Tolerance intervals and confidence intervals for the scale parameter of Pareto-Rayleigh distribution


In this paper we consider interval estimation in Pareto-Rayleigh distribution as an example of a Transformed-Transformer family of distribution defined by Alzaatreh et. al (2012). We construct confidence intervals (CIs) and tolerance intervals (TIs) using generalized variable (GV) approach by using maximum likelihood estimator (MLE) and modified maximum likelihood estimator (MMLE) as the likelihood equations are intractable (Tiku and Suresh (1992)). Performances of both intervals are studied using simulation and compared them with existing ones to check superiority of the proposed method. The confidence intervals and tolerance intervals are illustrated through real life data.

DOI Code: 10.1285/i20705948v10n1p29

Keywords: Transformed transformer (T-X) family, Pareto-Rayleigh distribution, MLE, MMLE, generalized pivotal quantity, confidence intervals, and tolerance intervals.


Atwood, C.L. (1984).Approximate tolerance intervals based on maximum likelihood estimator. Journal of American statistics association, 79,459-465.

Alzaatreh, A., Lee, C. and Famoye, F. (2012). A new method for generating families of continuous distributions. Submitted for publication.

Alzaatreh, A., Famoye, F. and Lee, C. (2012). Gamma-Pareto distribution and its applications. Journal of Modern Applied Statistical Methods, 11, 78-94.

Alzaatreh, A., Famoye, F. and Lee, C. (2013). Weibull-Pareto distribution and its applications. Communications in Statistics- Theory and Methods, 42, 1673-1691.

Akinsete, A., Famoye, F. and Lee, C. (2008). The beta-Pareto distribution. Statistics, 42, 547-563.

Cramer, H. (1946). Mathematical Methods of Statistics, Princeton University Press, Prinnceton, N.J.

Gulati, S and Mi, J. (2006). Testing for scale families using total variation distance. Journal of Statistical Computation and Simulation, 76(9), 773-792.

Guo, H. and Krishnamoorthy, K. (2005). Comparison between two quantiles: Normal and Exponential cases. Communications in Statistics Simulation and Computation, 34, 243-252.

Jordan, S. M. and Krishnamoorthy, K. (1996). Exact confidence intervals for the common mean of several normal populations. Biometrics, 52, 77-86.

Krishnamoorthy, K., Mathew, T. and Ramchandran, G. (2006). Generalized p-values and confidence intervals: A Novel approach for analyzing log normally distributed exposure data. Journal of Occupational and Environmental Hygiene, 3,642-650.

Krishnamoorthy, K. and Mathew, T (2003). Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals. Journal of Statistical Planning and Inference, 115,103-121.

Krishnamoorthy, K. and Mathew, T. (2004). One-Sided tolerance limits in balanced and unbalanced one-way random models based on generalized confidence limits. Technometrics, 46, 44-52.

Krishnamoorthy, K., Mukherjee, S. and Guo, H. (2007). Inference on reliability in two-parameter exponential stress-strength model. Metrika, 65, 261 - 273.

Krishnamoorthy, K. and Lian, X.(2012). Closed-form approximate tolerance intervals for some general linear models and comparison studies. Journal of Statistical Computation and Simulation, 82, 547-563.

Kumbhar, R. R. and Shirke, D. T. (2004). Tolerance limits for lifetime distribution of k-Unit parallel system. Journal of Statistical Computation and Simulation, 74, 201-213.

Kurian, K. M., Mathew, T. and Sebastian, G. (2008). Generalized confidence intervals for process capability indices in the one-way random model. Metrika, 67, 83-92.

Liao, C.T., Lin, T.Y. and Iyer, H.K. (2005). One and two sided tolerance intervals for general balanced mixed models and unbalanced one-way random models. Technometrics, 47,323-335.

Mahmoudi, E. (2011). The beta generalized Pareto distribution with application to lifetime data. Mathematics and Computers in Simulation, 81, 2414-2430.

Ng, C.K. (2007). Performance of the three methods of the interval estimation of coefficient of variation. Interstat.

Raqab, M.Z., Madi, M.T. and Kundu, D. (2008). Estimation of P(Y

Raqab, M.Z. and Kundu, D. (2005). Comparison of different estimators of P[Y

Schroeder, B., Damouras, S. and Gill, P. (2010). Understanding latent sector error and how to protect against them. ACM Transactions on Storage, 6, Article8.

Suresh, R.P (2004). Estimation of location and scale parameters in a two parameter exponential distribution from a censored sample. Statistical Methods, 6(1), 82-89.

Suresh, R.P (1997). On approximate likelihood estimators in censored normal samples. Gujarat Statistical Review, 24, 21-28.

Surles, J.G. and Padgett, W.J.(1998). Inference for reliability and stress-strength for a scaled Burr-type X distribution. Lifetime Data Analysis, 7,187-200.

Surles, J.G. and Padgett, W.J.(1998). Inference for P(Y

Tiku, M.L. (1967). Estimating the mean and standard deviation from a censored sample. Biometrica, 54, 155-165.

Tiku, M.L. (1968). Estimating the parameters of normal and logistic distribution from censored samples. Australian Journal of Statistic, 10, 64-74.

Tiku, M.L. and Suresh, R.P. (1992). A new method of estimation for location and scale parameters, Journal of Statistical Planning and Inference, 30, 281-292.

Tiku, M.L., Tan, W.Y. and Balkrishnan, N. (1986). Robust Inference, Marvel Delker, Inc, New York.

Tsui, K. and Weerahandi, S. (1989). Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. Journal of American Statistical Association, 84,602–607.

Vaughan, V.C. (1992). On Tiku-Suresh Method of Estimation. Communications in Statistics Theory Methods, 21, 451-469.

Verrill, S. and Johnson, R.A. (2007). Confidence bounds and hypothesis tests for normal distribution coefficients of variation. Communications in Statistics, 36, 2187-2206.

Weerahandi, S. (1993). Generalized confidence intervals. Journal of American Statistical Association. 88, 899–905.

Weerahandi, S. (1995). Exact Statistical methods for Data Analysis. Springer, New York.

Weerahandi, S. and Johnson R. A. (1992). Testing reliability in a stress-strength model when X and Y are normally distributed. Technometrics, 34, 83–91.

Yu, P. L.H., Sun, Y. and Sinha, B. K. (1999). On exact confidence intervals for the common mean of several normal populations. Journal of Statistical Planning Inference, 81, 263-277.

Full Text: pdf

Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.