# Parametric Inference: Karlin-Rubin Theorem

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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: $$ H_0:\theta\leq \theta_0\quad\mathrm{versus}\quad H_1:\theta>\theta_0. $$ 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: $$ \begin{aligned} \lambda(x)&=\frac{\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{(x-\theta_1)^2}{2}\right]}{\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{(x-\theta_0)^2}{2}\right]}\\ &=\exp \left[-\frac{x^2-2x\theta_1+\theta_1^2}{2}+\frac{x^2-2x\theta_0+\theta_0^2}{2}\right]\\ &=\exp\left[\frac{2x\theta_1-\theta_1^2-2x\theta_0+\theta_0^2}{2}\right]\\ &=\exp\left[\frac{2x(\theta_1-\theta_0)-(\theta_1^2-\theta_0^2)}{2}\right]\\ &=\exp\left[x(\theta_1-\theta_0)\right]\times\exp\left[-\frac{\theta_1^2-\theta_0^2}{2}\right] \end{aligned} $$ which is increasing as a function of $x$, since $\theta_1>\theta_0$.

Figure 1. Normal Densities with $\mu=1,2$. |

Figure 2. Likelihood Ratio of the Normal Densities. |

**Example 2**

Now consider testing the hypotheses, $H_0:\theta\geq \theta_0$ 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. |

Figure 4. Likelihood Ratio of the Beta Densities. |

### Reference

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