Math 541: Statistical Theory II
Statistical Inference and Method of Moment
Instructor: Songfeng Zheng
1
Statistical Inference Problems
In probability problems, we are given a probability distribution, and the purpose is to to
analyze the property (Mean, v
Math 541: Statistical Theory II
Connection between Method of Moment and Maximum Likelihood
Lecturer: Songfeng Zheng
In the parameter estimation problem, we have an i.i.d. random sample X1 , , Xn from the
probability distribution f (x|) with unknown parame
Math 541: Statistical Theory II
Maximum Likelihood Estimation
Lecturer: Songfeng Zheng
1
Maximum Likelihood Estimation
Maximum likelihood is a relatively simple method of constructing an estimator for an unknown parameter . It was introduced by R. A. Fish
v
sa
France with a Presidential election looming, and the outcome far from easy to predict.
We live in a world defined by the pace of change, and whilst the velocity of that change has not
always impacted upon our political institutions, many of which wou
v
sa
France with a Presidential election looming, and the outcome far from easy to predict.
We live in a world defined by the pace of change, and whilst the velocity of that change has not
always impacted upon our political institutions, many of which wou
v
sa
France with a Presidential election looming, and the outcome far from easy to predict.
We live in a world defined by the pace of change, and whilst the velocity of that change has not
always impacted upon our political institutions, many of which wou
v
sa
France with a Presidential election looming, and the outcome far from easy to predict.
We live in a world defined by the pace of change, and whilst the velocity of that change has not
always impacted upon our political institutions, many of which wou
v
sa
France with a Presidential election looming, and the outcome far from easy to predict.
We live in a world defined by the pace of change, and whilst the velocity of that change has not
always impacted upon our political institutions, many of which wou
Math 541: Statistical Theory II
Sucient Statistics and Exponential Family
Lecturer: Songfeng Zheng
1
Statistics and Sucient Statistics
Suppose we have a random sample X1 , , Xn taken from a distribution f (x|) which relies
on an unknown parameter in a par
4. Let
be an i.i.d. sample from a Rayleigh distribution with parameter > 0 :
f (x | ) =
Find the asymptotic variance of the MLE. Solution:
x
2
e x
2
/(2 2 )
We have log f ( x | ) = log x 2 log
x2 2 x2 , then log f ( x | ) = + 3 , and 2 2
2 2 3x 2 log f (
v
sa
France with a Presidential election looming, and the outcome far from easy to predict.
We live in a world defined by the pace of change, and whilst the velocity of that change has not
always impacted upon our political institutions, many of which wou