Unbiased estimation of the variance of the Graybill-Deal estimator of the common mean of several normal populations

An identity for the chi-squared distribution is used to derive an unbiased estimator of the variance of the familiar Graybill-Deal (1959) estimator of the common mean of several normal populations with possibly different unknown variances. This result appears to be new. It is observed that the unbiased estimator is a convergent series whose suitable truncation allows unbiased estimation up to any desired degree of accuracy.