91Ó°ÊÓ

\({X_1},...,{X_n}\) form a random sample from an exponential distribution for which the value of the parameter β is unknown. We need to determine if , the M.L.E. of β a minimal sufficient statistic.

A vector \(T = \left( {{T_1},...,{T_k}} \right)\) of statistics are minimal jointly sufficient statistics if the co-ordinates are jointly sufficient statistics and T is a function of every other jointly sufficient statistics.

The pdf of the exponential distribution is given by

\(f\left( {x|\beta } \right) = \beta {e^{ - \beta x}}\,\,\,\,\,\,\,\,\,\,\,\,\,x > 0\)

The likelihood function is given by

\(f\left( {x;\theta } \right) = {\theta ^n}{e^{ - \theta p}}\)where \(p = \sum\limits_{i = 1}^n {{x_i}} \)

Taking logarithm we get \(L\left( \theta \right) = \log f\left( {x;\theta } \right)\)

Now taking derivative and equating with zero we get

\(\frac{{\partial L\left( \theta \right)}}{{\partial \theta }} = \frac{n}{\theta } - p\).This implies \(\hat \theta = \beta = \frac{n}{p}\)

Since the M.L.E is also a sufficient statistic then it is also a minimal sufficient statistic.