91Ó°ÊÓ

Random variables \({X_{1,}}...{X_n}\) form a random sample of size n.it deals with the study of the sufficient statistic of the known parameter \(\theta \) from the distribution provided.the distribution of study is \(Unif\left( {1,2,3...\theta } \right)\)

The unknown parameters can take the only integer values, that is\(\theta \in \left\{ {1,2,3...} \right\}\)

The pdf of the given variable is\({f_x}\left( x \right) = \frac{1}{\theta },\;1 \le {X_i} \le \theta \)

Therefore the joint pdf is

\({f_{{x_1},...{x_n}}}\left( {{x_1},{x_2}...{x_n}} \right) = {\left( {\frac{1}{\theta }} \right)^n}I\left( {1,{X_{\left( 1 \right)}}} \right)I\left( {{X_{\left( n \right),}}\theta } \right)\)

Where

\(\begin{array}{l}{X_{\left( 1 \right)}} = \min \left\{ {{X_1},...{X_n}} \right\}\\{X_{\left( n \right)}} = \max \left\{ {{X_{1,}}...{X_n}} \right\}\end{array}\)

The sufficient statistic using NFFT, the following pdf can be factorised as:

\(\begin{array}{c}{f_{{x_1},...{x_n}}}\left( {{x_1},{x_2}...{x_n}} \right) = {\left( {\frac{1}{\theta }} \right)^n}I\left( {1,{X_{\left( 1 \right)}}} \right)I\left( {{X_{\left( n \right),}}\theta } \right)\\ = f\left( \theta \right)h\left( x \right)g\left( {\theta ,T\left( X \right)} \right)\end{array}\)

Where,

\(\begin{array}{c}f\left( \theta \right) = {\left( {\frac{1}{\theta }} \right)^n}\\h\left( x \right) = I\left( {1,{X_{\left( 1 \right)}}} \right)\\g\left( {\theta ,T\left( X \right)} \right) = I\left( {{X_{\left( n \right),}}\theta } \right)\end{array}\)

The above factorization shows that\({\left( {\frac{1}{\theta }} \right)^n}\)is the function of\(\theta \)only.

\(I\left( {1,{X_{\left( 1 \right)}}} \right)\)is the function of X only, while\(I\left( {{X_{\left( n \right),}}\theta } \right)\)is the only term which has the term of\(\theta \), through the sample\({X_{\left( n \right)}}\)only.

Therefore the function of the sample points which is linked with\(\theta \)is the lasrgest sample point or\({X_{\left( n \right)}}\).

Thus the sufficient statistic for unrestricted case is\({X_{\left( n \right)}}\),

Now since the parameter space of \(\theta \)is given in the question is discrete integer point and the distribution is uniform. Hence the sufficient statistic has also to be discrete integer point, which can be obtained as \(\hat \theta = round\left( {{X_{\left( n \right)}}} \right)\)