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.

DOI : 10.1051/ps:2005010
Classification : 62F10, 62F40
Mots-clés : bootstrap method, Matusia's measure, Morisita's measure, overlap coefficients, Taylor expansion, Weitzman's measure 
@article{PS_2005__9__206_0,
     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},
     mrnumber = {2148967},
     zbl = {1136.62378},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2005010/}
}
TY  - JOUR
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
SP  - 206
EP  - 219
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2005010/
DO  - 10.1051/ps:2005010
LA  - en
ID  - PS_2005__9__206_0
ER  - 
%0 Journal Article
%A Al-Saidy, Obaid
%A Samawi, Hani M.
%A Al-Saleh, Mohammad F.
%T Inference on overlap coefficients under the Weibull distribution : equal shape parameter
%J ESAIM: Probability and Statistics
%D 2005
%P 206-219
%V 9
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2005010/
%R 10.1051/ps:2005010
%G en
%F PS_2005__9__206_0
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. http://www.numdam.org/articles/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

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

[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

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

[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

[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

[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

[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

[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

[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

[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

[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

[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

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

[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

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

[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

[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

[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

Cité par Sources :