91Ó°ÊÓ

An unbiased estimator is a statistical measure used to estimate a population parameter, with the distinguishing feature that its expected value is equal to the true parameter value. This property ensures that across many samples, the estimator accurately targets the parameter. In other words, it avoids systematic errors. For example, in the exercise, we have estimates for variance: \(s^2\) and \(\hat{\sigma}^2\). To determine which is unbiased, we need to check if the expected value matches the true variance \(\sigma^2\). Since we know from the properties of the chi-square distribution that \(E(s^2) = \sigma^2\), \(s^2\) is unbiased. Meanwhile, \(\hat{\sigma}^2\) is less than \(\sigma^2\) on expectation, thus it is biased as \(E(\hat{\sigma}^2) eq \sigma^2\). This distinction is crucial in statistical analysis as unbiased estimators, like \(s^2\), provide more reliable estimates than biased ones.

The Mean Squared Error (MSE) is a key metric that combines both the variance of an estimator and its bias, squared. This serves as a comprehensive measure of an estimator's quality. Mathematically, it is expressed as \(MSE(\theta) = \text{Var}(\theta) + (\text{Bias}(\theta))^2\). An estimator with a lower MSE is generally preferred, as it indicates that the estimator has less error overall.

• For \(s^2\), the MSE is simplified to its variance because \(s^2\) is unbiased, meaning its bias is zero.
• However, \(\hat{\sigma}^2\) is not unbiased, as it is biased by \(\frac{-1}{n}\sigma^2\). Thus, the MSE of \(\hat{\sigma}^2\) includes both its variance and the square of its bias.
• When comparing \(s^2\) and \(\hat{\sigma}^2\), \(\hat{\sigma}^2\) has a smaller MSE, even though it is biased, because its variance is notably reduced compared to \(s^2\), making it preferable for finite sample sizes.

The chi-square distribution is a fundamental probability distribution in statistics, often arising in scenarios involving variance estimation. It is defined by the sum of the squares of independent standard normal variables and is characterized by its degrees of freedom, often denoted as \(r\). The chi-square distribution has important applications, particularly in hypothesis testing and estimating confidence intervals.

• In our given problem, the variance estimator \(\frac{(n-1)s^2}{\sigma^2}\) is distributed as \(\chi_{n-1}^2\), which tells us that the form of \(s^2\) directly aligns with the properties of the chi-square distribution.
• The expected value and variance of a chi-square-distributed random variable are \(r\) and \(2r\), respectively.
• This property helps in constructing unbiased estimators for variance, as seen with \(s^2\), allowing it to precisely estimate the variance under the chi-square framework.

Overall, understanding the chi-square distribution enables statisticians to derive meaningful estimators for variance and execute effective statistical tests.