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The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R p×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]
The theory of median-unbiased estimators was revived by George W. Brown in 1947: [8]. An estimate of a one-dimensional parameter θ will be said to be median-unbiased, if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates.
For each random variable, the sample mean is a good estimator of the population mean, where a "good" estimator is defined as being efficient and unbiased. Of course the estimator will likely not be the true value of the population mean since different samples drawn from the same distribution will give different sample means and hence different ...
which is an estimate of the covariance between variable and variable . The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector X {\displaystyle \textstyle \mathbf {X} } , a vector whose j th element ( j = 1 , … , K ) {\displaystyle (j=1,\,\ldots ,\,K)} is one of the ...
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
One way of seeing that this is a biased estimator of the standard deviation of the population is to start from the result that s 2 is an unbiased estimator for the variance σ 2 of the underlying population if that variance exists and the sample values are drawn independently with replacement. The square root is a nonlinear function, and only ...
where m x is the mean of the variate x and s xy is the covariance between x and y. To simplify the notation s xy will be used subsequently to denote the covariance between the variates x and y. Another estimator based on the Taylor expansion is [3]
Precisely which covariance matrix is of concern is a matter of context. Alternative estimators have been proposed in MacKinnon & White (1985) that correct for unequal variances of regression residuals due to different leverage. [11] Unlike the asymptotic White's estimator, their estimators are unbiased when the data are homoscedastic.