Likelihood Ratios
Rice, Chapter 9 discuss hypothesis testing and likelihood ratios.
A hypothesis is a statement about the statistical model, and in the case of parametric models a
statement about parameters. It is not a statement about random variables or
Statistics 3858 : Likelihood Ratio for Exponential Distribution
In these two example the rejection rejection region is of the form
cfw_x : 2 log (x) > c
for an appropriate constant c. For a size test, using Theorem 9.5A we obtain this critical value from
Statistics 3858b: Theory of Statistics
January, 2014
1. Instructor: R. J. Kulperger
e-mail: kulperger@stats.uwo.ca phone 661-3627. Oce - temporary due to renovations
WSC 276, but will move sometime this semester.
2. Text: J. A. Rice, Mathematical Statist
Statistics 3858 : Cramer-Rao and Sucient Statistics
1
Preliminary Comments
One of the methods to compare estimators is based of Mean Square Errors (MSE)
M SE() = E ( )2 ) = Var() + Bias()2
An estimator is unbiased if and only if for all
E () = .
Thus for
University of Western Ontario
Statistics 3858b Mid-Term Test
February 9, 2011 5:30-6:20 PM
Instructor: R. J. Kulperger
Instructions:
I. Make sure that your name and ID number are the front of your exam
booklet and question sheet. The exam questions are to
Neyman-Pearson Lemma
For two parameter values 0 and 1 consider the likelihood ratio
LR(x) =
f (x; 0 )
f (x; 1 )
(1)
The rejection region based on the likelihood ratio (1) is of the form
R = cfw_x : LR(x) < c
(2)
We use the notation P (X R|) in place of th
Statistics 3858 : Likelihood Ratio for Multinomial Models
Suppose X is multinomial on M categories, that is X Multinomial(n, p), where p = (p1 , p2 , . . . , pM )
A, and the parameter space is
M
A = cfw_p : pj 0,
pj = 1
j=1
The dimension of this paramet
FORMULAE SHEET
Some common pdfs and properties are given on this page. There are other common distributions
from the course that the student is expected to know.
1. The bivariate normal pdf is
f (x, y)
=
cfw_
[
1
1
(x X )2
exp
2
2)
2(1
X
2X Y 1 2
]
(x X
Delta Method
Often estimators are functions of other random variables, for example in the method
of moments. These functions of random variables can sometimes inherit a normal approximation from the underlying random variables.
Example : Method of Moments
Statistics 3858 : Construction of Two Common Types of
Estimators
Estimators are statistics used to estimate parameters or functions of parameters for a statistical model.
Here we consider nite parameter models. Let be the parameter space.
Consider the typ
Nonparametric Bootstrap
1
Parametric Bootstrap
Earlier in the course we used the parametric bootstrap. A brief overview of this method
is given below.
1. Xi , 1 = 1, . . . , n are iid from f (; ), . 0 is used to denote the true value of
the parameter.
2.
Statistics 3858b : Bayesian Methods
March 10, 2014
So far in our course we have viewed parameters as given numbers in a parameter space and the
distribution of the observable random variables as coming from a distribution f (; ) for one xed value
of . Thi
Statistics 3858 : Maximum Likelihood Estimators - Regular Case
Large Sample Theory
1
Regular Case
The regular maximum likelihood case is essentially the case where calculus methods apply in order to
calculate the MLE. These are written for the case of iid
University of Western Ontario
Statistics 3858b Term Test , brief solutions
March 19, 2014 , 5:30 - 6:20
1.
a) Dene Sucient Statistic and state the Factorization Theorem.
b) Consider the test of hypothesis setting with test statistic say T . The rejection
University of Western Ontario
Statistics 3858b Term Test
March 16, 2011 5:30-6:30 PM
Instructor: R. J. Kulperger
Instructions:
I. Make sure that your name and ID number are the front of your exam booklet and question
sheet. The exam questions are to be re
Introductory Examples
Rice, Section 10.2, page 378, gives a data set of the melting point of beeswax. This data is in the le
Chapter 10, beeswax.txt. Figure 1 gives a relative frequency histogram of the data. The histogram has
the general shape of a norma
Statistics 3858 : Statistical Models, Parameter Space and
Identiability
In an experiment or observational study we have data X1 , X2 , . . . , Xn . These we view as a observations
of random variables with some joint distribution.
Denition 1 A statistical
Statistics 3858 : Maximum Likelihood Estimators
1
Method of Maximum Likelihood
In this method we construct the so called likelihood function, that is
L() = L(; X1 , X2 , . . . , Xn ) = fn (X1 , X2 , . . . , Xn ; )
The function fn is either the joint pdf o
Properties of Estimators
We study estimators as random variables. In this setting we suppose X1 , X2 , . . . , Xn are random
variables observed from a statistical model F with parameter space .
In our usual setting we also then assume that Xi are iid with
Two sample normal problem.
The parameters are X , Y and 2 (recall there is a common variance assumption. Thus the parameter
space is
= cfw_ = (X , Y , 2 ) : X , Y R, 2 > 0.
The hypotheses to be tested are
H0 : X Y = and HA : X Y =
where is a xed (or giv
Statistics 3858b Assignment 5
Handout April 2, 2014 ; Due date: April 7, 2014
These problems are all from the course text unless otherwise stated.
1. Use the data from Problem 11.40 g. The eld present data is a sample
from one population distribution,say
Statistics 3858 : An Example of a Nonparametric Test
1
Non Parametric Methods : Introduction and Comments
Bootstrap
one method of nonparametric method that is used quite commonly
Nonparametric tests are sometimes are test statistics for particular hypothe