Question

How do we solve point estimation using
1)methods of moment
2)method of maximum likelihood estimation

Answer 1

Method of moments;

(i) Equate the first sample moments about the origin M₁ = $\frac {\sum x i}{n}=\bar x$  to the first theoretical moment (E(X))

(ii) Equate the second sample moments about the mean to the second theoretical moment about the mean $E[(X-\mu)^{2}]$

(iii) Continue equating corresponding sample moment about the mean M_k with the corresponding theoretical moment about the mean until you have equation equal to the number of parameters.$E[(X-\mu)^{k}]$ k=3,4, ...

(iv) Solve for the parameters and the resulting values are the method of moment estimators.

Method of Maximum Likelihood Estimation

Suppose having a random sample X₁,X₂ ,......,X_n where it is assumed the probability distribution depends on the same unknown parameter $\theta$ .

We find a point estimator  u(X₁,X_2, ......,X_n ) such that u(X₁,X_2, ...,X_n ) is  a good point estimate of the unknown parameter.

Let X₁,X₂ ,...,X_n be a random sample of a distribution depending on one or more unknown  parameters  $\theta _{1},\theta_{2},...,\theta_{m}$ with probability density  or mass function$f(x_{i};\theta_{1}, \theta_{2},...,\theta_{m})$ .  Suppose $(\theta_{1}, \theta_{2},...,\theta_{m})$ is restricted to the parameter space $\Omega$ . Then

(i) When regarded function of$\theta_{1},\theta_{2}, ...,\theta_{m}$ ,the joint probability density or mass function in $(x_{1},x_{2},...,x_{n})$:

$L(\theta_{1},\theta_{2},...,\theta_{m})=\prod f(x_{i};\theta_{1}, \theta_{2},...,\theta_{m}) [(\theta_{1},\theta_{2}, ..., \theta_{m})in \Omega]$ is the likelihood function

(ii)if:

$[u_{1}(x_1,x_{2},..x_n),u_2(x_1,x_2, ...,x_n),...,u_m(x_1,x_2,...,x_m)]$ is the m-tuple that maximizes the likelihood function ,then :

$\hat \theta=u_i(x_1, x_2,....,x_n)$

is the maximum likelihood estimator of  $\theta_i$ for i=1,2,..,m

(iii)The corresponding observed values in $[u_{1}(x_1,x_{2},..x_n),u_2(x_1,x_2, ...,x_n),...,u_m(x_1,x_2,...,x_m)]$ are the maximum likelihood estimates of $\theta_i$ for i=1,2,..,m