# Three Stage Estimation for the Mean of a One-Parameter Exponential Family

• M. AlMahmeed (Department of Quantitative Methods and Information systems, Kuwait University) ;
• A. Al-Hessainan (Kuwait University) ;
• Son, M.S. (Department of Mathematics and Statistics, University of Vermont) ;
• H. I. Hamdy (Department of Quantitative methods and Information Systems, Kuwait University)
• Published : 1998.08.01

#### Abstract

This article is concerned with the problem of estimating the mean of a one-parameter exponential family through sequential sampling in three stages under quadratic error loss. This more general framework differs from those considered by Hall (1981) and others. The differences are : (i) the estimator and the final stage sample size are dependent; and (ii) second order approximation of a continuously differentiable function of the final stage sample size permits evaluation of the asymptotic regret through higher order moments. In particular, the asymptotic regret can be expressed as a function of both the skewness $\rho$ and the kurtosis $\beta$ of the underlying distribution. The conditions on $\rho$ and $\beta$ for which negative regret is expected are discussed. Further results concerning the stopping variable N are also presented. We also supplement our theoretical findings wish simulation results to provide a feel for the triple sampling procedure presented in this study.

#### References

1. Proc. Camb. Phil. Soc v.45 Large sample theory of sequential estimation Anscombe, F. J.
2. Ann. Math. Statist v.36 On the asymptotic theory of fixed width sequential confidence intervals for the mean Chow, Y. S.;Robbins, H.
3. Ann. Statist. v.9 Asymptotic theory of triple sampling for sequential estimation of mean Hall, P.
4. Scand. J. of Stat. v.15 Remarks on the asymptotic theory of triple stage estimation of the normal mean Hamdym H. I.
5. South African Statist. J. v.31 Performance of fixed width intervals under type II error: the exponential case Hamdy, H. I.
6. Statistics v.28 A certain accelerated procedure to construct simultaneous confidence region: the exponential case Hamdy, H. I.;AlMahmeed, M.;Al-Zalzalah, Y.
7. Ann. Inst. Statis. Math. v.40 Triple stage point estimation for the exponential location parameter Hamdy, H. I.;Mukhopadhyay, N.;Costanza, M. C.;Son
8. Statistical LI On accelerating sequential procedures for point estimation: the normal case Hamdy, H. I.;Son, M. S.
9. Metron XLVII Three stage estimation procedure for the exponential location parameters Hamdy, H. I.;M. S. AlMahmeed, M.;Nigm, A.
10. Testing Statistical Hypotheses; 2ed. Lehmann, E. L.
11. J. Am. Stat. Assoc. v.83 Negative regret, optimal stopping, and the elimination of outliers Martinsek, A. T.
12. Ann Statist. v.10 Natural exponential families with quadratic variance function Morris, C. N.
13. Sequential Analysis v.6 Three-stage point estimation procedure for a normal mean Mukhopadhyay, N.;Hamdy, H. I.;AlMahmeed, M.;Costanza, M. C.
14. Ann Statist. v.5 Second order approximations for sequential point and interval estimation Woodroofe, M.
15. New Perspectives in Theoretical and Applied Statistics Asymptotically optimal sequential point estimation in three stages Woodroofe, M.