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On estimating the difference of location parameters of two negative exponential distributions. (English) Zbl 0543.62061
This paper is concerned with the problem of constructing two-stage and sequential fixed-width confidence intervals for the difference of location parameters in two negative-exponential distributions. First, the case of equal but unknown scale parameter is discussed and two-stage procedures along the lines of the first author, Metrika 27, 281-284 (1980; Zbl 0449.62028), and a pure sequential procedure are proposed. Then, two-stage and sequential procedures are derived for the case of unequal and unknown scale parameters. The asymptotic properties of the proposed sequential procedures are derived.
Reviewer: K.Uosaki

##### MSC:
 62L12 Sequential estimation 62F12 Asymptotic properties of parametric estimators 62F10 Point estimation
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##### References:
  Basu, On a sequential rule for estimating the location parameter of an exponential distribution, Naval Res. Logist. Quart. 18 pp 329– (1971) · Zbl 0227.62049  Chapman, Some two-sample tests, Ann. Math. Statist. 21 pp 601– (1950) · Zbl 0040.36602  Chow, On the asymptotic theory of fixed width sequential confidence intervals for the mean, Ann. Math. Statist. 36 pp 451– (1965) · Zbl 0142.15601  Ghosh, A two-stage procedure for the Behrens-Fisher problem, J. Amer. Statist. Assoc. 70 pp 457– (1975) · Zbl 0323.62028  Ghosh, On the distribution of the difference of twot-variables, J. Amer. Starur. Assoc. 70 pp 463– (1975)  Ghosh, M., and Mukhopadhyay, N. (1975). Asymptotic normality of stopping times in sequential analysis. Unpublished manuscript.  Ghosh, Sequential point estimation of the mean when the distribution is unspecified, Comm. Statist. A - Theory Methods 8 pp 637– (1979) · Zbl 0446.62085  Ghosh, Sequential point estimation of the difference of two normal means, Ann. Statist. 8 pp 221– (1980) · Zbl 0422.62075  Ghosh, Consistency and asymptotic efficiency of two stage and sequential estimation procedures, Sankhya Ser. A 43 pp 220– (1981) · Zbl 0509.62069  Ghurye, Note on sufficient statistics and two-stage procedures, Ann. Math. Stutist. 29 pp 155– (1958) · Zbl 0087.34104  Ghurye, Two-stage procedures for estimating the difference between means, Biometrika 41 pp 146– (1954) · Zbl 0055.12902  Gupta, On the smallest of several correlated F-statistics, Biometrika 49 pp 509– (1962) · Zbl 0124.11207  Guttman, Procedures for a best population problem when the criterion of bestness involves a fixed tolerance region, Ann. Inst. Statist. Math. 21 pp 149– (1969) · Zbl 0179.23804  Hayre, Sequential estimation of the difference between the means of two normal populations, Metrika 30 pp 101– (1983) · Zbl 0521.62066  Krishnaiah, P. R., and Armitage, J. V. (1964). Distribution of the Studentized smallest chi-square, with tables and applications. Report No. ARL 64-218. Aerospace Research Laboratory, Wright-Patterson Air Force Base, Dayton, Oho.  Louis, Optimal allocation in sequential tests comparing means of two Gaussian populations, Biometrika 62 pp 359– (1975) · Zbl 0321.62024  Mukhopadhyay, Sequential estimation of location parameter in exponential distributions, Calcutta Statist. Assoc. Bull. 23 pp 85– (1974) · Zbl 0342.62058  Mukhopadhyay, Sequential estimation of a linear function of means of three normal populations, J. Amer. Statist. Assoc. 71 pp 149– (1976) · Zbl 0331.62062  Mukhopadhyay, Remarks on sequential estimation of a linear function of two means: the normal case, Metrika 24 pp 197– (1977) · Zbl 0368.62063  Mukhopadhyay, N. (1980a). Stein’s two-stage procedure and exact consistency. Scand. Actuar. J., to appear. · Zbl 0493.62072  Mukhopadhyay, A consistent and asymptotically efficient two stage procedure t0 construct fixed width confidence intervals for the mean, Metrika 27 pp 281– (1980) · Zbl 0449.62028  Mukhopadhyay, On the asymptotic regret while estimating the location of an exponential distribution, Calcutta Statist. Assoc. Bull. 31 pp 207– (1982) · Zbl 0516.62077  O’Neill, A two-stage procedure for estimating the difference between the mean vectors of two multivariate normal distributions, Trabajos Estadist. Investigación Oper. 24 pp 123– (1973)  Robbins, A sequential analogue of the Behrens-Fisher problem, Ann. Marh. Statist. 38 pp 1384– (1967) · Zbl 0157.48105  Scheffé, On solutions of the Behrens-Fisher problem, based on the t-distribution, Ann. Math. Statist. 14 pp 35– (1943)  Scheffé, Practical solutions of the Behrens-Fisher problem, J. Amer. Stutist. Assoc. 65 pp 1501– (1970) · Zbl 0224.62009  Simons, On the cost of not knowing the variance when making a fixed-width confidence interval for the mean, Ann. Math. Statist. 39 pp 1946– (1968) · Zbl 0187.15805  Srivastava, On a sequential analogue of the Behrens-Fisher problem, J. Roy. Statist. Soc. Ser. B 32 pp 144– (1970) · Zbl 0209.50404  Stein, A two sample test for a linear hypothesis whose power is independent of the variance, Ann. Math. Statist. 16 pp 243– (1945) · Zbl 0060.30403  Stein, Some problems in sequential estimation. (Abstract.), Econometrica 17 pp 77– (1949)  Swanepoel, Fixed width confidence intervals for the location parameter of an exponential distribution, Commun. Statist. A-Theory Methods 11 pp 1279– (1982) · Zbl 0559.62030  Swanepoel, Fixed width confidence intervals for the truncation parameter of an unknown distribution function, South African Statist. J. 15 pp 161– (1982)  Wallace, R. A. Fisher; An Appreciation pp 119– (1980)  Woodroofe, Second order approximations for sequential point and interval estimation, Ann. Statist. 5 pp 984– (1977) · Zbl 0374.62081
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