zbMATH — the first resource for mathematics

Some properties of a scaled Burr type X distribution. (English) Zbl 1058.62017
Summary: Properties of a scaled Burr type X distribution are given. Closed-form expressions for the moments only exist for certain special cases, so upper and lower bounds for the first moment are given, as well as an approximation based on these bounds. Maximum likelihood estimation is considered, and the asymptotic properties of these estimators are discussed for i.i.d. samples, as well as for Types I and II censoring. Finally, an extension to a multivariate Burr type X distribution is introduced.

62E10 Characterization and structure theory of statistical distributions
62F12 Asymptotic properties of parametric estimators
62N05 Reliability and life testing
62E15 Exact distribution theory in statistics
Full Text: DOI
[1] Ahmad, K.E.; Fakhry, M.E.; Jaheen, Z.F., Empirical Bayes estimation of P(Y<X) and characterization of the burr-type X model, J. statist. plann. inference, 64, 297-308, (1997) · Zbl 0915.62001
[2] Bader, M.G., Priest, A.M., 1982. Statistical aspects of fibre and bundle strength in hybrid composites. In: Hayashi, T., Kawata, K., Umekawa, S. (Eds.), Progress in Science and Engineering Composites, Vol. ICCM-IV, Tokyo, pp. 1129-1136.
[3] Bhattacharyya, G.K., The asymptotics of maximum likelihood and related estimators based on type II censored data, J. amer. statist. assoc, 80, 398-404, (1985) · Zbl 0576.62042
[4] Burr, I.W., Cumulative frequency functions, Ann. math. statist, 13, 215-222, (1942) · Zbl 0060.29602
[5] Crowder, M., A distributional model for repeated failure time measurements, J. roy. statist. soc. B, 47, 447-452, (1985)
[6] Crowder, M., A multivariate distribution with Weibull connections, J. roy. statist. soc. B, 51, 93-107, (1989) · Zbl 0669.62029
[7] Jaheen, Z.F., Bayesian approach to prediction with outliers from the burr type X model, Microelectron. reliability, 35, 45-47, (1995) · Zbl 0852.62031
[8] Jaheen, Z.F., Empirical Bayes estimation of the reliability and failure rate functions of the burr type X failure model, J. appl. statist. sci, 3, 281-288, (1996) · Zbl 1054.62593
[9] Lawless, J.E., Statistical models and methods for lifetime data, (2003), Wiley Hoboken, NJ · Zbl 1015.62093
[10] Mudholkar, G.S.; Hutson, A.D., The exponential Weibull familysome properties and a flood data application, Commun. statist. theory methods, 25, 3059-3083, (1996) · Zbl 0887.62019
[11] Mudholkar, G.S.; Srivastava, D.K., Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE trans. reliability, 42, 299-302, (1993) · Zbl 0800.62609
[12] Mudholkar, G.S.; Srivastava, D.K.; Freimer, M., The exponentiated Weibull familya reanalysis of the bus-motor-failure data, Technometrics, 37, 436-445, (1995) · Zbl 0900.62531
[13] Sartawi, H.A.; Abu-Salih, M.S., Bayesian prediction bounds for the burr type X model, Commun. statist. theory methods, 20, 2307-2330, (1991) · Zbl 0850.62288
[14] Serfling, R.J., Approximation theorems of mathematical statistics, (1980), Wiley New York · Zbl 0456.60027
[15] Surles, J.G.; Padgett, W.J., Inference for P(Y<X) in the burr type X model, J. appl. statist. sci, 7, 225-238, (1998) · Zbl 0911.62092
[16] Surles, J.G.; Padgett, W.J., Inference for reliability and stress-strength for a scaled burr type X distribution, Lifetime data anal, 7, 182-200, (2001) · Zbl 0984.62082
[17] Takahasi, K., Note on the multivariate Burr’s distribution, Ann. inst. statist. math, 17, 257-260, (1965) · Zbl 0134.36703
[18] Wolstenholme, L.C., A nonparametric test of the weakest-link principle, Technometrics, 37, 169-175, (1995) · Zbl 0822.62088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.