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Lectorial Module 2.1

This document discusses various statistical estimation techniques, including: - Method of moments, which estimates population parameters by setting sample moments equal to theoretical population moments. Steps to find a method of moments estimator are described. - Maximum likelihood estimation, which finds estimates that maximize the likelihood function. Steps to derive a maximum likelihood estimator are outlined. - Bayesian estimation, which incorporates a prior distribution and calculates a posterior distribution based on observed data. Key terms in the Bayesian framework like prior, posterior, and joint density are explained. - An example of Bayesian estimation for a Bernoulli-Beta model is provided.

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Tharitha Murage
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65 views9 pages

Lectorial Module 2.1

This document discusses various statistical estimation techniques, including: - Method of moments, which estimates population parameters by setting sample moments equal to theoretical population moments. Steps to find a method of moments estimator are described. - Maximum likelihood estimation, which finds estimates that maximize the likelihood function. Steps to derive a maximum likelihood estimator are outlined. - Bayesian estimation, which incorporates a prior distribution and calculates a posterior distribution based on observed data. Key terms in the Bayesian framework like prior, posterior, and joint density are explained. - An example of Bayesian estimation for a Bernoulli-Beta model is provided.

Uploaded by

Tharitha Murage
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Estimation Techniques

Estimation Techniques

Module 2.1

School of Risk & Actuarial Studies


UNSW Business School

Lectorial notes

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Estimation Techniques

Quiz
Define the following concepts:
I Statistic

I Point estimate

I Estimator

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Estimation Techniques

Method of Moments

I How to find a method of moment estimator? Describe all


principal steps of the estimation procedure.

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Estimation Techniques

Method of Moments
Question: Consider X1 , X2 , . . . , Xn i.i.d. and Gamma(, )
find the MME of the parameters of the Gamma distribution.
(+r )
fX (x) = x 1 e x ;
() E [X r ] = r ()
   
MX (t) = E e tX = t ; Var (X ) = 2
.

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Estimation Techniques

Maximum Likelihood Estimation

I What is a likelihood function, L(|x)?


I Why do we need to introduce log-likelihood function, `(|x)?
I How to obtain a Maximum Likelihood Estimator? Describe all
principal steps of the estimation procedure and conditions.

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Estimation Techniques

Maximum Likelihood Estimation


Consider an i.i.d. sample X1 , X2 , ..., Xn from a Exp() with
observations x1 , x2 , ..., xn .
I What is the likelihood function?
I What is the domain and range of the likelihood function?
I Is the likelihood function continuous?
I Derive the maximum likelihood estimator.

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Estimation Techniques

Maximum Likelihood Estimation


Consider an i.i.d. sample X1 , X2 , ..., Xn from a Uniform(0, ) with
observations x1 , x2 , ..., xn .
I What is the likelihood function?
I What is the domain and range of the likelihood function?
I Is the likelihood function continuous?
I Derive the maximum likelihood estimator.

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Estimation Techniques

Bayesian Estimation
Under the Bayesian framework, explain the following terms
I probability distribution function
I joint density function
I prior distribution
I posterior distribution

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Estimation Techniques
Examples: Bayesian estimation

Example Bayesian estimation: Bernoulli-Beta


I Let X1 , X2 , . . . , XT be i.i.d. Bernoulli(), i.e.,
(Xi | = ) Bernoulli(). Assume the prior density of is
Beta(a, b) so that:

(a + b)
() = a1 (1 )b1 .
(a) (b)

I Find the Bayesian estimator, .


b

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