Let Y1<Y2<⋯<Yn be the order statistics of a random sample...

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample from a uniform distribution on $(0, \theta)$, where $\theta>0$. (a) Show that $\Lambda$ for testing $H_{0}: \theta=\theta_{0}$ against $H_{1}: \theta \neq \theta_{0}$ is $\Lambda=\left(Y_{n} / \theta_{0}\right)^{n}$, $Y_{n} \leq \theta_{0}$, and $\Lambda=0$ if $Y_{n}>\theta_{0}$ (b) When $H_{0}$ is true, show that $-2 \log \Lambda$ has an exact $\chi^{2}(2)$ distribution, not $\chi^{2}(1) .$ Note that the regularity conditions are not satisfied.

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample from a uniform distribution on $(0, \theta)$, where $\theta>0$.
(a) Show that $\Lambda$ for testing $H_{0}: \theta=\theta_{0}$ against $H_{1}: \theta \neq \theta_{0}$ is $\Lambda=\left(Y_{n} / \theta_{0}\right)^{n}$, $Y_{n} \leq \theta_{0}$, and $\Lambda=0$ if $Y_{n}>\theta_{0}$
(b) When $H_{0}$ is true, show that $-2 \log \Lambda$ has an exact $\chi^{2}(2)$ distribution, not $\chi^{2}(1) .$ Note that the regularity conditions are not satisfied.

Key Concepts

Likelihood Ratio Test (LRT)

The likelihood ratio test is a fundamental procedure in statistical hypothesis testing used to compare the fit of two nested models: one under the null hypothesis and one under the alternative. It involves the ratio of the maximum likelihood achieved under the null hypothesis to that under the alternative hypothesis. This test is particularly useful when the parameter of interest (in this case, the upper bound of the uniform distribution) appears in both the likelihoods, and when its distribution under the null hypothesis can be explicitly characterized or transformed into a standard distribution for inference.

Order Statistics in Uniform Distribution

Order statistics refer to the sorted values obtained from a random sample, and the maximum order statistic (often denoted as Y?) plays a key role in estimations and testing when sampling from a uniform distribution. For a sample from a uniform (0, ?) distribution, the maximum value is not only a natural estimator for ? (since it is the largest observed value), but its distribution is also well-understood and utilized in deriving tests and confidence intervals. Its distribution is pivotal when constructing test statistics like the likelihood ratio test in this context.

Transformation of Statistic and the Chi-Squared Distribution

In many hypothesis testing scenarios, especially those involving the likelihood ratio test, it is common to transform the test statistic (typically via a negative logarithm) such that its distribution converges to or can be exactly represented by a chi-squared distribution under the null hypothesis. Here, transforming the likelihood ratio statistic by taking -2 log ? leads to a result that follows an exact ?² distribution with 2 degrees of freedom, highlighting that the standard degrees-of-freedom results from regular cases may not hold under non-regular conditions.

Regularity Conditions in Likelihood Inference

Regularity conditions are technical assumptions underlying many asymptotic results in likelihood-based inference, such as those guaranteeing that the likelihood ratio statistic converges to a chi-squared distribution with a specific number of degrees of freedom. However, when these conditions are violated—as in the case of a parameter defining the support boundary of a distribution—standard results no longer apply directly. In this problem, the uniform distribution's parameter being an endpoint introduces non-standard behavior, which results in the likelihood ratio statistic following a ?²(2) distribution instead of the typical ?² with one degree of freedom.

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