Stat 543 I
Suppose first that \ is a discrete random variable with probability mass function 0 ÐBl)Ñ satisfying
standard regularity conditions for ) in @ an open interval in e.
(a) Define M\ Ð)! Ñ, the Fisher information in \ about the parameter ) , evaluated at )! − @Þ
(b) Suppose that for another open interval > in e, the differentiable increasing function 2 maps > onto
@. Consider the (reparameterized) model for \ defined by
1ÐBl# Ñ » 0 ÐBl2Ð#ÑÑ for # − >.
Express the Fisher information about the parameter # , evaluated at #! − >, in terms of the functions
M\ Ð)Ñ and 2Ð# Ñ. (Write 2w for the derivative of 2.) Argue carefully that your expression is correct.
Suppose now that ]" ß ]# ß ÞÞÞß ]8 are independent Bernoulli random variables. For known constants
>" ß ># ß ÞÞÞß >8 and unknown parameters ! − e and " − e we will suppose that
T Ò]3 œ "Ó œ
"
Þ
"  expÐ  Ð!  " >3 ÑÑ
(1)
This is equivalent to assuming that
lnŒ
T Ò]3 œ "Ó
 œ !  " >3 .
"  T Ò]3 œ "Ó
(2)
(c) Write out the log-likelihood function, _Ð!ß " Ñ, for this problem and a pair of equations that will
have to be solved in order to find maximum likelihood estimates of the parameters ! and " .
For a particular famous data set with 8 œ #$, two figures summarizing a likelihood analysis of this
problem are attached to this question. Use them in what follows. It will also be helpful to know that
s œ  Þ#$## and _Ð"&Þ!%%ß  Þ#$##Ñ œ  "!Þ"&).
MLE's for this data set are !
s œ "&Þ!%% and "
For " − e, let !‡ Ð" Ñ be a maximizer of _Ð!ß " Ñ considered as a function of !. Define
_‡ Ð" Ñ œ _Ð!‡ Ð" Ñß " Ñ Þ
Figure 1 is a plot of this function.
(d) Note that " œ ! in either (1) or (2) is the case where T Ò] œ "Ó doesn't depend upon >. Is H! À " œ !
a plausible hypothesis in the light of the Figure 1? Explain. (Hint: Consider a likelihood ratio test of
H! .)
(e) Give an approximate 90% confidence interval for " based on Figure 1.
In the real problem, the quantity :$" »
"
"expÐÐ!31"ÑÑ
was of vital interest. (This is the "success
probability" for a ] with > œ $".) It is possible to parameterize this problem in terms of :$" and, say,
:(# » "expÐÐ"!72"ÑÑ ( the "success probability" for a ] with > œ (#.)
(f) Give MLE's of :$" and :(# and argue carefully that your values are correct.
(g) The hope was that :$" was in fact small. Figure 2 is a plot outlining the region in Ð:$" ß :(# Ñ-space with
log-likelihood exceeding  "#Þ%'!&. Carefully interpret Figure 2 in the light of the hope about :$" .