Abstract
The concept of deficiency was introduced by Hodges and Lehmann [Hodges, J. L., Lehmann, E. L. (1970). Deficiency. Ann. Math. Statist. 41(3):783–801] which can be used to discriminate two asymptotically efficient (BAN) estimators. One-parameter exponential family of distributions depending on the parameter θ is considered. The mean square errors of the maximum likelihood (ML) estimator and the uniformly minimum variance unbiased (UMVU) estimator of an estimable function g(θ) are obtained up to the order n −2, where n is the sample size. The deficiency between the two estimators is considered. The restriction imposed by Hwang and Hu [Hwang, T.-Y., Hu, C.-Y. (1990). More comparisons of MLE with UMVUE for exponential families. Ann. Inst. Statist. Math. 42(1):65–75] is relaxed. An illustrative example is discussed.
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