Let 𝐗=(X, Y) be a pair of random variables with unknown...

Let $\mathbf{X}=(X, Y)$ be a pair of random variables with unknown means and covariances. Consider the following estimator for the covariance of $X$ and $Y$ : $$ \hat{K}_{X, Y}=\frac{1}{n-1} \sum_{j=1}^n\left(X_j-\bar{X}_n\right)\left(Y_j-\bar{Y}_n\right) . $$ (a) Find the expected value of this estimator. (b) Explain why the estimator approaches the estimator in Problem 8.8 for $n$ large. Hint: See Eq. (8.15).

Let $\mathbf{X}=(X, Y)$ be a pair of random variables with unknown means and covariances. Consider the following estimator for the covariance of $X$ and $Y$ :

$$
\hat{K}_{X, Y}=\frac{1}{n-1} \sum_{j=1}^n\left(X_j-\bar{X}_n\right)\left(Y_j-\bar{Y}_n\right) .
$$

(a) Find the expected value of this estimator.
(b) Explain why the estimator approaches the estimator in Problem 8.8 for $n$ large. Hint: See Eq. (8.15).

Key Concepts

Covariance

Covariance is a measure of how much two random variables change together. It quantifies the joint variability of the variables and is defined as the expected value of the product of their deviations from their respective means. In statistics, understanding and calculating covariance is fundamental for assessing the relationship between variables.

Sample Covariance

Sample covariance is an estimator of the true covariance based on observed data. When the population parameters are unknown, sample covariance is calculated using the deviations of each observation from the sample means. This concept is essential in statistics as it provides a practical method for estimating the covariance from a dataset.

Unbiased Estimator

An unbiased estimator is a statistical estimator whose expected value is equal to the parameter it estimates. In the context of covariance estimation, using a divisor of (n - 1) instead of n corrects for the bias that arises from using sample means rather than the true means. This adjustment ensures that the estimator, over many samples, accurately reflects the true covariance.

Asymptotic Behavior

Asymptotic behavior refers to the properties of an estimator as the sample size becomes large. For large n, the differences between using n and (n - 1) as the normalization constant for the sample covariance become negligible. Thus, the sample covariance estimator converges to the simpler form where the bias correction is less significant, aligning with estimators used in limiting cases or in problems that assume large samples.

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