### Comparisons of Methods of Estimation for a New Pareto-type Distribution

#### Abstract

Bourguignon et al. (2016) introduced a new Pareto-type distribution to model income and reliability data. The aim of this paper is to estimate the parameters of this distribution from both frequentist and Bayesian view points. The maximum likelihood estimates, method of moment estimates, percentile estimates, least square and weighted least square estimates and maximum product of spacing estimates are considered as frequentist estimates. We have also considered the Bayes estimates of the unknown parameters and the associated credible intervals. The Bayes estimates are computed using an importance sampling method. To evaluate the performance of the different estimates, a Monte Carlo simulation study is carried out. Some real life data sets have been analyzed for illustrative purposes.

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M. V. AARSET (1987). How to identify bathtub hazard rate. IEEE Transactions on Reliability, 36, pp. 106–108.

M. ABDI, A. ASGHARZADEH, H. S. BAKOUCH, Z. ALIPOUR (2019). A new compound gamma and Lindley distribution with application to failure data. Austrian Journal of Statistics, 48, pp. 54–75.

M. R. ALKASABEH, M. Z. RAQAB (2009). Estimation of the generalized logistic distribution parameters: Comparative study. Statistical Methodology, 6, pp. 262–279.

B. C. ARNOLD (1983). Pareto Distributions. International Cooperative Publishing House, Fairland, Maryland, USA.

B. C. ARNOLD, S. J. PRESS (1983). Bayesian inference for Pareto populations. Journal of Econometrics, 21, pp. 287–306.

B. C. ARNOLD, S. J. PRESS (1989). Bayesian estimation and prediction for Pareto data. Journal of the American Statistical Association, 84, pp. 1079–1084.

A. ASGHARZADEH, R. REZAIE, M. ABDI (2011). Comparisons of methods of estimation for the half-logistic distribution. Selcuk Journal of Applied Mathematics, Special Issue, pp. 93–108.

H. S. BAKOUCH, S. DEY, P. L. RAMOS, F. LOUUZADA (2017). Binomial-exponential 2 distribution: Different estimation methods with weather applications. Trends in Applied and Computational Mathematics, 18, no. 2, pp. 233–251.

M. BOURGUIGNON, H. SAULO, R. N. FERNANDEZ (2016). A new Pareto-type distribution with applications in reliability and income data. Physica A, 457, pp. 166–175.

M. H. CHEN, Q. M. SHAO (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8, pp. 69–92.

R. C. H. CHENG, N. A. K. AMIN (1979). Maximum product of spacings estimation with applications to the lognormal distribution. Tech. rep., Department of Mathematics, University of Wales.

R. C. H. CHENG, N. A. K. AMIN (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society, Series B, no. 3, pp. 394–403.

C. DAGUM (2006). Wealth distribution models: Analysis and applications. Statistica, 66, no. 3, pp. 235–268.

R .DEMICHELI, G. BONADONNA,W. J. HRUSHESKY, M. W. RETSKY, P. VALAGUSSA (2004). Menopausal status dependence of the timing of breast cancer recurrence after surgical removal of the primary tumour. Breast Cancer Research, 6, no. 6, pp. 689–696.

S. DEY, T. DEY, D. KUNDU (2014). Two-parameter Rayleigh distribution: Different methods of estimation. American Journal of Mathematical and Management Sciences, 33, no. 1, pp. 55–74.

D. DYER (1981). Structural probability bounds for the strong Pareto laws. Canadian Journal of Statistics, 9, pp. 71–77.

R. D. GUPTA, D. KUNDU (2001). Generalized exponential distribution: Different methods of estimation. Journal of Statistical Computation and Simulation, 69, pp. 315–338.

N. L. JOHNSON, S. KOTZ, N. BALAKRISHNAN (1994). Continuous Univariate Distributions, vol. 1. Wiley, New York.

J. KAO (1959a). Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, 13, pp. 15–22.

J. KAO (1959b). A graphical estimation of mixed Weibull parameters in life testing electron tubes. Technometrics, 1, pp. 389–407.

D. KUNDU, M. Z. RAQAB (2005). Generalized Rayleigh distribution: Different methods of estimation. Computational Statistics and Data Analysis, 49, no. 1, pp. 187–200.

C. LAI, M. XIE (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.

T. LWIN (1972). Estimation of the tail of the Paretian law. Skand Aktuarietidskr, 55, pp. 170–178.

N. R. MANN, R. E. SCHAFER, N. D. SINGPURWALLA (1974). Methods for Statistical Analysis of Reliability and Life Data. Wiley, New York.

W. B. NELSON (1970). Statistical methods for accelerated life test data the inverse power law model. Technical report 71-c011, General Electric Company.

M. E. J. NEWMAN (2005). Pareto distributions and Zipfs law. Contemporary Physics, 46, no. 5, pp. 323–351.

V. PARETO (1964). Cours d’économie politique, vol. 1. Librairie Droz, Lausanne.

B. RANNEBY (1984). The maximum spacing method, an estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, 11, pp. 93–112.

P. G. SANKARAN, N. U. NAI, P. JOHN (2014). A family of bivariate Pareto distributions. Statistica, 74, no. 2, pp. 199–215.

J. SWAIN, S. VENKATRAMAN, J. WILSON (1988). Least squares estimation of distribution function in Johnson's translation system. Journal of Statistical Computation and Simulation, 29, pp. 271–297.

M. L. TIKU, A. D. AKKAYA (2004). Robust Estimation and Hypothesis Testing. New Age International (P) Limited, New Delhi.

L. ZANINETTI, M. FERRARO (2008). On the truncated Pareto distribution with applications. Central European Journal of Physics, 6, no. 1, pp. 1–6.

DOI: 10.6092/issn.1973-2201/9331