91Ó°ÊÓ

A random variableis a factor with an undetermined number or a program that gives numbers to each study's results.

The first step of estimating parameters\(\theta \) using the maximum is the estimator.

M.L.E:

Let random variables\({x_1},{x_2},...,{x_n}\) have joint pdf or pmb

\(f\left( {{x_1},{x_2},...,{x_n};{\theta _1},{\theta _2},...,{\theta _m}} \right)\)

Let random variables have joint pdf or pmb

where the parameters \({\theta _{i,}}i = 1,2,...,m\) are unknown. When function f is a function of parameters\({\theta _{i,}}i = 1,2,...,m\) , it is called the likelihood function. Values\(\widehat {{\theta _i}}\) that maximize the likelihood function are the maximum likelihood estimates or equally values \(\widehat {{\theta _i}}\)for

\(f\left( {{x_1},{x_2},...,{x_n};\widehat {{\theta _1}},{{\widehat \theta }_2},...,{\theta _m}} \right) \ge f\left( {{x_1},{x_2},...,{x_n};{\theta _1},{\theta _2},...,{\theta _m}} \right)\)

For every\({\theta _{i,}}i = 1,2,...,m\) . By substituting\({X_i}\,\,\) with\({x_i}\) , the maximum likelihood estimators are obtained.

The sampling is from the Exponential Distribution function is

\(f\left( {{x_1},{x_2},...,{x_n};\beta } \right) = {\beta ^n}\,{e^{\beta y}},\)

\(y = \sum\limits_{i = 1}^n {\,{x_i}} \)

By maximizing the log of the likelihood function, the maximum likelihood estimates are more accessible to

\(\begin{aligned}{l}L\left( \beta \right) &= \log \,ff\left( {{x_1},{x_2},...,{x_n};\alpha ,\beta } \right)\\ &= n\,\log \beta - \beta y\end{aligned}\)

From which, by finding the partial derivative and equating it with zero, the maximum is obtained.

\(\frac{\partial }{{\partial \beta }}L\left( \beta \right) = n\frac{1}{\beta } - y\)

which yields

\(\widehat \beta = \frac{n}{y} = \frac{1}{{{{\overline x }_n}}}\)

The M.L. estimator

\(\widehat \beta = \frac{1}{{{{\overline x }_n}}} = \widehat {\theta .}\)

The distribution \(\widehat F\)in the bootstrap would be the exponential distribution with parameter \(1/\overline X \). To use bootstrap, let the simulation size be from the distribution\(\widehat F\) .

The variance of the sample average V of simulations would be the bootstrap estimate of the variance of \(\overline X \) .

The variance of a single observation from the distribution\(\widehat F\) is the variance of\({\overline X ^2}\) the sample average of V simulations is \(1/n\) times larger, and thus the bootstrap estimate is

\(\frac{{{{\overline X }^2}}}{n}\)

Hence,

\(\frac{{{{\overline X }^2}}}{n}\)