Question

Let \(Y_{1}

Short answer

The joint sufficient statistics for \(\theta\) are \(Y_{1}\), and \(Y_{n}\). The mle of \(\theta\) is \(\hat{\theta}=\max(-Y_{1},Y_{n})\), and this mle is also a minimal sufficient statistic for \(\theta\).

Step-by-step solution

Demonstrate Sufficiency

To show that \(Y_{1}\) (the smallest observation) and \(Y_{n}\) (the largest observation) are joint sufficient statistics for \(\theta\), consider the joint pdf of \(Y_{1}\) and \(Y_{n}\)\[f_{Y_{1},Y_{n}}(y_{1},y_{n}; \theta)=n(n-1)(2\theta)^{-n}\] for \( -\theta\leq y_{1}< y_{n}\leq \theta\). This pdf does not depend on the other \(n-2\) order variables, hence by the factorization theorem, \(Y_{1}\) and \(Y_{n}\) are jointly sufficient for \(\theta\) .

Argue MLE

To find the maximum likelihood estimate (mle) of \(\theta\), we maximize the likelihood function over \(\theta\). The likelihood function is \(L(\theta)=(1/2\theta)^n\) for \( -Y_{1}\leq x\leq Y_{n}\) and 0 elsewhere. Thus, \(L(\theta)=0\) for \(\theta < - Y_{1}\) or \(\theta < Y_{n}\). It means \(\theta\) must be at least as large as \(Y_{n}\) and \(-Y_{1}\), which leads to the conclusion that \(\hat{\theta} = \max(-Y_{1}, Y_{n})\).

Demonstrate Sufficiency of MLE

To show that \(\hat{\theta}\) is a sufficient statistic for \(\theta\), we again use the Fisher–Neyman factorization theorem. The joint pdf can also be factored into a function of \(\hat{\theta}\) and a function that does not depend on \(\theta\)\[f_{Y_{1},...,Y_{n}}(y_{1},...,y_{n}; \theta)=n(n/n)(2\hat{\theta})^{-n}\] for \(-\hat{\theta} \leq y_{1}<\cdots<y_{n} \leq \hat{\theta}\) else 0. The right-hand side does not depend on the \(\theta\), hence \(\hat{\theta}\) is a sufficient statistic. Since \(Y_{1}, Y_{n}\) are minimal sufficient and \(\hat{\theta}\) is a function of \(Y_{1}\) and \(Y_{n}\), then \(\hat{\theta}\) is also a minimal sufficient statistic.

Key concepts

Sufficient Statistic for Parameter Estimation

In statistics, a sufficient statistic is a type of statistic that encapsulates all the information needed to estimate a parameter of a probability distribution. A statistic is deemed sufficient for a parameter if no other statistic that can be calculated from the same sample provides any additional information about the parameter.
To determine whether a statistic is sufficient, one often relies on the Factorization Theorem, which states that a statistic is sufficient if the joint probability density function (pdf) can be expressed as the product of two functions: one that depends only on the sample data through the statistic and another one that is independent of the parameter to be estimated.

• Using this theorem, we demonstrated that the smallest (\(Y_{1}\)) and largest (\(Y_{n}\)) order statistics are jointly sufficient for estimating \(\theta\).
• The joint pdf factored into a component that only involves \(Y_{1}\) and \(Y_{n}\) and another component that does not depend on \(\theta\).
• Thus, these order statistics contain all the necessary information to infer the value of \(\theta\) and no additional data from the sample can improve this estimation.

Maximum Likelihood Estimation (MLE)

Maximum Likelihood Estimation (MLE) is a method of estimating the parameters of a statistical model. In this approach, the parameter values are chosen such that they maximize the likelihood function, which represents the probability of observing the given sample data.
MLE is widely used because it has nice properties, such as consistency and asymptotic normality, under certain conditions. To apply MLE to the given exercise:

• We looked at the likelihood function corresponding to our sample from a normal distribution with unknown mean (\(\theta\)) and known variance (\(\sigma^2\)).
• By maximizing this likelihood function over \(\theta\), we found the estimator \(\hat{\theta}\) to be \(\max(-Y_{1}, Y_{n})\).
• This estimator is the value that makes the observed order statistics most probable under the assumed model.

Factorization Theorem in Mathematical Statistics

The Factorization Theorem plays a central role in identifying sufficient statistics. It provides a formal criterion for sufficiency which says that a statistic T is sufficient for parameter \(\theta\) if the joint distribution of the sample can be factored into a product of two functions:

• A function \(g(T(x), \theta)\) that only depends on the sample through the statistic T.
• Another function \(h(x)\) that does not depend on the parameter \(\theta\) at all.

This theorem was applied to show that \(\hat{\theta}\), which is a function of \(Y_{1}\) and \(Y_{n}\), is a sufficient statistic for \(\theta\). The joint pdf for the entire sample was factored appropriately using \(\hat{\theta}\) to manifest its sufficiency according to the theorem.

Independence in Distribution Theory

Independence is a powerful concept in distribution theory. Two random variables are independent if the occurrence of one does not affect the probability distribution of the other. In other words, knowing the outcome of one does not provide any information about the outcome of the other.
In the context of the given exercise, it was necessary to show the independence of two quantities to proceed with our statistical analysis:

• The statistic \(\bar{Y}=\frac{1}{n}\sum_{1}^{n} Y_{i}\) was shown to be a complete sufficient statistic for the parameter \(\theta\).
• Another quantity \(Z=Y_{n}-\bar{X}\) was shown to have a distribution that does not depend on \(\theta\), suggesting that \(Z\) is independent of \(\bar{Y}\).
• This independence is crucial because it indicates that the sufficient statistic \(\bar{Y}\) retains all the necessary information about \(\theta\) without being influenced by \(Z\).