Let 𝐗=(X, Y) be a pair of random variables with known means, μ1 and μ2. Consider the following e

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

Let $\mathbf{X}=(X, Y)$ be a pair of random variables with known means, $\mu_1$ and $\mu_2$. Consider the following estimator for the covariance of $X$ and $Y$ : $$ \hat{C}_{X, Y}=\frac{1}{n} \sum_{j=1}^n\left(X_j-\mu_1\right)\left(Y_j-\mu_2\right) $$ (a) Find the expected value and variance of this estimator. (b) Explain the behavior of the estimator as $n$ becomes large.

Let $\mathbf{X}=(X, Y)$ be a pair of random variables with known means, $\mu_1$ and $\mu_2$. Consider the following estimator for the covariance of $X$ and $Y$ :

$$
\hat{C}_{X, Y}=\frac{1}{n} \sum_{j=1}^n\left(X_j-\mu_1\right)\left(Y_j-\mu_2\right)
$$

(a) Find the expected value and variance of this estimator.
(b) Explain the behavior of the estimator as $n$ becomes large.

Key Concepts

Covariance

Covariance measures the degree to which two random variables change together. It is defined as the expected product of deviations of each random variable from their respective means, providing insight into the joint variability of the variables. Estimation of covariance from data involves capturing this joint variability through sample observations.

Estimator Unbiasedness and Variance

When constructing estimators for parameters such as covariance, a key concept is unbiasedness, which means that the expected value of the estimator equals the true parameter value. The estimator provided is unbiased because its expected value is the true covariance. Additionally, the variance of an estimator quantifies the spread of its sampling distribution and is computed using properties of the underlying random variables, often resulting in a variance that scales inversely with the sample size.

Consistency and the Law of Large Numbers

Consistency of an estimator is the property that as the number of samples increases, the estimator converges in probability to the true parameter value. This behavior is underpinned by the Law of Large Numbers, which ensures that sample moments converge to their corresponding population moments. In this context, as the sample size n increases, the variance of the covariance estimator decreases, and the estimator becomes increasingly reliable.

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