91Ó°ÊÓ

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\(_{i = 1}^n\log {X_i}\)