## The Annals of Statistics

- Ann. Statist.
- Volume 10, Number 4 (1982), 1137-1147.

### Semi Tail Upper Bounds on the Class of Admissible Estimators in Discrete Exponential Families with Applications to Poisson and Negative Binomial Distributions

#### Abstract

Admissibility problems involving simultaneous estimation in discrete exponential families are studied by solving difference inequalities. It is shown that if an estimator is admissible under the loss function $L_m(\mathbf{\theta, a)} = \sum^p_{i = 1} \theta^{m_i}_i (\theta_i - a_i)^2$, then in the tail (i.e., for large values of the observations), this estimator has to be less than certain bounds. Specific bounds, called Semi Tail Upper Bounds (STUB), are given here. These STUBs are not only of theoretical interest, but also are sharp enough that they establish many new results. Two of the most interesting ones are: (i) the establishment of Brown's conjecture concerning inadmissibility of some of the estimators proposed by Clevenson and Zidek (1975), and (ii) the establishment of inadmissibility of Hudson's (1978) estimator which improves upon the uniformly minimum variance unbiased estimator in Negative Binomial families.

#### Article information

**Source**

Ann. Statist., Volume 10, Number 4 (1982), 1137-1147.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345979

**Digital Object Identifier**

doi:10.1214/aos/1176345979

**Mathematical Reviews number (MathSciNet)**

MR673649

**Zentralblatt MATH identifier**

0526.62005

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C15: Admissibility

Secondary: 62F10: Point estimation 62H99: None of the above, but in this section 39A10: Difference equations, additive

**Keywords**

Admissibility difference inequality discrete exponential family loss function negative binomial distribution Poisson distribution

#### Citation

Hwang, Jiunn Tzon. Semi Tail Upper Bounds on the Class of Admissible Estimators in Discrete Exponential Families with Applications to Poisson and Negative Binomial Distributions. Ann. Statist. 10 (1982), no. 4, 1137--1147. doi:10.1214/aos/1176345979. https://projecteuclid.org/euclid.aos/1176345979