where and are the times required to calculate the estimators and respectively in this schemet is considered more efficient than if its relative time variance product is smaller we discuss next two systematic approaches for constructing sound estimators method of moments suppose xn are outcomes from an iid sample xn ~iid ( th)where th [th thk ]is unknown the moments of the sampling distribution can be easily estimated namelyif ( th)then the -th moment of xthat is ur (theth (assuming it exists)can be estimated through the sample -th momentn ni= xir the method of moments involves choosing the estimate th of th such that each of the first sample and true moments are matchedn ur ( th) = sample -th moment method of moments in generalthis set of equations is nonlinear and so its solution often has to be found numerically example (sample mean and sample variancesuppose the data is given by { xn }where the {xi form an iid sample from general distribution with mean and variance matching the first two moments gives the set of equations xi un = xi = the method of moments estimates for and are therefore the sample mean sample mean and xi =xn = ( ( (xi ) = = ( the corresponding estimator for uxis unbiased howeverthe estimator for is biasedc ( )/ an unbiased estimator is the sample variance es sample variance = (xi ) = its square roots is called the sample standard deviation example (sample covariance matrixthe method of moments can also be used to estimate the covariance matrix of random vector in particularlet the xn sample standard deviation