91Ó°ÊÓ

Gibbs Samplingis a Monte Carlo Markov Chain method for estimating complex joint distributions that draw an instance from the distribution of each variable iteratively based on the current values of the other variables.

\(\left( {1.} \right)\,\)Pick starting values \({x_2}^{\left( 0 \right)}\) for \(\,{x_2}\) , and let \(\,\,i = 0\,\)

\(\left( {2.} \right)\,\)let be simulated values from the conditional distribution of \(\,{x_1}\)given that \(\,\,{X_1} = {x_2}^{\left( i \right)}\)

\(\left( {3.} \right)\,\)Let\(\,{x_2}^{\left( {i + 1} \right)\,\,}\,\) be a simulated value from the conditional distribution of \(\,{x_2}\) given that \(\,{X_1} = {x_1}^{\left( {i + 1} \right)}\)

\(\left( {4.} \right)\)Repeat steps \(\,2.\,\,and\,3.\) \(\,i\) where\(\,i + 1\)

There is a total of \(n + m\) random variables from the exponential distribution with parameter \(\theta \) , from which the first n can be observed. For the censored m, random variables\(\,{X_n} + i \le c,i = 1,2,...,m\) can be observed. It is true that

\(\Pr \left( {{X_n} + i \le c} \right) = 1{ - ^{ - c\theta }}\)

Because \(\,{X_n} + i\) they are the exponential distribution with parameters.

Let be \({x_1},{x_2},{x_n}\)i.i.d. random variables from an exponential distribution with an unknown parameter \(\theta > 0\). If the prior distribution \(\theta \)is gamma distribution with parameters \(\alpha ,{\rm{ }}\beta > 0\), then the posterior distribution of \(\theta \)the given is\({X_1} = {x_1},{X_2} = {x_2},{X_n} = {x_n}\) gamma distribution where

\({\alpha _1} = \alpha + n\)

And

\({\beta _1} = \beta + \sum\limits_{i = 1}^n {{x_i}.} \)

Next, the posterior distribution is approximate to the likelihood function times the prior distribution, which is

\({\xi _{nm}} = {\theta ^{n + \alpha - 1}}{\left( {1 - {e^{ - c\theta }}} \right)^m}\,{e^{ - \theta \,\sum\nolimits_{i = 1}^n {{x_i}} }}\)

For\(\theta < x,c\) , it follows that the conditional distribution of\({x_{n + i\,\,}}\)given\({x_{n + i\,\,}} \le c\) is obtained as the following quotient of two p.d.f.'s

\({f_1}\,\left( {x\mid \theta ,{x_{n + i}} \le c} \right) = \frac{{\theta {e^{ - \theta x}}}}{{1 - {e^{ - \theta c}}}},\)

Using this p.d.f. in the likelihood function, the product of the likelihood function times the prior distribution \(\theta > 0,0 < {x_i} < c,i = 1,2,...,n + m\,\) becomes

\(\theta {e^{n + \alpha - 1}}{\left( {1 - {e^{ - c\theta }}} \right)^m}\,{e^{ - \theta }}\sum\nolimits_{i = n}^{n + m} {{x_i}} \)

This is the p.d.f. of the gamma distribution with parameters

\(\begin{aligned}{l}{\alpha _g} = n + m + \alpha \\{\beta _g}\sum\nolimits_{i = n}^{n + m} {{x_i}} \end{aligned}\)

as a function of\(\theta \) , and it looks like the p.d.f\({f_\alpha }\left( {x\mid \theta \mid {x_{n + i\,\,}} \le c} \right)\)

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

There are a total of \(n + m\)random variables from the exponential distribution with parameter, from which the first n can be observed. For the censored m, random variables\(\,{X_n} + i \le c,i = 1,2,...,m\) can be observed. It is true that

Because \(\,{X_n} + i\) they are from the exponential distribution with parameter \(\theta ,i = 1,2,...,m\).

Next, the posterior distribution is approximate to the likelihood function times the prior distribution, which is

\({\xi _{nm}} = {\theta ^{n + \alpha - 1}}{\left( {1 - {e^{ - c\theta }}} \right)^m}\,{e^{ - \theta \,\sum\nolimits_{i = 1}^n {{x_i}} }}\)

For\(\theta < x,c\) , it follows that the conditional distribution of \(\,{X_n} + i\) given \({x_{n + i\,\,}} \ge c\)is obtained as the following quotient of two p.d.f.'s

\({f_1}\,\left( {x\mid \theta ,{x_{n + i}} \ge c} \right) = \frac{{\theta {e^{ - \theta x}}}}{{1 - {e^{ - \theta c}}}},\)

Or equally

\({\xi _{nm}} = {\theta ^{n + }}{m^{ + \alpha - 1}}\,{e^{ - \theta \,\sum\nolimits_{i = 1}^{n + m} {{x_i}} }}\)

This is the p.d.f. of the gamma distribution with parameters

\(\begin{aligned}{l}{\alpha _g} = n + m + \alpha \\{\beta _g} = \sum\limits_{i = 1}^{n + m} {{x_i}} ,\end{aligned}\)

As a function \(\theta \)of, and it looks like the p.d.f. \({f_\alpha }\left( {x\mid \theta \mid {x_{n + i\,\,}} \ge c} \right)\)

The Gibbs Sampling algorithm starts by picking a starting value for\(\theta \) , e.g., M.L.E., and then simulate observations from the p.d.f. \({f_1}\,\). The quantile function \(0 < q < 1\)is obtained from

\(q = {\theta ^m}\,{e^{ - \theta \left( {x - c} \right)}}\)

which yields

\({F^{ - 1}}\left( q \right) = c - \frac{{\log \left( {1 - q} \right)}}{\theta }\)

Use the quantile function to simulate observations from as

\(x = c - \frac{{\log \left( {1 - q} \right)}}{\theta }\)

U is a random variable from the uniform distribution on interval\(\left( {0,1} \right)\).

Finally, simulate a new value of \(\theta \) with the given parameters above.

There is an easier way for simulation without using the Gibbs Sampling algorithm

Hence,

\(\begin{aligned}{}\left( {\left( a \right)} \right){\rm{ }}Find{\rm{ }}Pr\left( {Xn + i{\rm{ }} \le {\rm{ }}C} \right);\\\left( {\left( b \right)} \right){\rm{ Find}}\,\,\,\,\,Pr(Xn + i{\rm{ }} \ge {\rm{ }}C);\end{aligned}\)