Jackknife Maximum Likelihood Estimates for a Bivariate Normal Distribution with Missing Data
Keywords:
biased estimator, bivariate normal distribution, Jackknife’s method, missing data, normal distributionAbstract
In this paper, the estimator of bivariate normal distribution for incomplete data called the J_Anderson estimator is proposed. The first r pairs of ( X1,X2 ) were distributed as a bivariate normal distribution with mean vector ( µ1, µ2 ) and a variance covariance matrix w = , whereas the rest n−r observations of X1 were distributed as a normal distribution with mean µ1 and variance
. The factored maximum likelihood estimators,
,
,
and
, were proposed by Anderson. In addition, the Anderson’s method was found that
and
were biased estimators of
and
respectively. In this study, Jackknife’s method of bias reduction was introduced to modify
and
. In this case, the proposed estimators were denoted by
and
. Moreover, the simulation study found that the absolute relative biases of E
And E
were smaller than those of E
And E
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.
