θ? Obtain an unbiased estimator ofθbased onYn.5.2 ConvergenceinDistribution.......................
In the last section, we introduced the concept of convergence in probability. With
this concept, we can formally say, for instance, that a statistic converges to a pa-
rameter and, furthermore, in many situations we can show this without having to
obtain the distribution function of the statistic. But how close is the statistic to the
estimator? For instance, can we obtain the error of estimation with some credence?
The method of convergence discussed in this section, in conjunction with earlier
results, gives us affirmative answers to these questions.
Definition 5.2.1(Convergence in Distribution).Let{Xn}be a sequence of random
variables and letXbe a random variable. LetFXnandFXbe, respectively, the cdfs
ofXnandX.LetC(FX)denote the set of all points whereFXis continuous. We
say thatXnconverges in distributiontoXif
lim
n→∞
FXn(x)=FX(x), for allx∈C(FX).We denote this convergence by
Xn
D
→X.Remark 5.2.1.This material on convergence in probability and in distribution
comes under what statisticians and probabilists refer to asasymptotic theory.Of-
ten, we say that the distribution ofX is theasymptotic distributionor the
limiting distributionof the sequence{Xn}. We might even refer informally to