362 Maximum Likelihood Methods(b)Approximate the mle by plotting the function in Part (a). Make use of the
following R code which assumes that the data are in the R vectorx:
theta=seq(-6,6,.001);lfs<-c()
for(th in theta){lfs=c(lfs,sum(log((x-th)^2+1)))}
plot(lfs~theta)6.1.9.Let the table
x 01 2 345
Frequency 714121363represent a summary of a random sample of size 55 from a Poisson distribution.
Find the maximum likelihood estimator ofP(X= 2). Use the R functiondpoisto
find the estimator’s realization for the data in the table.
6.1.10.LetX 1 ,X 2 ,...,Xnbe a random sample from a Bernoulli distribution with
parameterp.Ifpis restricted so that we know that^12 ≤p≤1, find the mle of this
parameter.
6.1.11.LetX 1 ,X 2 ,...,Xnbe a random sample from aN(θ, σ^2 ) distribution, where
σ^2 is fixed but−∞<θ<∞.
(a)Show that the mle ofθisX.(b)Ifθis restricted by 0≤θ<∞, show that the mle ofθisθ̂=max{ 0 ,X}.6.1.12.LetX 1 ,X 2 ,...,Xnbe a random sample from the Poisson distribution with
0 <θ≤2. Show that the mle ofθiŝθ=min{X, 2 }.
6.1.13.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution with one of
two pdfs. Ifθ=1,thenf(x;θ=1)=√^12 πe−x(^2) / 2
,−∞<x<∞.Ifθ=2,then
f(x;θ=2)=1/[π(1 +x^2 )],−∞<x<∞.Findthemleofθ.
6.2 Rao–Cram ́erLowerBoundandEfficiency
In this section, we establish a remarkable inequality called theRao–Cram ́erlower
bound, which gives a lower bound on the variance of any unbiased estimate. We then
show that, under regularity conditions, the variances of the maximum likelihood
estimates achieve this lower bound asymptotically.
As in the last section, letXbe a random variable with pdff(x;θ),θ∈Ω, where
the parameter space Ω is an open interval. In addition to the regularity conditions
(6.1.1) of Section 6.1, for the following derivations, we require two more regularity
conditions, namely,
Assumptions 6.2.1 (Additional Regularity Conditions). Regularity conditions
(R3) and (R4) are given by
(R3) The pdff(x;θ)is twice differentiable as a function ofθ.