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A family of pdfs or pmfs \${g(t|theta):thetainTheta}\$ 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:thetaleq 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):thetainTheta}\$ 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:thetaleq theta_0quadmathrm{versus}quad H_1:theta>theta_0. \$\$ Then \$theta_1>theta_0\$, and the likelihood ratio test statistics would be \$\$ lambda(x)=frac{f(x|theta_1)}{f(x|theta_0)}. \$\$ And we say that the null hypothesis is rejected if \$lambda(x)>k\$. To see if the distribution of the sample has MLR property, we simplify the above equation as follows: \$\$ begin{aligned} lambda(x)&=frac{frac{1}{sqrt{2pi}}expleft[-frac{(x-theta_1)^2}{2}right]}{frac{1}{sqrt{2pi}}expleft[-frac{(x-theta_0)^2}{2}right]}\ &=exp left[-frac{x^2-2xtheta_1+theta_1^2}{2}+frac{x^2-2xtheta_0+theta_0^2}{2}right]\ &=expleft[frac{2xtheta_1-theta_1^2-2xtheta_0+theta_0^2}{2}right]\ &=expleft[frac{2x(theta_1-theta_0)-(theta_1^2-theta_0^2)}{2}right]\ &=expleft[x(theta_1-theta_0)right]timesexpleft[-frac{theta_1^2-theta_0^2}{2}right] end{aligned} \$\$ which is increasing as a function of \$x\$, since \$theta_1>theta_0\$.  Figure 1. Normal Densities with \$mu=1,2\$.
By illustration, consider Figure 1. The plot of the likelihood ratio of these models is monotone increasing as seen in Figure 2, where rejecting \$H_0\$ if \$lambda(x)>k\$ is equivalent to rejecting it if \$Tgeq t_0\$.  Figure 2. Likelihood Ratio of the Normal Densities.
And by factorization theorem the likelihood ratio test statistic can be written as a function of the sufficient statistics since the term, \$h(x)\$ will be cancelled out. That is, \$\$ lambda(t)=frac{g(t|theta_1)}{g(t|theta_0)}. \$\$ And by Karlin-Rubin theorem, the rejection region \$R={t:t>t_0}\$ is a uniformly most powerful level-\$alpha\$ test. Where \$t_0\$ satisfies the following: \$\$ begin{aligned} mathrm{P}(T>t_0|theta_0)&=mathrm{P}(Tin R|theta_0)\ alpha&=1-mathrm{P}(Xleq t_0|theta_0)\ 1-alpha&=int_{-infty}^{t_0}frac{1}{sqrt{2pi}}expleft[-frac{(x-theta_0)^2}{2}right]operatorname{d}x end{aligned} \$\$ Hence the quantile of the \$1-alpha\$ probability, which is \$z_{alpha}\$ is equal to \$t_0\$, that is \$z_{alpha}=t_0\$, and thus we reject \$H_0\$ if \$T>z_{alpha}\$.

Example 2
Now consider testing the hypotheses, \$H_0:thetageq theta_0\$ versus \$H_1:theta< theta_0\$ using the sample \$X\$ (single observation) from Beta(\$theta\$, 2), and to be more specific let \$theta_0=4\$ and \$theta_1=3\$. Can we apply Karlin-Rubin? Of course! Visually, we have something like in Figure 3.  Figure 3. Beta Densities Under Different Parameters.
Note that for this test, \$theta_1 k\$ if and only if \$T < t_0\$. Where \$t_0\$ satisfies the following equations: \$\$ begin{aligned} mathrm{P}(T < t_0|theta_0)&=mathrm{P}(X < t_0|theta_0)\ alpha&=int_{0}^{t_0}frac{Gamma(theta_0+2)}{Gamma(theta_0)Gamma(2)}x^{theta_0-1}(1-x)^{2-1}operatorname{d}x\ alpha&=int_{0}^{t_0}frac{Gamma(6)}{Gamma(4)Gamma(2)}x^{3}(1-x)operatorname{d}x. end{aligned} \$\$ Hence the quantile of the \$alpha\$ probability, \$x_{alpha}=t_0\$. And thus we reject \$H_0\$ if \$T < x_{alpha}\$.  Figure 4. Likelihood Ratio of the Beta Densities.

### Reference

1. Casella, G. and Berger, R.L. (2001). Statistical Inference. Thomson Learning, Inc.