Question

We consider a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) from a distribution with pdf \(f(x ; \theta)=(1 / \theta) \exp (-x / \theta), 0

Short answer

The joint pdf of the order statistics is \(L(\theta) = \frac{n!}{(r!(n - r)!)} \frac{\exp \left(- \sum_{i=1}^{r} Y_i / \theta\right)}{\theta^n} (Y_r^{n - r} - Y_{r-1}^{n - r})\), the MLE \(\hat{\theta}\) is \(\frac{\sum_{i=1}^{r} Y_i }{n}\), the mgf of \(\hat{\theta}\) is \(\left( 1 - \frac{t}{n} \right)^{-n}\) for \(t < n\), the pdf of \(\hat{\theta}\) is \(\frac{(n \theta)^{n-1} e^{-n \theta}}{(n - 1)!}\) for \(\theta > 0\), and \(\hat{\theta}\) is a sufficient statistic.

Step-by-step solution

Determine the Joint pdf of Order Statistics

The joint pdf of the order statistics \(Y_1, Y_2, ..., Y_r\) is given by \( L(\theta) = \frac{n!}{(r!(n - r)!)} \frac{\exp \left(- \sum_{i=1}^{r} Y_i / \theta\right)}{\theta^n} (Y_r^{n - r} - Y_{r-1}^{n - r}), \text{ where } 0< Y_1 < Y_2 < \cdots < Y_r < \infty. \)

Compute MLE by Maximizing Likelihood Function

The derivative of \(L(\theta)\) with respect to \(\theta\) is set to zero and solved for \(\theta\) to find the MLE. The solution \(\hat{\theta}\) is given by \(\hat{\theta} = \frac{\sum_{i=1}^{r} Y_i }{n}\).

Find the mgf and pdf of the MLE

The moment generating function (mgf) and pdf of a standard Gamma distribution are used, which are given by \( M_{\hat{\theta}}(t) = \left( 1 - \frac{t}{n} \right)^{-n} \) for \( t < n \) and \( f_{\hat{\theta}}(\theta) = \frac{(n \theta)^{n-1} e^{-n \theta}}{(n - 1)!} \) for \( \theta > 0 \).

Verify if the MLE is a Sufficient Statistic

Per the Factorization Theorem, a statistic is sufficient for parameter \(\theta\) if the joint pdf or pmf can be factored into a product of two factors: one that depends on \(\theta\) and \(Y_1, Y_2, \ldots, Y_r\) and one that only depends on \(Y_1, Y_2, \ldots, Y_r\). On applying the theorem, it is seen that this factorization is possible for the given pdf and hence, \(\hat{\theta}\) is a sufficient statistic.

Key concepts

Order Statistics

Order statistics are incredibly helpful when dealing with data samples. Imagine you have a random sample, and you want to know not just any data point but the smallest, second smallest, and so on. That's precisely what order statistics help with. They are essentially the sorted values from your sample. For example, if you observed values 2, 5, and 3, your order statistics would be 2, 3, and 5.
Order statistics are crucial when not all data points are available, say in life testing situations. Here, only a few of the smallest observations might be accessible. Compute metrics and analyses using only these values. This exercise asks you to record the joint probability density function (pdf) of the first few order statistics and utilize it to find the Maximum Likelihood Estimation (MLE).

• The joint pdf of order statistics takes into account the distribution of your data as well as the specific order of the sample points.
• In our problem, it's represented by a complex formula involving factorials and exponentials.
• This structure helps when deriving the MLE, ensuring we consider the unique traits of our ordered data.

Moment Generating Function

A moment generating function (mgf) is a powerful tool in probability theory. It encodes all the moments of a random variable. Moments are like the average, variance, and more, describing the shape and spread of your data.
For our problem, the mgf assists in understanding the distribution of the MLE, \( \hat{\theta} \). The mgf is derived from the Gamma distribution relevant to our MLE and helps in identifying its statistical properties. Here’s what you need to know about mgf:

• The mgf of \( \hat{\theta} \) simplifies to \( M_{\hat{\theta}}(t) = (1 - \frac{t}{n})^{-n} \) for \( t < n \).
• The formula shows how the generating function behaves as 't' varies, encapsulating the behaviour of all possible moments.
• This helps in probability analysis, making MLE-based decisions more robust by knowing the distribution it's drawn from.

Sufficient Statistic

A sufficient statistic is central to understanding and summarizing data in a way that maximizes information about a parameter. It captures all necessary information without redundant data.
In our scenario, the parameter of interest is \( \theta \). The statistic \( \hat{\theta} \) must encapsulate all we need to draw conclusions about \( \theta \). Here’s how sufficient statistics work:

• The Factorization Theorem plays a key role. It states that a function can be split into one dependent on the data and parameter, and another only on the data.
• If such factorization is possible, the statistic is sufficient.
• For \( \hat{\theta} \), it passes this factorization test, indicating it's a sufficient statistic for \( \theta \).

By ensuring all information about the parameter \( \theta \) is captured efficiently, sufficient statistics simplify analysis and can reduce complex data into a single informative value.