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Zur Parameter- und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang

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Summary

Some of the many methods developed for estimating parameters or percentage points of the Weibull distribution are compared.

It is shown that the known estimation of the reciprocal shape parameter with the aid of a straight line in the extremal probability paper is rather biased for small sample sizes. To avoid the bias, correction factors are given, and the efficiency of the resulting unbiased estimator is calculated for sample sizesn=2, 3, …, 9.

Results ofJ. Lieblein concerning the double exponential distribution are slightly modified in order to get best linear unbiased estimators for parameters and for the logarithms of percentage points of the Weibull distribution. Other methods are shortly discussed and a median-unbiased estimator for the shape parameter is derived.

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Literatur

  • Chapman, D. G. u.H. Robbins: “Minimum Variance Estimation without regularity Assumptions”. Ann. Math. Statist. (1951)22, 581–585.

    MathSciNet  Google Scholar 

  • McCool, J. I.: “The Construction of good Linear Unbiased Estimates from the Best Linear Estimates for a smaller Sample Size.” Technometrics (1965)7, 543–552.

    Article  MathSciNet  Google Scholar 

  • Dubey, S. D.: „On some Statistical Inferences for Weibull laws“. Ann. Math. Statist. (1962)33, 1504–1505 (abstract); Naval Research Logistics Quarterly,13, 227–251.

    Google Scholar 

  • —— “Some Percentile Estimators for Weibull Parameters” Technometrics (1967)9, 119–129.

    Article  MathSciNet  Google Scholar 

  • Govindarajulu, Z., u.M. Joshi: “Best Linear Unbiased Estimation of Location and Scale Parameters of Weibull Distribution using ordered Observations”. Case Institute of Technology (1962), Statistical Laboratory Rep. No. 102, Cleveland 6, Ohio.

  • Gumbel, E. J.: „Statistics of Extremes“ Columbia University Press, New York, 1960.

    Google Scholar 

  • Leone, F. C., Y. H. Rutenberg, C. W. Topp: “Order Statistics and Estimators for the Weibull Distribution” Case Institute of Technology (1960), Statistical Laboratory Rep. No. 1026, Cleveland 6, Ohio.

  • Lieblein, J.: “On the Exact Evaluation of the Variances and Covariances of Order Statistics in Samples from the Extreme-Value Distribution” Ann. Math. Statist. (1953)24, 282–287.

    MathSciNet  Google Scholar 

  • —— “On Moments of Order Statistics from the Weibull Distribution” Ann. Math. Statist. (1955)26, 330–333.

    MathSciNet  Google Scholar 

  • -- “A new Method of Analyzing Extreme Value Data”. National Advisory Commitee for Aeronautics (1954), Technical Note 3053.

  • Lloyd, E. H.: “Least Squares Estimation of Location and Scale Parameters Using Order Statistics”. Biometrika (1952)39, 87–95.

    MathSciNet  Google Scholar 

  • Menon, M. V.: “Estimation of the Shape and Scale Parameters of the Weibull Distribution”. Technometrics (1963)5, 175–182.

    Article  MATH  MathSciNet  Google Scholar 

  • Schreiber, H. H.: „Lebensdauerversuche mit Wälzlagern — ihre Aussagegenauigkeit und Planung“. Wälzlagertechnik (1963)3; Berichte der Firma FAG Kugelfischer Georg Schäfer & Co. Schweinfurt.

  • Smirnow, N. W., u.I. W. Dunin-Barkowski: „Mathematische Statistik in der Technik“. (Übersetzung aus dem Russischen.) Deutscher Verlag der Wissenschaften Berlin. 1963.

  • “Table of the First Moment of Ranked Extremes”. National Applied Mathematics Laboratories, Project S 50–39 NACA and National Bur. Standards 1951.

  • Uhlmann, W.: „Statistische Schätzverfahren für Dauerfestigkeits-Versuche“. Deutsche Luft- und Raumfahrt. Forschungsbericht 67–10, Deutsche Versuchsanstalt für Luft- und Raumfahrt, München, 1967.

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Vogt, H. Zur Parameter- und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang. Metrika 14, 117–131 (1969). https://doi.org/10.1007/BF02613646

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  • DOI: https://doi.org/10.1007/BF02613646

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