Inference on overlap coefficients under the Weibull distribution : equal shape parameter
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 206-219.

In this paper we consider three measures of overlap, namely Matusia’s measure ρ, Morisita’s measure λ and Weitzman’s measure Δ. These measures are usually used in quantitative ecology and stress-strength models of reliability analysis. Herein we consider two Weibull distributions having the same shape parameter and different scale parameters. This distribution is known to be the most flexible life distribution model with two parameters. Monte Carlo evaluations are used to study the bias and precision of some estimators of these overlap measures. Confidence intervals for the measures are also constructed via bootstrap methods and Taylor series approximation.

Classification : 62F10,  62F40
Mots clés : bootstrap method, Matusia's measure, Morisita's measure, overlap coefficients, Taylor expansion, Weitzman's measure
     author = {Al-Saidy, Obaid and Samawi, Hani M. and Al-Saleh, Mohammad F.},
     title = {Inference on overlap coefficients under the {Weibull} distribution : equal shape parameter},
     journal = {ESAIM: Probability and Statistics},
     pages = {206--219},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2005},
     doi = {10.1051/ps:2005010},
     zbl = {1136.62378},
     mrnumber = {2148967},
     language = {en},
     url = {}
AU  - Al-Saidy, Obaid
AU  - Samawi, Hani M.
AU  - Al-Saleh, Mohammad F.
TI  - Inference on overlap coefficients under the Weibull distribution : equal shape parameter
JO  - ESAIM: Probability and Statistics
PY  - 2005
DA  - 2005///
SP  - 206
EP  - 219
VL  - 9
PB  - EDP-Sciences
UR  -
UR  -
UR  -
UR  -
DO  - 10.1051/ps:2005010
LA  - en
ID  - PS_2005__9__206_0
ER  - 
Al-Saidy, Obaid; Samawi, Hani M.; Al-Saleh, Mohammad F. Inference on overlap coefficients under the Weibull distribution : equal shape parameter. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 206-219. doi : 10.1051/ps:2005010.

[1] L.J. Bain and C.E. Antle, Estimation of parameters in Weibull the distribution. Technometrics 9 (1967) 621-627. | Zbl 0152.36904

[2] L.J. Bain and M. Engelhardt, Statistical analysis of reliability and life-testing models. Marcel Dekker (1991). | Zbl 0724.62096

[3] D.B. Brock, T. Wineland, D.H. Freeman, J.H. Lemke and P.A. Scherr, Demographic characteristics, in Established Population for Epidemiologic Studies of the Elderly, Resource Data Book, J. Cornoni- Huntley, D.B. Brock, A.M. Ostfeld, J.O. Taylor and R.B. Wallace Eds. National Institute on Aging, NIH Publication No. 86- 2443. US Government Printing Office, Washington, DC (1986).

[4] T.E. Clemons and Bradley Jr., A nonparametric measure of the overlapping coefficient. Comp. Statist. Data Analysis 34 (2000) 51-61. | Zbl 1052.62514

[5] A.C. Cohen, Multi-censored sampling in three-parameter Weibull distribution. Technometrics 17 (1974) 347-352. | Zbl 0307.62065

[6] P.M. Dixon, The Bootstrap and the Jackknife: describing the precision of ecological Indices, in Design and Analysis of Ecological Experiments, S.M. Scheiner and J. Gurevitch Eds. Chapman & Hall, New York (1993) 209-318.

[7] K.N. Do and P. Hall, On importance resampling for the bootstrap. Biometrika 78 (1991) 161-167.

[8] B. Efron, Bootstrap methods: another look at the jackknife. Ann. Statist. 7 (1979) 1-26. | Zbl 0406.62024

[9] W.T. Federer, L.R. Powers and M.G. Payne, Studies on statistical procedures applied to chemical genetic data from sugar beets. Technical Bulletin, Agricultural Experimentation Station, Colorado State University 77 (1963).

[10] P. Hall, On the removal of Skewness by transformation. J. R. Statist. Soc. B 54 (1992) 221-228.

[11] H.L. Harter and A.H. Moore, Asymptotic variances and covariances of maximum-likelihood estimators, from censored samples, of the parameters of the Weibull and gamma populations. Ann. Math. Statist. 38 (1967) 557-570. | Zbl 0168.17502

[12] H.I. Ibrahim, Evaluating the power of the Mann-Whitney test using the bootstrap method. Commun. Statist. Theory Meth. 20 (1991) 2919-2931.

[13] M. Ichikawa, A meaning of the overlapped area under probability density curves of stress and strength. Reliab. Eng. System Safety 41 (1993) 203-204.

[14] H.F. Inman and E.L. Bradley, The Overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities. Comm. Statist. Theory Methods 18 (1989) 3851-3874. | Zbl 0696.62131

[15] F.C. Leone, Y.H. Rutenberg and C.W. Topp, Order statistics and estimators for the Weibull population. Tech. Reps. AFOSR TN 60-489 and AD 237042, Air Force Office of Scientific Research, Washington, DC (1960).

[16] J. Lieblein and M. Zelen, Statistical investigations of the fatigue life of deep groove ball bearings. Research Paper 2719. J. Res. Natl. Bur Stand. 57 (1956) 273-316.

[17] R. Lu, E.P. Smith and I.J. Good, Multivariate measures of similarity and niche overlap. Theoret. Population Ecol. 35 (1989) 1-21. | Zbl 0665.92021

[18] N. Mann, Point and Interval Estimates for Reliability Parameters when Failure Times have the Two-Parameter Weibull Distribution. Ph.D. dissertation, University of California at Los Angeles, Los Angeles, CA (1965).

[19] N. Mann, Results on location and scale parameters estimation with application to Extreme-Value distribution. Tech. Rep. ARL 670023, Office of Aerospace Research, USAF, Wright-Patterson AFB, OH (1967a).

[20] N. Mann, Tables for obtaining the best linear invariant estimates of parameters of the Weibull distribution. Technometrics 9 (1967b) 629-645.

[21] N. Mann, Best linear invariant estimation for Weibull distribution. Technometrics 13 (1971) 521-533. | Zbl 0226.62099

[22] K. Matusita, Decision rules based on the distance for problem of fir, two samples, and Estimation. Ann. Math. Statist. 26 (1955) 631-640. | Zbl 0065.12101

[23] J.I. Mccool, Inference on Weibull Percentiles and shape parameter from maximum likelihood estimates. IEEE Trans. Rel. R-19 (1970) 2-9.

[24] S.N. Mishra, A.K. Shah and J.J. Lefante, Overlapping coefficient: the generalized t approach. Commun. Statist. Theory Methods (1986) 15 123-128. | Zbl 0602.62020

[25] M. Morisita, Measuring interspecific association and similarity between communities. Memoirs of the faculty of Kyushu University. Series E. Biology 3 (1959) 36-80.

[26] M.S. Mulekar and S.N. Mishra, Overlap Coefficient of two normal densities: equal means case. J. Japan Statist. Soc. 24 (1994) 169-180. | Zbl 0818.62100

[27] M.S. Mulekar and S.N. Mishra, Confidence interval estimation of overlap: equal means case. Comp. Statist. Data Analysis 34 (2000) 121-137. | Zbl 1054.62502

[28] D.N.P. Murthy, M. Xie and R. Jiang, Weibull Models. John Wiley & Sons (2004). | MR 2013269 | Zbl 1047.62095

[29] M. Pike, A suggested method of analysis of a certain class of experiments in carcinogenesis. Biometrics 29 (1966) 142-161.

[30] B. Reser and D. Faraggi, Confidence intervals for the overlapping coefficient: the normal equal variance case. The statistician 48 (1999) 413-418.

[31] P. Rosen and B. Rammler, The laws governing the fineness of powdered coal. J. Inst. Fuels 6 (1933) 29-36.

[32] H.M. Samawi, G.G. Woodworth and M.F. Al-Saleh, Two-Sample importance resampling for the bootstrap. Metron (1996) Vol. LIV No. 3-4. | Zbl 0896.62039

[33] H.M. Samawi, Power estimation for two-sample tests using importance and antithetic r resampling. Biometrical J. 40 (1998) 341-354. | Zbl 1008.62584

[34] E.P. Smith, Niche breadth, resource availability, and inference. Ecology 63 (1982) 1675-1681.

[35] P.H.A. Sneath, A method for testing the distinctness of clusters: a test of the disjunction of two clusters in Euclidean space as measured by their overlap. Math. Geol. 9 (1977) 123-143.

[36] D.R. Thoman, L.J. Bain and C.E. Antle, Inference on the parameters of the Weibull distribution. Technometrics 11 (1969) 445-460. | Zbl 0179.48501

[37] W. Weibull, A statistical theory of the strength of materials. Ing. Vetenskaps Akad. Handl. 151 (1939) 1-45.

[38] W. Weibull, A statistical distribution function of wide application. J. Appl. Mech. 18 (1951) 293-297. | Zbl 0042.37903

[39] M.S. Weitzman, Measures of overlap of income distributions of white and Negro families in the United States. Technical paper No. 22. Department of Commerce, Bureau of Census, Washington, US (1970).

[40] J.S. White, The moments of log-Weibull Order Statistics. General Motors Research Publication GMR-717. General Motors Corporation, Warren, Michigan (1967). | Zbl 0174.22904

Cité par Sources :