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

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample of size $n$ from the uniform distribution over the closed interval $[-\theta, \theta]$ having pdf $f(x ; \theta)=(1 / 2 \theta) I_{[-\theta, \theta]}(x) .$ (a) Show that $Y_{1}$ and $Y_{n}$ are joint sufficient statistics for $\theta$. (b) Argue that the mle of $\theta$ is $\hat{\theta}=\max \left(-Y_{1}, Y_{n}\right)$ (c) Demonstrate that the mle $\hat{\theta}$ is a sufficient statistic for $\theta$ and thus is a minimal sufficient statistic for $\theta$.

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample of size $n$ from the uniform distribution over the closed interval $[-\theta, \theta]$ having pdf $f(x ; \theta)=(1 / 2 \theta) I_{[-\theta, \theta]}(x) .$
(a) Show that $Y_{1}$ and $Y_{n}$ are joint sufficient statistics for $\theta$.
(b) Argue that the mle of $\theta$ is $\hat{\theta}=\max \left(-Y_{1}, Y_{n}\right)$
(c) Demonstrate that the mle $\hat{\theta}$ is a sufficient statistic for $\theta$ and thus is a minimal sufficient statistic for $\theta$.

Key Concepts

Minimal Sufficient Statistic

A minimal sufficient statistic is the most reduced form of a sufficient statistic, meaning it is a function of any other sufficient statistic. It captures the full information about the parameter with the least possible dimension or complexity. Finding a minimal sufficient statistic is valuable because it simplifies the analysis and can lead to more efficient inference procedures.

Maximum Likelihood Estimation (MLE)

Maximum likelihood estimation is a method used to estimate the parameters of a statistical model by maximizing the likelihood function, which measures how probable the observed data is for different parameter values. MLEs are popular because, under regularity conditions, they have desirable properties such as consistency, efficiency, and asymptotic normality.

Sufficient Statistic

A sufficient statistic is a function of the sample data that contains all the information needed to estimate a parameter. The factorization theorem is a fundamental tool used to verify sufficiency; it states that if the joint probability density (or mass) function can be factored into a product where one factor depends on the data only through the statistic and the other factor does not depend on the parameter, then the statistic is sufficient.

Order Statistics

Order statistics refer to the sorted values from a random sample. They are important because they capture the extremal (minimum and maximum) as well as intermediate values of the data, which often provide key information about the underlying distribution. In many problems, including estimation and inference, they simplify analyses and can lead to optimal or sufficient statistics due to their structured properties.

Please give Ace some feedback