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Example sheet 4. Statistical Modelling: Mathematical Tripos, Part IIC

Summary: Example sheet 4. Statistical Modelling: Mathematical
Tripos, Part IIC
P.M.E.Altham, Statistical Laboratory, University of Cambridge.
January 18, 2005
These questions are all adapted from recent Part IIA Tripos questions. There may
be some overlap between questions, eg on basic bookwork matters.
You are not intended to do all these questions for 1 supervision. Be selective. Keep some questions
for later revision purposes.
1998/4/13M (long question) Suppose that Y 1 ; :::; Yn are independent random variables, and that
Y i has probability density function
f(y i j i ; ) = exp[(y i  i b( i ))= + c(y i ; )]:
Assume that E(Y i ) =  i , and that there is a known link function g such that
g( i ) = T x i ; where x i is known and is unknown:
Show that
(a) E(Y i ) = b 0 ( i ),
(b) var(Y i ) = b 00 ( i ) = V i say, and hence
(c) if `( ; ) is the log-likelihood function from the observations (y 1 ; :::; yn ) then
@`( ; )


Source: Altham, Pat - Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge


Collections: Mathematics