Suppose that x1 and x2 are random samples of observations...

Suppose that $x_1$ and $x_2$ are random samples of observations from a population with mean $\mu$ and variance $s^2$. Consider the following three point estimators, $X, Y$, $Z$, of $\mu$ : $$ X=\frac{1}{2^2}+\frac{1}{2^2} \quad Y=\frac{1}{4} x_1+\frac{3}{4} x_2 \quad Z=\frac{1}{3} x_1+\frac{2}{3} x_2 $$ a. Show that all three estimators are unbiased. b. Which of the estimators is the most efficient? c. Find the relative efficiency of $X$ with respect to each of the other two estimators.

Suppose that $x_1$ and $x_2$ are random samples of observations from a population with mean $\mu$ and variance $s^2$. Consider the following three point estimators, $X, Y$, $Z$, of $\mu$ :
$$
X=\frac{1}{2^2}+\frac{1}{2^2} \quad Y=\frac{1}{4} x_1+\frac{3}{4} x_2 \quad Z=\frac{1}{3} x_1+\frac{2}{3} x_2
$$
a. Show that all three estimators are unbiased.
b. Which of the estimators is the most efficient?
c. Find the relative efficiency of $X$ with respect to each of the other two estimators.

Key Concepts

Efficiency

Efficiency refers to the quality of an estimator in terms of its variance; among unbiased estimators, the one with the smallest variance is considered the most efficient. An efficient estimator makes the best use of the data by providing estimates that vary less from one sample to the next, resulting in more consistent and reliable parameter estimates.

Relative Efficiency

Relative efficiency is a measure used to compare the performance of two unbiased estimators by taking the ratio of their variances. It quantifies how much more (or less) efficient one estimator is compared to another. This concept is useful in selecting between different estimation methods by considering which provides a lower variance in the estimates.

Unbiased Estimator

An unbiased estimator is a statistic used to estimate a parameter such that its expected value is equal to the parameter it estimates. In other words, on average, the estimator yields the true value of the parameter over repeated samples. This property is fundamental in statistics as it ensures that there is no systematic error in the estimation process.

Variance Calculation for Linear Combinations

When estimators are defined as linear combinations of sample observations, their variances can be calculated by applying the rules for the variance of weighted sums of independent random variables. This calculation is essential in comparing the efficiencies of different estimators, as it allows for the determination of which estimator has the smallest variance among unbiased candidates.

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