Question

let x1, . . . , xn denote an iid random sample from a population with mean µ and unknown variance σ 2 . create an unbiased estimator for µ 2 .

Answer 1

To develop an unbiased estimator for µ², one should apply the Central Limit Theorem and account for the distribution's variance and the properties of the sample, which may require advanced statistical techniques beyond simply squaring the sample mean.

Step-by-step explanation:

To create an unbiased estimator for µ², we need to focus on sample means with the use of the Central Limit Theorem. An unbiased estimator assumes the expected value of the estimator matches the true value of the parameter. For a sample with a mean x, using the sample mean x squared as an estimator for µ² may not be unbiased due to the unknown variance σ². Instead, a correction factor is generally needed. The steps would involve utilizing the sample means to derive this estimator, applying properties of expectation, and possibly leveraging the concept of moments or moment-generating functions to address the bias.

The process would not be straightforward as collecting terms and squaring the sample mean x. However, the general notion is that by accounting for the distribution's variance and the properties of the sample, one can derive an expression that estimates µ² without bias. This could entail more advanced statistical techniques and requires a good understanding of statistical estimation theory.