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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]
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 ...
For two jointly distributed real-valued random variables and with finite second moments, the covariance is defined as the expected value (or mean) of the product of their deviations from their individual expected values: [3] [4]: 119
The covariance matrix can be used to compute portfolio variance. RiskMetrics assumes that the market is driven by risk factors with observable covariance. The risk factors are represented by time series of prices or levels of stocks, currencies, commodities, and interest rates.
The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test. In Bayesian statistics, the Fisher information plays a role in the derivation of non-informative prior distributions according to Jeffreys ...
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of one or more categorical independent variables (IV) and across one or more continuous variables.
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z ( x ) on a domain D , a covariance function C ( x , y ) gives the covariance of the values of the random field at the two ...