A family of pdfs or pmfs \{g(t|\theta):\theta\in\Theta\} for a univariate random variable T with real-valued parameter \theta has a monotone likelihood ratio (MLR) if, for every \theta_2>\theta_1, g(t|\theta_2)/g(t|\theta_1) is a monotone (nonincreasing or nondecreasing) function of t on \{t:g(t|\theta_1)>0\;\text{or}\;g(t|\theta_2)>0\}. Note that c/0 is defined as \infty if 0< c. Consider testing H_0:\theta\leq \theta_0 versus H_1:\theta>\theta_0. Suppose that T is a sufficient statistic for \theta and the family of pdfs or pmfs \{g(t|\theta):\theta\in\Theta\} of T has an MLR. Then for any t_0, the test that rejects H_0 if and only if T >t_0 is a UMP level \alpha test, where \alpha=P_{\theta_0}(T >t_0). Example 1 To better understand the theorem, consider a single observation, X, from \mathrm{n}(\theta,1), and test the following hypotheses: $$ H_0:\theta\leq \theta_0\quad\mathrm{versus}\quad H_1:\theta>\theta...
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