Chapter 8.3. Maximum Likelihood Estimation
Prof. Tesler
Math 283
October 14, 2013
Prof. Tesler
8.3 Maximum Likeilihood Estimation
Math 283 / October 14, 2013
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Estimating parameters
Let Y be a random variable with a distribution of known type but
unk

Maximum Likelihood &
Method of Moments
Estimation
Patrick Zheng
01/30/14
1
Introduction
Goal: Find a good POINT estimation of population
parameter
Data: We begin with a random sample of size n
taken from the totality of a population.
We shall estimate the

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Binomial Distribution Examples
I. Emily hits 60% of her free throws in basketball games. She had 25 free throws in last weeks game. Use
this information to answer the next two questions.
1. What is the average number of hits?
(a)
(b)
(c)
(d)
10
1

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2016-2017, 1st term
Preliminary: Reviews of Sets, Counting, and Calculus
A.1
Elementary Set Theory
Set
a precisely specified collection of objects.
Element
the objects in the set.
We may specify a set by listing its ele

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Some mathematical facts related to the Fisher Information
1. The Fisher Information I is the variance of the score function S X ;
d
log f X ;
d
and can be evaluated as
d2
I E
log f X ; .
2
d
Proof:
Consider
S x;

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Cauchy-Schwartz Inequality
Let X and Y be random variables with finite second moments, then
E XY 2 E X 2 E Y 2 .
The equality holds if and only if either P Y 0 1 or P X aY 1 for some constant a, i.e. if
and only if X

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Chapter III
3.1
2016-2017, 1st term
Mathematical Expectation
Expected Value
Example 3.1
Consider the following two games. In each game, three fair dice will be rolled.
Game 1: If all the dice face up with same number, t

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Independence between the sample mean and sample variance from normal sample
iid
Suppose that X 1 , X 2 ,., X n ~ N , 2 , then X and S 2 are independent.
Proof:
The joint pdf of X 1 , X 2 ,., X n is given by
n
x 2

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Chapter II
2.1
Random
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Distributions
and
2016-2017, 1st term
Probability
Random Variables
Definition
A random variable X : is a numerical valued function defined on a sample
space. In other words, a number X ,

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Chapter I
1.1
2016-2017, 1st term
Probability
Classical Probability
Definition
Suppose a single trial in a chance situation can have one of N equally likely
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Formal Derivation of Poisson Process
Denote N t as the number of occurrences of a specific event E within time interval 0, t .
Assume the following postulates:
1. Independence The number of times E occurs in non-over

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Properties of Cumulative Distribution Function
1. F x is nondecreasing.
Proof:
Suppose a b , then X a X b and therefore P X a P X b , i.e.
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Properties of Probability
1. P 0
Since , these two events are disjoint. By axiom 3, P P P and the
result follows from the fact that .
2. P Ac 1 P A
Obviously A Ac . By axiom 2, P A Ac P 1 . Since A Ac , these two
eve

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Wilks Theorem
(Here only the one-dimensional parameter space is considered. For multidimensional parameters
version, see the book Mathematical Statistics by Bickel and Doksum.)
For testing H 0 : 0 vs H 1 : 0 based on