be iid copies of -dimensional random vector with mean vector and covariance matrix we assume the moment estimator for isas in the casex ( xn )/ as the covariance matrix can be written (see ( )as ( )( )the method of moments yields the estimator (xi )(xi )sn = sample covariance matrix similar to the one-dimensional case ( )replacing the factor / with /( gives an unbiased estimatorcalled the sample covariance matrix likelihood function log-likelihood function ( maximum likelihood method the concept of likelihood is central in statistics it describes in precise way the information about model parameters that is contained in the observed data let be (randomdata object that is modeled as draw from the pdf ( th(discrete or continuouswith parameter vector th th let be an outcome of the function (th : ( th)th this called the likelihood function of thbased on the (naturallogarithm of the likelihood function is called the log-likelihood function and is often denoted by lower case note that (th tand ( thhave the same formulabut the first is viewed as function of th for fixed twhere the second is viewed as function of for fixed th the concept of likelihood is particularly useful when is modeled as an iid sample { xn from some pdf in that casethe likelihood of the data { xn }as function of this given by the product (th tmaximum likelihood estimator = (xi th( let be an observation from ( th)and suppose that ( thtakes its largest value at th th in way this th is our best estimate for thas it maximizes the probability (densityfor the observation it is called the maximum likelihood estimate (mleof th note that th = th(tis function of the corresponding random variablealso denoted th is the maximum likelihood estimator (also abbreviated as mlemaximization of (th tas function of th is equivalent (when searching for the maximizerto maximizing the log-likelihood (th )as the natural logarithm is an increasing function this is often easierespecially when is an iid sample from some sampling distribution for examplefor of the form ( )we have (th tn = ln (xi th