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On Inference of Overlapping Coefficients in Two Lomax Populations Using Different Sampling Methods

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Abstract

This paper investigates point and interval estimation for some well-known measures of overlap. Two types of sampling procedures, namely, Simple Random Sample and Ranked Set Sample from two Lomax populations with different shape parameters are considered. Simulation studies are conducted to get insight on the performance of the proposed estimators. Taylor series approximations as well as bootstrap method are used to construct confidence intervals for those measures.

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References

  • Al-Saidy, O., Samawi, H.M., Al-Saleh, M.F., 2005. Inference on overlapping coefficients under the Weibull distribution: equal shape parameter. ESAIM Probability and Statistics (P&S), 9, 206–219.

    Article  Google Scholar 

  • Al-Saleh, M.F., Samawi, H.M., 2007. Inference on overlapping coefficients in two exponential populations. Journal of Modern Statistical Science, 6(2), 503–516.

    Google Scholar 

  • Arnold, B.C., 1983. The Pareto Distributions, International Co-operative Publishing House, Fairland, MD.

    MATH  Google Scholar 

  • Balkema, A.A., de Haan, L., 1974. Residual life at great age. Annals of Probability, 2, 792–804.

    Article  MathSciNet  Google Scholar 

  • Bradley, E.L., Piantadosi, S., 1982. The overlapping coefficient as a measure of agreement between distributions. Technical Report, Department of Biostatistics and Biomathematics, University of Alabama at Birmingham, Birmingham, AL.

    Google Scholar 

  • Bryson, M.C., 1974. Heavy-tailed distributions: properties and tests. Technometrics, 16, 61–68.

    Article  MathSciNet  Google Scholar 

  • Chahkandi, M., Ganjali, M., 2009. On some lifetime distributions with decreasing failure rate. Comput. Statist. Data Anal., 53, 4433–4440.

    Article  MathSciNet  Google Scholar 

  • Clemons, T.E., 1996. The overlapping coefficient for two normal probability functions with unequal variances. Unpublished Thesis, Department of Biostatistics, University of Alabama at Birmingham, Birmingham, AL.

    Google Scholar 

  • Clemons, T.E., Bradley E.L., 2000. A nonparametric measure of the overlapping coefficient. Comp. Statist. And Data Analysis, 34, 51–61.

    Article  Google Scholar 

  • Dubey, S.D., 1970. Compound gamma, beta and F distributions. Metrika, 16, 27–31.

    Article  MathSciNet  Google Scholar 

  • Efron, B., 1979. Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7(1), 1–26.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Harner, E.J., Whitmore, R.C., 1977. Multivariate measures of niche overlap using discriminant analysis. Theoret. Population Biol., 12, 21–36.

    Article  MathSciNet  Google Scholar 

  • Hui, T.P., Modarres, R., Zheng, G., 2000. Bootstrap confidence interval estimation of mean via ranked set sampling linear regression. Journal of Statistical Computation and Simulation, 75(7), 543–553.

    Article  MathSciNet  Google Scholar 

  • Ibrahim, H.I., 1991. Evaluating the power of the Mann-Whitney test using the bootstrap method. Commun. Statist. Theory Meth., 20, 2919–2931.

    Article  MathSciNet  Google Scholar 

  • Ichikawa, M., 1993. A meaning of the overlapped area under probability density curves of stress and strength. Reliab. Eng. System Safety, 41, 203–204.

    Article  Google Scholar 

  • Inman, H.F., Bradley, E.L., 1989. The Overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities. Comm. Statist. Theory and Methods, 18, 3851–3874.

    Article  MathSciNet  Google Scholar 

  • Kaur, A., Patil, G.P., Sinha, A.K., Tailie, C., 1995. Ranked set sampling: An annotated bibliography. Environmental and Ecological Statistics, 2, 25–54.

    Article  Google Scholar 

  • Lomax, K., 1954. Business failures: Another example of the analysis of failure data. J. Amer. Statist. Assoc., 49, 847–852.

    Article  Google Scholar 

  • MacArthur, R.H., 1972. Geographical Ecology. Harper and Row, New York.

    Google Scholar 

  • Matusita, K., 1955. Decision rules based on distance, for problems of fit, two samples and estimation. Annals of Mathematical Statistics, 26, 631–640.

    Article  MathSciNet  Google Scholar 

  • McIntyre, G.A., 1952. A method for unbiased selective samplings using ranked sets. Australian Journal of Agricultural Research, 3, 385–390.

    Article  Google Scholar 

  • Mizuno S., Yamaquchi T., Fukushima A., Matsuyama Y. and Ohashi Y. (2005). Overlap coefficient for assessing the similarity of pharmacokinetic data between ethnically different populations. Clin. Trials, 2, 174–181.

    Article  Google Scholar 

  • Morisita, M., 1959. Measuring of interspecific association and similarity between communities. Memoirs of the Faculty of Science of Kyushu University, Series E., Biology, 3, 65–80.

    Google Scholar 

  • Mulekar, M.S., Gonzales, S., Aryal, S., 2001. Estimation and inference for the overlap of two exponential distributions. 2001 Proceedings of American Statistical Association, Joint Statistical Meetings.

    Google Scholar 

  • Mulekar, M.S., Mishra, S.N., 1994. Overlap coefficient of two normal densities: equal means case. J. Japan Statist. Soc., 24, 169–180.

    MathSciNet  MATH  Google Scholar 

  • Mulekar, M.S., Mishra, S.N., 2000. Confidence interval estimation of overlap: equal means case. Comp. Statist.and Data Analysis, 34, 121–137.

    Article  Google Scholar 

  • Muttlak, H.A., 1997. Median ranked set sampling. Journal of Applied Statistical Science, 6, 245–255.

    MATH  Google Scholar 

  • Patil, G.P., Sinha, A.K., Taillie, C., 1999. Ranked set sampling: a bibliography. Environmental and Ecological Statistics, 6, 91–98.

    Article  Google Scholar 

  • Reiser, B., Faraggi, D., 1999. Confidence intervals for the overlapping coefficient: the normal equal variance case. The statistician, 48, Part 3, 413–418.

    Google Scholar 

  • Samawi, H.M., Ahmed, M.S., Abu Dayyeh, W., 1996. Estimating the population mean using extreme ranked set sampling. Biometrical Journal, 38, 577–586.

    Article  Google Scholar 

  • Samawi, H.M., Al-Sakeer, O.A.M, 2001. On the estimation of the distribution function using extreme and median ranked set sampling. Biometrical Journal, 43, 29–45.

    Article  MathSciNet  Google Scholar 

  • Samawi, H.M., Al-Saleh, M.F., 2008. Inference on overlapping coefficients in two exponential populations using ranked set sample. Communication of the Korean Statistical Society, 15(2), 147–159.

    Google Scholar 

  • Samawi, H., Pararai, M., 2009. An optimal bivariate ranked set sample design for the matched pairs sign test. Journal of Statistical Theory and Practice, 3(2), 393–406.

    Article  MathSciNet  Google Scholar 

  • Samawi H.M., Woodworth G.G, Lemke J., 1998. Power estimation for two-sample tests using importance and antithetic re-sampling. Biometrical J., 40(3), 341–354.

    Article  Google Scholar 

  • Schmid, F., Schmidt, A., 2006. Nonparametric estimation of the coefficient of overlapping-theory and empirical application. Computational Statistics & Data Analysis, 50, 1583–1596.

    Article  MathSciNet  Google Scholar 

  • Slobodchikoff, C.N., Schulz, W.C., 1980. Measures of nice overlap. Ecology, 61, 1051–1055.

    Article  Google Scholar 

  • Sneath, P.H.A., 1977. A method for testing the distinctness of clusters: a test of the disjunction of two clusters in Euclidean space as measured by their overlap. J. Int. Assoc. Math. Geol., 9, 123–143.

    Article  Google Scholar 

  • Tadikamalla, P.R., 1980. A look at the Burr and related distributions. International Statistical Review, 48, 337–344.

    Article  MathSciNet  Google Scholar 

  • Weitzman, M.S., 1970. 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. U.S.

    Google Scholar 

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Correspondence to Amal Helu.

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Helu, A., Samawi, H. On Inference of Overlapping Coefficients in Two Lomax Populations Using Different Sampling Methods. J Stat Theory Pract 5, 683–696 (2011). https://doi.org/10.1080/15598608.2011.10483739

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  • DOI: https://doi.org/10.1080/15598608.2011.10483739

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