Jackknife Maximum Likelihood Estimates for a Bivariate Normal Distribution with Missing Data

In this paper, the estimator of bivariate normal distribution for incomplete data called the J_Anderson estimator is proposed. The first r pairs of (X-1, X-2) were distributed as a bivariate normal distribution with mean vector (mu(1,) mu(2) ) and a variance covariance matrix W = [(sigma 12)(sigma 12) (sigma 22) (sigma 12)] whereas the rest n - r observations of X-1 were distributed as a normal distribution with mean mu(1) and variance sigma(2)(1). The factored maximum likelihood estimators, (mu) over cap (1And) ,(mu) over cap (2And) , (sigma) over cap (2)(1And) and (sigma) over cap (2)(2And) , were proposed by Anderson. In addition, the Anderson's method was found that (sigma) over cap (2)(1And) and (sigma) over cap (2)(2And) were biased estimators of sigma(2)(1) and sigma(2)(2) respectively. In this case, the proposed estimators were denoted by (sigma) over cap (2)(1J-And) and (sigma) over cap (2)(2J-And). Moreover, the simulation study found that the absolute relative biases of E((sigma) over cap (2)(1J-And)) and E((sigma) over cap (2)(2J-And)) were smaller than those of E((sigma) over cap (2)(1And)) and E((sigma) over cap (2)(2And)) respectively whatever sample size and percentage of missing data. Additionally, the mean square error of the proposed estimators seemed to decrease when the sample size was large whatever percentage of missing data.