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*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 $0< c$.

**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

Figure 4. Likelihood Ratio of the Beta Densities. |

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