376 Maximum Likelihood Methods(a)Obtain a histogram of the data using the argumentpr=T.Overlaythepdfof
aβ(4,1) pdf. Comment.(b)Using the results of Exercise 6.2.12, compute the maximum likelihood estimate
based on the data.(c)Using the confidence interval found in Part (c) of Exercise 6.2.12, compute
the 95% confidence interval forθbased on the data. Is the confidence interval
successful?6.2.14.Consider sampling on the random variableXwith the pdf given in Exercise
6.2.9.(a)Obtain the corresponding cdf and its inverse. Show how to generate observa-
tions from this distribution.(b)Write an R function that generates a sample onX.(c)Generate a sample of size 50 and compute the unbiased estimate ofθdiscussed
in Exercise 6.2.9. Use it and the Central Limit Theorem to compute a 95%
confidence interval forθ.6.2.15.By using expressions (6.2.21) and (6.2.22), obtain the result for the one-step
estimate discussed at the end of this section.
6.2.16. LetS^2 be the sample variance of a random sample of sizen>1from
N(μ, θ), 0<θ<∞,whereμis known. We knowE(S^2 )=θ.
(a)What is the efficiency ofS^2?(b)Under these conditions, what is the mleθ̂ofθ?(c)What is the asymptotic distribution of√
n(θ̂−θ)?6.3 MaximumLikelihoodTests
In the last section, we presented an inference for pointwise estimation and confidence
intervals based on likelihood theory. In this section, we present a corresponding
inference for testing hypotheses.
As in the last section, letX 1 ,...,Xnbe iid with pdff(x;θ)forθ∈Ω. In this
section,θis a scalar, but in Sections 6.4 and 6.5 extensions to the vector-valued
case are discussed. Consider the two-sided hypothesesH 0 :θ=θ 0 versusH 1 : θ =θ 0 , (6.3.1)whereθ 0 is a specified value.