A construction of the UMVU estimator for simple quadratic natural exponential families

This paper presents a construction of the uniformly minimum variance unbiased (UMVU) estimator of real-valued functions for the simple quadratic natural exponential families on . A polynomial expansion of the estimator is derived and a condition for its existence is given. The exact variance of the UMVU estimator is calculated. It is also shown that the series of the multidimensional Bhattacharyya bounds converges to this variance. These results are extensions of Morris (Ann. Statist. 11 (1983) 515) and Blight and Rao (Biometrika 61 (1974) 137). Some illustrations are indicated. Non-i.i.d. and biased cases are also discussed.