You plan to use a rod to lay out a square, each side of...

You plan to use a rod to lay out a square, each side of which is the length of the rod. The length of the rod is $\mu$, which is unknown. You are interested in estimating the area of the square, which is $\mu^{2}$. Because $\mu$ is unknown, you measure it $n$ times, obtaining observations $X_{1}, X_{2}, \ldots, X_{n} .$ Suppose that each measurement is unbiased for $\mu$ with variance $\sigma^{2}$. (a) Show that $\bar{X}^{2}$ is a biased estimate of the area of the square. (b) Suggest an estimator that is unbiased.

You plan to use a rod to lay out a square, each side of which is the length of the rod. The length of the rod is $\mu$, which is unknown. You are interested in estimating the area of the square, which is $\mu^{2}$. Because $\mu$ is unknown, you measure it $n$ times, obtaining observations $X_{1}, X_{2}, \ldots, X_{n} .$ Suppose that each measurement is unbiased for $\mu$ with variance $\sigma^{2}$.
(a) Show that $\bar{X}^{2}$ is a biased estimate of the area of the square.
(b) Suggest an estimator that is unbiased.

Key Concepts

Variance of the Sample Mean

The sample mean is a fundamental estimator used to estimate the true mean of a population because it is unbiased. Its variance, which quantifies the estimator's precision, is equal to the variance of the individual measurements divided by the number of observations. This decreasing variance with increasing sample size is a core concept in statistical estimation.

Bias Correction in Nonlinear Transformations

When estimating a nonlinear function of a parameter (for instance, ?² when the parameter ? is estimated by the sample mean), the nonlinearity can introduce bias even if the original estimator is unbiased. The correction typically involves calculating the expected value of the transformed estimator (which includes an additional bias term related to the variance) and then adjusting the estimator by subtracting this bias term to make it unbiased.

Unbiased Estimation

An estimator is considered unbiased if its expected value is equal to the parameter it estimates. This means that, over many samples, the estimator will hit the true value on average, ensuring that there is no systematic overestimation or underestimation in the process.

Estimator Bias

Estimator bias is the difference between the expected value of an estimator and the true value of the parameter being estimated. When an unbiased estimator is transformed by a nonlinear function—such as squaring—the resulting estimator may not retain the unbiased property, leading to systematic error, or bias, in the estimation.

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