Question

Suppose that \({X_1},.....,{X_n}\)form a random sample from the \({\chi ^2}\) distribution with unknown degrees of freedom\(\theta \)\((\theta = 1,2,...)\), and it is desired to test the following hypotheses at a given level of significance\({\alpha _0}\left( {0 < {\alpha _0} < 1} \right)\):

\(\begin{array}{l}{H_0}:\;\;\;\theta \le 8,\\{H_1}:\;\;\;\theta \ge 9.\end{array}\)

Show that there exists a UMP test, and the test specifies rejecting \({H_0}\)if \(\sum\limits_{i = 1}^n {log} {X_i} \ge k\) for some appropriate constant\(k\).

Short answer

Proved and showing that the family has an MLR property in\(_{i = 1}^n\log {X_i}\)

Step-by-step solution

To Show that the UMP testat \({\alpha _0}\left( {0 < {\alpha _0} < 1} \right)\)specifies rejecting\({H_0}\)if\(\sum\limits_{i = 1}^n {log} {X_i} \ge k\)

Take\({\theta _1} > {\theta _2}\). Then the likelihood ratio takes the value

\(\frac{{f\left( {{\bf{x}}\mid {\theta _1}} \right)}}{{f\left( {{\bf{x}}\mid {\theta _2}} \right)}} = {\left| {\frac{{{2^{\frac{{{\theta _2}}}{2}}}\Gamma \frac{{{\theta _2}}}{2}}}{{{2^{\frac{{{\theta _1}}}{2}}}\Gamma \frac{{{\theta _1}}}{2}}}} \right|^n}\)\({_{i = 1}^n}{x_i}^{\frac{{{\theta _1} - {\theta _2}}}{2}}\)

Which is an increasing function of\(_{i = 1}^n{X_i}\), or equivalently of\(T = _{i = 1}^n\log {X_i}\).

Thus the chi squared family has an MLR property in\(T\). Hence we reject the null for large values of\(T\), where the cutoff for being large is determined by the level condition. Thus, \(\exists k\)depending on level \({\alpha _0}\) such that we reject the null when\(T \ge k\).

Follows by showing that the family has an MLR property in\(_{i = 1}^n\log {X_i}\)