The farm experts at Fanny Farmers Farm plan to lay out a...

The farm experts at Fanny Farmers Farm plan to lay out a square plot of land with side $\mu$ by using a long rod of length $\mu$ so the area of the plot is $\mu^2$. Unfortunately, the length of the rod is not known exactly, so $n$ independent measurements by $n$ independent farmers are taken, yielding the values $x_1, x_2, \ldots, x_n$. We assume that each $X_i$ has mean $\mu$ and variance $\sigma^2$. (a) Show that $(\bar{X})^2$ is not an unbiased estimator of $\mu^2$, the area of the field. (b) For what value of $k$ is the estimator $(\bar{X})^2-k \times S^2$, where $S^2$ is the sample variance, an unbiased estimator for $\mu^2$ ?

The farm experts at Fanny Farmers Farm plan to lay out a square plot of land with side $\mu$ by using a long rod of length $\mu$ so the area of the plot is $\mu^2$. Unfortunately, the length of the rod is not known exactly, so $n$ independent measurements by $n$ independent farmers are taken, yielding the values $x_1, x_2, \ldots, x_n$. We assume that each $X_i$ has mean $\mu$ and variance $\sigma^2$.
(a) Show that $(\bar{X})^2$ is not an unbiased estimator of $\mu^2$, the area of the field.
(b) For what value of $k$ is the estimator $(\bar{X})^2-k \times S^2$, where $S^2$ is the sample variance, an unbiased estimator for $\mu^2$ ?

Key Concepts

Bias Correction

Bias correction involves modifying an estimator by subtracting or adding a term so that its expectation exactly equals the parameter being estimated. In the problem, adjusting (X?)² by subtracting k times the sample variance S² is a method of bias correction. By determining the appropriate constant k, one can eliminate the additional bias introduced by the variance term and achieve an unbiased estimator for ?².

Unbiased Estimation

An estimator is unbiased if its expected value equals the true value of the parameter being estimated. In the context of this problem, the estimator (X?)² is evaluated to see whether its expectation is equal to ?², the true area. If the expectation deviates from ?² by a constant or term related to sample variance, the estimator is not considered unbiased.

Sample Mean

The sample mean (X?) is the arithmetic average of independent measurements and serves as an estimator for the population mean ?. Its properties, particularly its expectation being equal to ? and its variance being ?²/n, are fundamental when analyzing the bias in functions of the sample data, such as the square of the sample mean.

Sample Variance

The sample variance (S²) is an estimator for the population variance ?² based on the collected sample data. It provides a measure of dispersion among the individual measurements and is used to adjust for the bias in derived estimators. Its role becomes critical when correcting the bias of quadratic estimators like (X?)².

Expectation of a Function of Random Variables

When dealing with functions of random variables, such as squaring the sample mean, one must consider not only the mean but also the variance. For instance, E[(X?)²] equals ?² plus an extra term related to the variance, specifically ?²/n. This extra bias term illustrates why (X?)² is not an unbiased estimator of ?², and motivates the need for a corrective factor.

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