557: M ATHEMATICAL S TATISTICS II
M ETHODS OF E VALUATING E STIMATORS
An estimator, T (X ), of can be evaluated via its statistical properties. Typically, two aspects are
considered:
Expectation
Variance
either in terms of nite n behaviour, or the hypot
MATH 557 - A SSIGNMENT 2 S OLUTIONS
1 (a) The joint pdf for X1 , . . . , Xn is 1 1 fX | (x|) = n exp where x(1) = mincfw_x1 , . . . , xn . Thus
n T n
(xi ) I(x(1) ,) ()
i=1
< <
T (X ) =
i=1
Xi , X(1)
= T1 (X ), T2 (X )
T
,
say, is a sufcient statistic for
557: M ATHEMATICAL S TATISTICS II
B AYESIAN I NFERENCE AND D ECISION M AKING
1
Bayesian Inference
1.1 Introduction and Terminology
In Bayesian analysis, is treated as a random variable with a prior density encapsulating the beliefs
about before the data a
MATH 557 - A SSIGNMENT 1 S OLUTIONS
1 We construct the joint pmf/pdf in each case, and inspect the required conditional pdf. Note that 1-1 transformations of the statistics are also sufficient. (a) For the fX |, (x|, ) = suggesting the sufficient statisti
557: M ATHEMATICAL S TATISTICS II L ARGE S AMPLE AND A SYMPTOTIC R ESULTS - III
Behaviour of the Likelihood Ratio Test Statistic
In the test of H0 : = 0 H1 : = 0 using the likelihood ratio test, suppose that, in fact = T . Then, under conditions A0-A4, 2
557: M ATHEMATICAL S TATISTICS II T HE EM A LGORITHM
The EM Algorithm is a method for producing the maximum likelihood estimates in incomplete data problems, that is, models formulated for data that are only partially observed. Suppose that random variabl
557: M ATHEMATICAL S TATISTICS II
I NTERVAL E STIMATION - E XAMPLES
Example 1 : Inverting a Test Statistic
Suppose that X1 , . . . , Xn Normal(, 2 ) for 2 known. A condence interval can be constructed by
recalling the UMP unbiased test at level of
H0 : =
557: M ATHEMATICAL S TATISTICS II T HE EM A LGORITHM : G ENETICS OF H UMAN B LOOD G ROUPS
In human genetics, the genotype at a genomic locus is a pair of alleles corresponding to small segments of DNA lying on the two chromosomal strands. The phenotype is
557: M ATHEMATICAL S TATISTICS II N ON - PARAMETRIC M AXIMUM L IKELIHOOD
Suppose that X1 , . . . , Xn are a random sample from a distribution with cdf FX that is not specified using a parametric model, that is, the whole function FX (x) = Pr[X x] -<x<
is
557: M ATHEMATICAL S TATISTICS II
L ARGE S AMPLE AND A SYMPTOTIC R ESULTS
We now assess the properties of statistical procedures when the sample size n becomes large (large
sample theory), or in the limit as n tends to innity (asymptotic theory).
5.1 Poin
557: M ATHEMATICAL S TATISTICS II I NTERVAL E STIMATION
For a random sample X1 , . . . , Xn from parametric probability model fX| (x|), an interval estimator for scalar parameter comprises a pair of statistics, (L(X ), U (X ), such that for all x X , L(x)
557: M ATHEMATICAL S TATISTICS II
H YPOTHESIS T ESTING : E XAMPLES
Example 1 Suppose that X1 , . . . , Xn N (, 1). To test
H0 : = 0
H1 : = 1
the most powerful test at level is based on the statistic
(x) =
fX | (x|1)
fX | (x|0)
1 n
(xi 1)2
2 i=1
1 n 2
(2)n
557: M ATHEMATICAL S TATISTICS II
H YPOTHESIS T ESTING
A statistical hypothesis test is a decision rule that takes as an input observed sample data and returns
an action relating to two mutually exclusive hypotheses that reect two competing hypothetical s
557: M ATHEMATICAL S TATISTICS II
C OMPLETE S TATISTICS IN THE E XPONENTIAL FAMILY
Suppose that f is a one-parameter natural Exponential Family distribution in canonical form, written
using the tilting formulation as
f (x|) = f (x) expcfw_x K()
for pdf f