Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a distribution that is \(N(0, \theta)\). Then \(Y=\sum X_{i}^{2}\) is a complete sufficient statistic for \(\theta\). Find the MVUE of \(\theta^{2}\).

Short answer

The Minimum Variance Unbiased Estimator (MVUE) for \(\theta^{2}\) is \(\theta^{2}Y / n\).

Step-by-step solution

Understand the given conditions

We are given \(X_{1}, X_{2}, \ldots, X_{n}\) as random samples from a normally distributed population where mean is 0 and variance is \(\theta\). Each of these random variables follows N(0,\(\theta\)). Also, we know \(Y=\sum X_{i}^{2}\) is a complete and sufficient statistic for \(\theta\), implying it contains all the information about the parameter \(\theta\) that is contained in the sample.

Form the Unbiased Estimator

We need to find an unbiased estimator of \(\theta^{2}\). By quadratic property of normal distribution, the sum of squares of independently and identically distributed sample from a normal distribution follows a chi-square distribution i.e., \(Y = X_{1}^{2} + X_{2}^{2} + ... + X_{n}^{2} = \chi^{2}(n)\). Here, the degrees of freedom are equal to the sample size 'n'.

Calculate the estimate

Let's denote the estimator of \(\theta^{2}\) as g(Y). We don't know g yet, but assuming it's a linear function of Y, we can denote it as g(Y) = aY. Now since we know that Y is chi-square distributed, we know that E(Y) = n. We can use this to find a. For g(Y) to be an unbiased estimator, we need E(g(Y)) = \(\theta^{2}\). Given g(Y) = aY and E(Y) = n, we equate a*n to \(\theta^{2}\) to get a = \(\theta^{2} / n\). Therefore, the unbiased estimator g(Y) = \(\theta^{2}Y / n\).

Verify Completeness

For g(Y) to be MVUE, it not only needs to be unbiased but also complete. A statistic T(Y) is said to be complete for the parameter θ if for every function g: E(g(T(Y))) = 0 implies that P(g(T(Y))) = 0. As Y is said to be a complete statistic and g is a function of Y, the function g(Y) is complete.

Conclusion

Based on the steps performed, we can ascertain that the estimator \(g(Y) = \(\theta^{2}Y / n\) is the Minimum Variance Unbiased Estimator (MVUE) for \(\theta^{2}\).

Key concepts

Statistical Inference

Statistical inference is a critical concept in statistics that involves making judgments about a population based on a sample. The essence is to draw conclusions from data subject to random variation. For example, if you want to understand the average height of a tree species, you wouldn't measure every tree in the world. Instead, you’d sample a few trees and make an inference about the population from this sample.

There are two main types of statistical inferences: estimation and hypothesis testing. Estimation involves using sample data to estimate a population parameter. In our exercise, the parameter of interest is \(\theta^2\), which is related to the variance of a normal distribution. To infer this parameter, we use an estimator derived from the sample.

An unbiased estimator means our estimates, on average, hit the true population parameter. The goal is often to find not just any unbiased estimator, but the best one in terms of having minimum variance among all unbiased estimators. This is where the concept of Minimum Variance Unbiased Estimator (MVUE) comes into play.

Chi-square Distribution

The chi-square distribution is an essential concept in statistics, especially in the context of variance and hypothesis testing. It is a continuous probability distribution that is used primarily in goodness-of-fit testing and in the confidence intervals estimation for a population standard deviation of a normal distribution when the mean is known.

In our exercise, we encounter the chi-square distribution through the sum of squared standard normal variables. If \(X_1, X_2, \ldots, X_n\) are independent random samples from a normal distribution with a mean of 0 and variance \(\theta\), then \(Y = \sum X_{i}^{2}\) follows a chi-square distribution with 'n' degrees of freedom, denoted as \(\chi^{2}(n)\).

This characteristic of the chi-square distribution is crucial because it provides a foundation to calculate the unbiased estimator for \(\theta^2\). The expected value of a chi-squared variable is its degrees of freedom, which in the case of our problem is 'n'. This connection allows us to formulate an unbiased estimator for \(\theta^2\) by equating the expected value of the proposed estimator to the desired parameter.

Sufficient Statistics

Sufficient statistics are a set of descriptive measures that capture all available information about a parameter of interest from the data. The formal definition states that a statistic is sufficient for a parameter if no other statistic that can be calculated from the same sample provides any additional information about the parameter.

In the context of our exercise, we dealt with the sum of the squares of the sample, \(Y = \sum X_{i}^{2}\), which is labeled as a sufficient statistic for the parameter \(\theta\). This means that \(Y\) contains all the information needed to estimate \(\theta\); no additional sample data or statistics would improve the estimate of \(\theta\) if we already know \(Y\).

Moreover, the concept of sufficiency is ground-breaking because it can significantly reduce the complexity of statistical analysis. Instead of dealing with the entire sample, we can focus on just this one summary statistic, making subsequent analyses like finding the MVUE far more straightforward and computationally efficient.