Suppose that we seek an intuitive estimator for a The

Chapter 11, Problem 54E

(choose chapter or problem)

Suppose that we seek an intuitive estimator for

                \(p=\frac{\operatorname{cov}(x, y)}{{ }^{\sigma} x^{\sigma} y}\),

a The method-of-moments estimator of \(\operatorname{cov}(x, y)=E\left[\left(x-\mu_{x}\right)\left(y-\mu_{y}\right)\right]\) is

\(\operatorname{Cov}(x, y)=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)\)

Show that the method-of-moments estimators for the standard deviations of X and Y are

\(\hat{\sigma}_{x}=-\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\) and \(\hat{\sigma}_{y}=\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}}\)

b Substitute the estimators for their respective parameters in the definition of ρ and obtain the method-of-moments estimator for ρ. Compare your estimator to r , the maximum-likelihood estimator for ρ presented in this section.

Equation transcription:

Text transcription:

p=\frac{{cov}(x, y)}{^{\sigma} x^{\sigma} y}

{cov}(x, y)=E[(x-mu{x})(y-\mu{y})]

\operatorname{Cov}(x, y)=\frac{1}{n} sum{i=1}^{n}(x{i}-\bar{x})(y{i}-\bar{y})

hat{\sigma}{x}=-sqrt{\frac{1}{n} sum{i=1}^{n}(x{i}-\bar{x})^{2}}

hat{\sigma}{y}=sqrt{\frac{1}{n} sum{i=1}^{n}(y{i}-\bar{y})^{2}}