Suppose that X1 and X2 are independent random variables...

Suppose that $X_{1}$ and $X_{2}$ are independent random variables having a common mean $\mu$. Suppose also that $\operatorname{Var}\left(X_{1}\right)=\sigma_{1}^{2}$ and $\operatorname{Var}\left(X_{2}\right)=\sigma_{2}^{2} .$ The value of $\mu$ is unknown and it is proposed to estimate $\mu$ by a weighted average of $X_{1}$ and $X_{2}$. That is, $\lambda X_{1}+(1-\lambda) X_{2}$ will be used as an estimate of $\mu$, for some appropriate value of $\lambda$. Which value of $\lambda$ yields the estimate having the lowest possible variance? Explain why it is desirable to use this value of $\lambda$.

Key Concepts

Linear Combination of Independent Random Variables

When estimating a parameter, it is common to form a linear combination of independent random variables. The linearity means that the expected value of the combination is the weighted sum of the individual expected values and, due to independence, the variance of the combination is the sum of the variances weighted by the squares of the coefficients. This principle underlies many statistical estimation and prediction techniques.

Variance Minimization

In estimation theory, one of the primary goals is to reduce the uncertainty or variability of an estimator. Minimizing the variance of an estimator, while maintaining unbiasedness, is important because it directly affects the precision and reliability of the estimates. A lower variance implies that the estimator is more concentrated around the true parameter value.

Optimal Weighting and Inverse Variance Weighting

The idea behind optimal weighting is to assign more weight to observations that are more reliable, which in this context means having lower variance. By choosing weights inversely proportional to the variances of the individual random variables, the combined estimator minimizes overall variance. This strategy ensures that components with higher precision contribute more significantly to the final estimate.

Calculus-Based Optimization in Estimation

Calculus plays an important role in optimization problems in statistics. By expressing the variance of the estimator as a function of a weight parameter and then differentiating this function with respect to that parameter, one can find the optimum by setting the derivative to zero. This process yields the exact weight that minimizes the estimator's variance, ensuring an efficient use of available data.

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