2020
19
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0
228
Accurate Inference for the Mean of the Poisson-Exponential Distribution
2
2
Although the random sum distribution has been well-studied in probability theory, inference for the mean of such distribution is very limited in the literature. In this paper, two approaches are proposed to obtain inference for the mean of the Poisson-Exponential distribution. Both proposed approaches require the log-likelihood function of the Poisson-Exponential distribution, but the exact form of the log-likelihood function is not available. An approximate form of the log-likelihood function is then derived by the saddlepoint method. Inference for the mean of the Poisson-Exponential distribution can either be obtained from the modified signed likelihood root statistic or from the Bartlett corrected likelihood ratio statistic. The explicit form of the modified signed likelihood root statistic is derived in this paper, and a systematic method to numerically approximate the Bartlett correction factor, hence the Bartlett corrected likelihood ratio statistic is proposed. Simulation studies show that both methods are extremely accurate even when the sample size is small.
1
19
Wei
Lin
Wei
Lin
Department of Mathematics and Statistics, Thompson Rivers University, 805 TRU Way, Kamloops, British Columbia, Canada V2C 0C8.
becky@utstat.toronto.edu
Xiang
Li
Xiang
Li
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3.
lovedaershou@126.com
Augustine
Wong
Augustine
Wong
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3.
august@yorku.ca
Bartlett Correction
p-value Punction
Saddlepoint Approximation
Signed Likelihood Root.
[Alya, A. M., and Low, H. C. (2013), Saddlepoint approximation to cumulative distribution function for poisson-exponential distribution. Modern Applied Science, 7, 26-32.##Barbour, A. D., and Chryssaphinou, O. (2001), Compound poisson approximation: A user’s guide. The Annals of Applied Probability, 11, 964–1002.##Barbour, A. D., Johnson, O., Kontoyiannis, I., and Madiman, M. (2010), Compound poisson approximation via information functionals. Electron. J. Prob., 15, 1344–1369.##Barndorff-Nielsen, O. E. (1986), Inference on full and partial parameters based on the standardized signed log likelihood ratio. Biometrika, 73, 307–322.##Barndorff-Nielsen, O. E. (1991), Modified signed log likelihood ratio. Biometrika, 78, 557–564.##Bartlett, M. S. (1937), Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London, 160, 268–282.##Bero, R. (2003), Ckerstan’s method for compound poisson approximation. The Annals of Probability, 32,1754–1771.##Daniels, H. E. (1954), Saddlepoint approximations in statistics. Annals of Mathematical Statistics, 25, 631–650.##Daniels, H. E. (1987), Tail probability approximations. International Statistical Review, 55, 37–48.##Fraser, D. A. S. (2017), p-values: The insight to modern statistical inference. Annual Review of Statistics and Its Application, 4, 1–14.##Fraser, D. A. S., and Reid, N. (1995), Ancillaries and third-order significance. Utilitas Mathematica, 7, 33–53.##Fraser, D. A. S., Reid, N., and Wu, J. (1999), A simple general formula for tail probabilities for frequentist and bayesian inference. Biometrika, 86, 249–264.##Ghribi, A. and Masmoudi, A. (2013), Acompound poisson model for learning discrete bayesian networks. Acta Mathematica Scientia, 33, 1767–1784.##Kalbfleisch, J. G. (1985), Probability and Statistical Inference Volumne 2: Statistical Inference (2nd Edition). Springer-Verlag, New York.##Lugannani, R., and Rice, S. (1980), Saddlepoint approximation for the distribution of sums of random variables. Advances in Applied Probability, 12, 475–490.##Murphy, S. A., and Van der Vaart, A. M. (2000), On profile likelihood. Journal of the American Statistical Association, 95, 449–465.##Reid, N. (2010), Likelihood inference. Wiley Interdisciplinary Reviews: Computational Statistics, 2(5), 517–525.##Thmazella , V. L. D., Cancho, V. G., and Louzada V. (2013), Bayesian reference analysis for the poisson-exponential lifetime distribution. Chilean Journal of Statistics, 4, 99–113.## ##]
Jackknifed Liu-type Estimator in Poisson Regression Model
2
2
The Liu estimator has consistently been demonstrated to be an attractive shrinkage method for reducing the effects of multicollinearity. The Poisson regression model is a well-known model in applications when the response variable consists of count data. However, it is known that multicollinearity negatively affects the variance of the maximum likelihood estimator (MLE) of the Poisson regression coefficients. To address this problem, a Poisson Liu estimator has been proposed by numerous researchers. In this paper, a Jackknifed Liu-type Poisson estimator (JPLTE) is proposed and derived. The idea behind the JPLTE is to decrease the shrinkage parameter and, therefore, improve the resultant estimator by reducing the amount of bias. Our Monte Carlo simulation results suggest that the JPLTE estimator can bring significant improvements relative to other existing estimators. In addition, the results of a real application demonstrate that the JPLTE estimator outperforms both the Poisson Liu estimator and the maximum likelihood estimator in terms of predictive performance.
21
37
Ahmed
Alkhateeb
Ahmed
Alkhateeb
Department of Operation Research and Intelligent Techniques, University of Mosul, Mosul, Iraq
ahmednazih17@gmail.com
Zakariya
Algamal
Zakariya
Algamal
Department of Statistics and Informatics, College of Computer science & Mathematics, University of Mosul, Mosul, Iraq
zakariya.algamal@uomosul.edu.iq
Multicollinearity
Liu Estimator
Poisson Regression Model
Shrinkage
Monte Carlo Simulation.
[Algamal, Z. Y. (2012), Diagnostic in poisson regression models. Electronic Journal of Applied Statistical Analysis, 5(2), 178-186.##Algamal, Z. Y. (2018), Biased estimators in Poisson regression model in the presence of multicollinearity: A subject review. Al-Qadisiyah Journal for Administrative and Economic Sciences, 20(1), 37-43.##Algamal, Z. Y.(2018), A new method for choosing the biasing parameter in ridge estimator for generalized linear model. Chemometrics and Intelligent Laboratory Systems, Elsevier, 183, 96-101.##Algamal, Z. Y. and Alanaz, M. M.(2018), Proposed methods in estimating the ridge regression parameter in Poisson regression model. Electronic Journal of Applied Statistical Analysis, 11, 506-515.##Arashi, M., Kibria, B. G., Norouzirad, M. and Nadarajah, S. (2014), Improved preliminary test and Stein-rule Liu estimators for the ill-conditioned elliptical linear regression model. Journal of Multivariate Analysis, Elsevier, 126, 53-74.##Arashi, M., Nadarajah, S. and Akdeniz, F. (2017), The distribution of the Liu-type estimator of the biasing parameter in elliptically contoured models. Communications in Statistics-Theory and Methods, Taylor and Francis, 46, 3829-3837.##Arashi, M., Norouzirad, M., Ahmed, S. E. and Yüzba and scedil, and imath, B. (2018), Rank-based Liu regression. Computational Statistics, Springer, 33, 1525-1561.##Asar, Y., and Genç, A. (2017), A New Two-Parameter Estimator for the Poisson Regression Model. Iranian Journal of Science and Technology, Transactions A: Science, 42(2), 793-803.##Batah, F. S. M., Ramanathan, T. V., and Gore, S. D. (2008), The efficiency of modified jackknife and ridge type regression estimators: a comparison. Surveys in Mathematics and its Applications, 3, 111-122.##Cameron, A. C., and Trivedi, P. K. (2013), Regression analysis of count data. Cambridge university press, 53.##De Jong, P., and Heller, G. Z. (2008), Generalized linear models for insurance data. Cambridge University Press Cambridge, 10.##Akdeniz Duran, E., and Akdeniz, F. (2012), Efficiency of the modified jackknifed Liutype estimator. Statistical Papers, 53(2), 265-280.##Hinkley, D.V. (1977), Jackknifing in unbalanced situations. Technometrics, 19(3), 285-292.##Hoerl, A. E., and Kennard, R. W. (1970), Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1), 55-67.##Karbalaee, Mohammad H and Tabatabaey, Seyed Mohammad M and Arashi, Mohammad, (2019), On the preliminary test generalized Liu estimator with series of stochastic restrictions. Journal of The Iranian Statistical Society, 18(1), 113-131.##Karbalaee, M. H., Tabatabaey, S. M. M. and Arashi, M. (2019), On the Preliminary Test Generalized Liu Estimator with Series of Stochastic Restrictions. Journal of The Iranian Statistical Society, 18, 113-131.##Liu, K. (1993), A new class of blased estimate in linear regression. Communications in Statistics-Theory and Methods, 22(2), 393-402.##Kibria, B. M. G. (2003), Performance of some new ridge regression estimators. Communications in Statistics - Simulation and Computation, 32(2), 419-435.##Kibria, B. G., Mansson, K., and Shukur, G. (2015), A simulation study of some biasing parameters for the ridge type estimation of Poisson regression. Communications in Statistics-Simulation and Computation, 44(4), 943-957.##Liu, K. (2003), Using Liu-type estimator to combat collinearity. Communications in Statistics-Theory and Methods , 32(5), 1009-1020.##Mansson, K., Kibria, B. G., Sjolander, P., and Shukur, G. (2012), Improved Liu estimators for the Poisson regression model. International Journal of Statistics and Probability, 1(1), 2.##Månsson, K., and Shukur, G. (2011), A Poisson ridge regression estimator. Economic Modelling, 28(4), 1475-1481 .##Montgomery, D. C., Peck, E. A., and Vining, G. G. (2015), Introduction to linear regression analysis. New York: John Wiley and Sons.##Nyquist, H. (1988), Applications of the jackknife procedure in ridge regression. Computational Statistics and Data Analysis, 6(2), 177-183.##Quenouille, M. H. (1956), Notes on bias in estimation. Biometrika, 43(3/4), 353-360.##Rashad, N. K. and Algamal, Z. Y. (2019), A New Ridge Estimator for the Poisson Regression Model. Iranian Journal of Science and Technology, Transactions A: Science, Springer., 43, 2921-2928.##Singh, B., Chaubey, Y., and Dwivedi, T. (1986), An almost unbiased ridge estimator. Sankhy¯a: The Indian Journal of Statistics, Series B, 342-346.##Tukey, J. (1958), Bias and confidence in not quite large samples. Ann. Math. Statist., 29(614).##Türkan, S., and Özel, G. (2015), A new modified Jackknifed estimator for the Poisson regression model. Journal of Applied Statistics, 43(10), 1892-1905.##Türkan, S., and Özel, G. (2017), A Jackknifed estimators for the Negative Binomial regression model. Communications in Statistics - Simulation and Computation, 47(6), 1845-1865.##Yıldız, N. (2018), On the performance of the Jackknifed Liu-type estimator in linear regression model. Communications in Statistics-Theory and Methods, 47(9), 2278-2290.## ##]
Bounds for CDFs of Order Statistics Arising from INID Random Variables
2
2
In recent decades, studying order statistics arising from independent and not necessary identically distributed (INID) random variables has been a main concern for researchers. A cumulative distribution function (CDF) of these random variables (Fi:n) is a complex manipulating, long time consuming and a software-intensive tool that takes more and more times. Therefore, obtaining approximations and boundaries for Fi:n and other theoretical properties of these variables, such as moments, quantiles, characteristic function, and some related probabilities, has always been a main chal- lenge. Recently, Bayramoglu (2018) provided a new definition of ordering, by point to point ordering Fi’s (D-order) and showed that these new functions are CDFs and also, the corresponding random variables are independent. Thus, he suggested new CDFs (F[i]) that can be used as an alternative of Fi:n. Now with using, just F[1], and F[n], we have found the upper and lower bounds of Fi:n. Furthermore, specially a precisely approximation for F1:n and Fn:n (F1;n:n). Also in many cases approximations for other CDFs are derived. In addition, we compare approximated function with those oered by Bayramoglu and it is shown that our results of these proposed functions are far better than D-order functions.
39
57
جابر
کاظم پور
Jaber
Kazempoor
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
kazempoor.jaber@mail.um.ac.ir
آرزو
حبیبی راد
Arezou
Habibirad
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
ahabibi@um.ac.ir
خیرالله
اخلی
Kheirolah
Okhli
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.
kh.okhli@mail.um.ac.ir
Approximation
bounds
cumulative distribution function
independent and not necessary identically distributed
order statistics.
[Ahsanullah, M., Nevzorov, V. B., and Shakil, M. (2013), An introduction to order statistics.##Arnold, B. C. and Balakrishnan, N. (2012), Relations, bounds and approximations for order statistics, Springer Science & Business Media, volume 53.##Arnold, B. C., Balakrishnan, N., and Nagaraja, H. N. (1992), A first course in order statistics, volume 54, Siam.##Bairamov, I. and Tavangar, M. (2015), Residual lifetimes of k-out-of-n systems with exchangeable components. Journal of The Iranian Statistical Society, 14(1), 63–87.##Balakrishnan, N., Bendre, S., and Malik, H. (1992), General relations and identities for order statistics from non-independent non-identical variables. Annals of the Institute of Statistical Mathematics, 44(1), 177–183.##Balakrishnan, N. and Sultan, K. (1998), 7 recurrence relations and identities for moments of order statistics. Handbook of Statistics, 16, 149–228.##Bayramoglu, I. (2018), A note on the ordering of distribution functions of inid random variables. Journal of Computational and Applied Mathematics, 343, 49–54.##David, H. A. and Nagaraja, H. N. (2004), Order statistics. Encyclopedia of Statistical Sciences.##Reiss, R.-D. (2012), Approximate distributions of order statistics: with applications to nonparametric statistics. Springer Science & Business Media.##Team, R. C. (2018), R: A language and environment for statistical computing.## ##]
A New Proof of FDR Control Based on Forward Filtration
2
2
For multiple testing problems, Benjamini and Hochberg (1995) proposed the false discovery rate (FDR) as an alternative to the family-wise error rate (FWER). Since then, researchers have provided many proofs to control the FDR under different assumptions. Storey et al. (2004) showed that the rejection threshold of a BH step-up procedure is a stopping time with respect to the reverse filtration generated by the p-values and proposed a new proof based on the martingale theory. Following this work, martingale methods have been widely used to establish FDR control in various settings, but have been primarily applied to reverse filtration only. However, forward filtration can be more amenable for generalized and adaptive FDR controlling procedures. In this paper, we present a new proof, based on forward filtration, for step-down FDR controlling procedures that start from small p-values and update the rejection regions as larger p-values are observed.
59
68
Ahmad
Ehyaei
Ahmad
Ehyaei
Sharif University of Technology, Tehran, Iran.
ahmad.ehyaei@gmail.com
Kasra
Alishahi
Kasra
Alishahi
Sharif University of Technology, Tehran, Iran.
alishahi@sharif.ir
Ali
Shojaei
Ali
Shojaei
University of Washington, USA.
ashojaie@uw.edu
Benjamini-Hochberg
False Discovery Rate
Step-down Methods
[Benditkis, J., P. Heesen, and A. Janssen (2018). The false discovery rate (fdr) of multiple tests in a class room lecture. Statistics & Probability Letters, 134, 29–35.##Benditkis, J. and A. Janssen (2017). Finite sample bounds for expected number of false rejections under martingale dependence with applications to FDR. Electronic Journal of Statistics, 11(1), 1827–1857.##Benjamini, Y. and Y. Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. Series B. Methodological, 57(1), 289–300.##Benjamini, Y. and W. Liu (1999). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence. Journal of Statistical Planning and Inference, 82(1), 163–170.##Benjamini, Y. and D. Yekutieli (2001). The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29(4), 1165–1188.##Blanchard, G. and É. Roquain (2009). Adaptive false discovery rate control under independence and dependence. Journal of Machine Learning Research, 10(Dec), 2837–2871.##Cai, T. T. and W. Sun (2009). Simultaneous testing of grouped hypotheses: Finding needles in multiple haystacks. Journal of the American Statistical Association, 104(488), 1467–1481.##Heesen, P., A. Janssen, et al. (2015). Inequalities for the false discovery rate (fdr) under dependence. Electronic Journal of Statistics, 9(1), 679–716.##Klenke, A. (2013). Probability Theory: A Comprehensive Course. Universitext. Springer London.##Li, A. and R. F. Barber (2019). Multiple testing with the structure-adaptive benjamini– hochberg algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81(1), 45–74.##Peña, E. A., J. D. Habiger, and W. Wu (2011). Power-enhanced multiple decision functions controlling family-wise error and false discovery rates. Annals of statistics, 39(1), 556.##Ploner, A., S. Calza, A. Gusnanto, and Y. Pawitan (2006). Multidimensional local false discovery rate for microarray studies. Bioinformatics, 22(5), 556–565.##Ramdas, A., J. Chen, M. J. Wainwright, and M. I. Jordan (2017). Dagger: A sequential algorithm for fdr control on dags. arXiv preprint arXiv:1709.10250.##Romano, J. P., A. M. Shaikh, and M. Wolf (2008). Control of the false discovery rate under dependence using the bootstrap and subsampling. Test, 17(3), 417.##Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. The Annals of Statistics, 30(1), 239–257.##Storey, J. D., J. E. Taylor, and D. Siegmund (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(1), 187–205.##Sun, W. and T. T. Cai (2007). Oracle and adaptive compound decision rules for false discovery rate control. Journal of the American Statistical Association, 102(479), 901–912.## ##]
On Conditional Inactivity Time of Failed Components in an (n-k+1)-out-of-n System with Nonidentical Independent Components
2
2
In this paper, we study an (n-k+1)-out-of-n system by adopting their components to be statistically independent though nonidentically distributed. By assuming that at least m components at a fixed time have failed while the system is still working, we obtain the mixture representation of survival function for a quantity called the conditional inactivity time of failed components in the system. Moreover, this quantity for (n-k+1)-out-of-n system, in one sample with respect to k and m and in two samples, are stochastically compared.
69
83
Farkhondeh Alsadat
Sajadi
Farkhondeh Alsadat
Sajadi
Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran.
f.sajadi@sci.ui.ac.ir
Mohammad Hossein
Poursaeed
Mohammad Hossein
Poursaeed
Department of Statistics, Lorestan University, Khoramabad, Iran.
poursaeed.m@lu.ac.ir
Sareh
Goli
Sareh
Goli
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.
s.goli@cc.iut.ac.ir
Conditional Inactivity Time
Nonidentical Components
Residual Lifetime.
[Asadi , M. ( 2006 ), On the mean past lifetime of the components of a parallel system. Journal of Statistical Planning and Inference, 136( 4 ), 1197-1206 .##Eryılmaz, S. (2013), On residual lifetime of coherent systems after the r failure. Statistical Papers, 54, 243-250.##Gurler, S. and Bairamov, I. (2009), Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions. Applied Mathematical Modelling, 33, 1116-1125.##Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999), The “signature” of a coherent system and its application to comparison among systems. Naval Research Logistics, 46, 507-523.##Kochar, S. and Xu, M. (2010), On residual lifetimes of k-out-of-n systems with nonidentical components. Probability in the Engineering and Informational Sciences, 24, 109-127.##Khaledi, B. E. and Shaked, M. (2007), Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference, 137, 1173-1184.##Li, X. and Zhao, P. ( 2006), Some aging properties of the residual life of k-out-of-n systems. IEEE Transactions on Reliability, 55, 535-541.##Li, X. and Zhao, P. ( 2008), Stochastic comparison on general inactivity time and general residual life of k-out-of-n system. Communication in Statistics- Simulation and Computation, 37, 1005-1019.##Li, X. and Zhang, Z. (2008), Some stochastic comparisons of conditional coherent systems. Applied Stochastic Models in Business and Industry, 24, 541-549.##Navarro, J., Ruiz,J. M. and Sandoval, C. J. (2005), A note on comparisons among coherent systems with dependent components using signatures. Statistics and Probability Letters, 72, 179-185.##Navarro, J., Aguila, Y., Sordo, M. A. and Suarez-Llorens, A. (2013), Stochastic ordering properties for systems with dependent identically distributed components. Applied Stochastic Models in Business and Industry, 29, 264-278.##Navarro, J., Samaniego, F.J. and Balakrishnan, N. (2013a), Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components. Advances in Applied Probability, 45, 1011-1027.##Sadegh, M. K. (2008), Mean past and mean residual Life of parallel system with nonidentical components. Communications in Statistics Theory and Methods, 37, 1137-1145.##Salehi, E. T. , Asadi, M. and Eryılmaz, S. (2012), On the mean residual lifetime of consecutive k-out-of-n systems. Test, 21, 93-115.##Salehi, E. and Tavangar M. (2019), Stochastic comparisons on conditional residual lifetime and inactivity time of coherent systems with exchangeable components, Statistics and Probability Letters, 145, 327-337.##Tavangar, M. (2016), Conditional inactivity time of components in a coherent operating system. IEEE Transactions on Reliability, 65, 359-369.##Tavangar, M. and Asadi, M. (2010) , A study on the mean past lifetime of the components of (n - k + 1)-out-of-n system at the system level. Metrika, 72(1), 59-73.##Zhang, Z. (2010), Ordering conditional general coherent systems with exchangeable components. Journal of Statistical Planning and Inference. 140, 454-460.##Zhao, P. Li, X. and Balakrishnan, N. (2008), Conditional ordering of $k$-out-of-$n$ systems with independent but nonidentical components. Journal of Applied Probability, 45, 1113-1125.## ##]
Sequential-Based Approach for Estimating the Stress-Strength Reliability Parameter for Exponential Distribution
2
2
In this paper, two-stage and purely sequential estimation procedures are considered to construct fixed-width confidence intervals for the reliability parameter under the stress-strength model when the stress and strength are independent exponential random variables with different scale parameters. The exact distribution of the stopping rule under the purely sequential procedure is approximated using the law of large numbers and Monte Carlo integration. For the two-stage sequential procedure, explicit formulas for the distribution of the total sample size, the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress-strength model are provided. Moreover, it is shown that both proposed sequential procedures are finite, and in exceptional cases, the exact distribution of stopping times is degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally using real data, the procedures are clearly illustrated.
85
120
Ashkan
Khalifeh
Ashkan
Khalifeh
Department of Statistics, Yazd University, Yazd, Iran.
khalifeh68@yahoo.com
Eisa
Mahmoudi
Eisa
Mahmoudi
Department of Statistics, Yazd University, Yazd, Iran.
emahmoudi@yazd.ac.ir
Ali
Dolati
Ali
Dolati
Department of Statistics, Yazd University, Yazd, Iran.
adolati@yazd.ac.ir
Law of Large Numbers
Purely Sequential Sampling
Stopping Rule
Two-stage Sequential Sampling.
[Awad, A. M., Azzam, M. M., and Hamdan, M. A. (1981). Some inference results on Pr(X < Y) in the bivariate exponential model. Communications in Statistics - Theory and Methods, 10, 2515-2525.##Bandyopadhyay, U., Das, R., and Biswas, A. (2003). Fixed width confidence interval of P(X < Y) in partial sequential sampling scheme. Sequential Analysis, 22, 75-93.##Bapat, S. R. (2018). Purely sequential fixed accuracy confidence intervals for P(X < Y) under bivariate exponential models. American Journal of Mathematical and Management Sciences, 37, 386-400.##Beg, M. A. (1980). On the estimation of Pr(Y < X) for the two-parameter exponential distribution. Metrika, 27, 29-34.##Birnbaum, Z. W. (1956). On a use of the mann-whitney statistic. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Ed. J. Neyman. Berkeley: University of California, pp 13-17.##Birnbaum, Z.W. and McCarty, R. C. (1958).Adistribution-free upper confidence bound for P(Y < X), based on independent samples of X and Y. The Annals of Mathematical Statistics, 29, 558-562.##Chao, A. (1982). On comparing estimators of Pr(Y < X) in the exponential case. IEEE Transactions on Reliability, 31, 389-392.##Chiodo, E. (2014). Model robustness analysis of a bayes stress-strength reliability estimation with limited data. In 2014 International Symposium on Power Electronics, Electrical Drives, Automation and Motion. New York: IEEE, pp 1140-1145.##Cramer, E. and Kamps, U. (1997). The umvue of P(X < Y) based on type-II censored samples from weinman multivariate exponential distributions. Metrika, 46, 93-93.##Domma, F. and Giordano, S. (2013). A copula-based approach to account for dependence in stress-strength models. Statistical Papers, 54, 807-826.##Enis, P. and Geisser, S. (1971). Estimation of the probability that Y < X. Journal of the American Statistical Association, 66, 162-168.##Ferguson, T. (1996). A course in large sample theory. New York: Chapman and Hall.##Freund, J. E. (1961). A bivariate extension of the exponential distribution. Journal of the American Statistical Association, 56, 971-977.##Govindarajulu, Z. (1974). Fixed-width confidence intervals for P(X < Y). In Reliability and biometry: statistical analysis of lifelength, Eds. E. F. Proschan and R. J. Serfling. Philadelphia: SIAM, pp 747-757.##Govindarajulu, Z. (2004). Sequential Estimation. Singapore: World Scientific. Johnson, R. A. (1988). 3 stress-strength models for reliability. In Quality Control and Reliability, Eds. P.R. Krishnaiah and C.R. Rao. Amsterdam: Elsevier, pp 27-54.##Kao, E. (1997). An Introduction to Stochastic Processes. Scituate :Duxbury.##Kelley, G. D., Kelley, J. A., and Schucany, W. R. (1976). Ecient estimation of P(Y < X) in the exponential case. Technometrics, 18, 359-360.##Kotz, S., Lumel’skii, I., and Pensky, M. (2003). The Stress-strength Model and Its Generalizations: Theory and Applications. Singapore: World Scientific.##Kundu, D. and Gupta, R. D. (2006). Estimation of P(Y < X) for weibull distributions. IEEE Transactions on Reliability, 55, 270-280.##Lim, D. L., Isogai, E., and Uno, C. (2004). Two-sample fixed width confidence intervals for a function of exponential scale parameters. Far East Journal of Theoretical Statistics, 14, 215-227.##Mahmoudi, E., Khalifeh, A., and Nekoukhou, V. (2018). Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution. Sequential Analysis. doi: 10.1080/07474946.2019.1649347.##Marshall, A. W. and Olkin, I. (1967). A generalized bivariate exponential distribution. Journal of Applied Probability, 4, 291-302.##Mirjalili, M., Torabi, H., Nadeb, H., and Bafekri. F., S. (2016). Stress-strength reliability of exponential distribution based on type-I progressively hybrid censored samples. Journal of Statistical Research of Iran, 13, 89-105.##Mukhopadhyay, N. and Zhuang, Y. (2016). On fixed-accuracy and bounded accuracy confidence interval estimation problems in fisher’s "nile" example. Sequential Analysis, 35, 516-535.##Nadarajah, S. and Kotz, S. (2006). Reliability for some bivariate exponential distributions. Mathematical Problems in Engineering, 2006, 1-14.##Patowary, A. N., Hazarika, J., and Sriwastav, G. L. (2013). Interference theory of reliability: a review. International Journal of System Assurance Engineering and Management, 4, 146-158.##Sathe, Y. and Shah, S. (1981). On estimating P(X > Y) for the exponential distribution. Communications in Statistics - Theory and Methods, 10, 39-47.##Stein, C. (1945). A two-sample test for a linear hypothesis whose power is independent of the variance. The Annals of Mathematical Statistics, 16, 243-258.##Tong, H. (1974). A note on the estimation of Pr(Y < X) in the exponential case. Technometrics, 16, 625-625.##Xia, Z., Yu, J., Cheng, L., Liu, L., and Wang, W. (2009). Study on the breaking strength of jute fibres using modified weibull distribution. Composites Part A: Applied Science and Manufacturing, 40, 54-59.## ##]
The Weibull Topp-Leone Generated Family of Distributions: Statistical Properties and Applications
خانواده توزیع وایبل تاپ-لئون تعمیم یافته: ویژگی های آماری و کاربردها
2
1
توزیع های آماری در توصیف و پیش بینی پدیده های دنیای واقعی بسیار مفید هستند. انتخاب توزیع آماری مناسب برای مدل سازی داده ها بسیار مهم است. در این مقاله، یک کلاس جدید از توزیع های طول عمر پیشنهاد شده به نام خانواده وایبل تاپ-لئون تعمیم یافته (WTLG) پیشنهاد داده می شود. خانواده پیشنهاد شده از توزیع ها با ترکیب توزیع وایبل با توزیع تاپ- لئون ساخته می شود که می تواند انعطاف بیشتری نسبت به توزیع های طول عمر شناخته شده، داشته باشد. همچنین چندین خواص آماری این خانواده از جمله چگالی و تابع نرخ خطر، رفتار مجانبی، نمایش آمیخته، چولگی و کشیدگی، گشتاورها، تابع مولد گشتاور بیان شده است. برای برآورد پارامترهای آن از روش های مختلف استفاده شده است. مقایسه عملکرد برآوردگرهای پیشنهاد شده در عمل به صورت عددی بررسی شده است. به علاوه آزمون نسبت درسنمایی ماکسیمم را برای این خانواده انجام شده است. در ادامه به کمک شبیه سازی عملکرد برآوردگر درستنمایی ماکزیمم را با محاسبه اریبی و میانگین توان دوم خطا مورد بررسی قرار داده می شود. در انتها بر پایه ی دو نوع داده واقعی انعطاف پذیری خانواده توزیع های هدف نشان داده شده است.
2
Statistical distributions are very useful in describing and predicting real world phenomena. Consequently, the choice of the most suitable statistical distribution for modeling given data is very important. In this paper, we propose a new class of lifetime distributions called the Weibull Topp-Leone Generated (WTLG) family. The proposed family is constructed via compounding the Weibull and the Topp-Leone distributions. It can provide better fits and is very flexible in comparison with the various known lifetime distributions. Several general statistical properties of the WTLG family are studied in details including density and hazard shapes, limit behavior, mixture representation, skewness and kurtosis, moments, moment generating function, incomplete moment. Different methods have been used to estimate its parameters. The performances of the estimators are numerically investigated. We have discussed inference on the new family based on the likelihood ratio statistics for testing some lifetime distributions. We assess the performance of the maximum likelihood estimators in terms of the biases and mean squared errors by means of a simulation study. The importance and flexibility of the new family are illustrated by means of two applications to real data sets.
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161
حمید
کرمی کبیر
Hamid
Karamikabir
Department of Statistics, Persian Gulf University, Bushehr, Iran
h_karamikabir@yahoo.com
محمود
افشاری
Mahmoud
Afshari
Department of Statistics, Persian Gulf University, Bushehr, Iran
afshar@pgu.ac.ir
هیثم
یوسف
Haitham M.
Yousof
Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt
haitham.yousof@fcom.bu.edu.eg
مراد
علیزاده
Morad
Alizadeh
Department of Statistics, Persian Gulf University, Bushehr, Iran
moradalizadeh78@gmail.com
غلامحسین
همدانی
Gholamhossien
Hamedani
Department of Mathematics, Statistics and Computer Science , Marquette University, USA
g.hamedani@mu.edu
Generating Function
Lifetime Distributions
Maximum Likelihood Estimation
Quantile Function
Topp-Leone Distribution
Weibull Distribution.
تابع مولد
توزیع های طول عمر
برآورد ماکسیمم درستنمایی
توزیع تاپ-لئون
توزیع وایبل.
[Afify, A. Z., Yousof, H. M. and Nadarajah, S. (2017), The beta transmuted-H family of distributions: properties and applications. Stasistics and its Inference, 10, 505-520.##Alizadeh, M., Ghosh, I., Yousof, H. M., Rasekhi, M. and Hamedani G. G. (2017), The generalized odd generalized exponential family of distributions: properties, characterizations and applications. Journal of Data Science, 16, 443-446.##Alizadeh, M., Korkmaz, M. C., Almamy, J. A. and Ahmed, A. A. E. (2018), Another odd log-logistic logarithmic class of continuous distributions. Journal of Statisticians: Statistics and Actuarial Sciences, 11(2), 55-72.##Anderson, T. W. and Darling, D. A. (1952), Asymptotic theory of certain" goodness of fit" criteria based on stochastic processes. The annals of mathematical statistics, 193-212.##Alzaatreh, A., Lee, C. and Famoye, F. (2013), A new method for generating families of continuous distributions. Metron, 71, 63-79.##Alzaatreh, A., Famoye, F. and Lee, C. (2014), The gamma-normal distribution: Properties and applications. Computational Statistics and Data Analysis, 69, 67-80.##Brito, E., Cordeiro, G. M., Yousof, H. M., Alizadeh, M. and Silva , G. O. (2017), Topp-Leone odd log-logistic family of distribution. Journal of Statistical Computation and Simulation, 87(15), 3040-3058.##Bourguignon, Silva M., R. B. and Cordeiro, G. M. (2014), The Weibull-G Family of Probability Distributions. Journal of Data Science, 12, 53-68.##Choi, K. and Bulgren,W. (1968), Anestimation procedure for mix- tures of distributions. Journal of the Royal Statistical Society. Series B (Methodological), 444-460.##Cooray, K. (2006), Generalization of the Weibull distribution: the odd Weibull family. Statistical Modelling, 6, 265-277.##Cooray, K. and Ananda, M. M. (2008), A generalization of the half-normal distribution with applications to lifetime data. Communications in Statistics-Theory and Methods, 37(9), 1323-1337.##Cordeiro, G. M., Ortega, E. M. M. and Nadarajah, S. (2010), The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 347, 1399-1429.##Cordeiro, G. M., Ortega, E. M. and da Cunha, D. C. C. (2013), The exponentiated generalized class of distributions. Journal of Data Science, 11, 1-27.##Dey. S., Mazucheli, J. and Nadarajah.S. (2017), Kumaraswamy distribution: different methods of estimation. Computational and Applied Mathematics, 1-18.##Eugene, N., Lee, C., and Famoye, F. (2002), Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31, 497-512.##Famoye, F., Lee, C. and Olumolade, O. (2005), The beta-Weibull distribution. Journal of Statistical Theory and Applications, 4(2), 121-136.##Fonseca, M. B. and Franca, M. G. C. (2007), A influoencia da fertilidade do solo e caracterizaca da fixacao biologica de N2 para o crescimento de Dimorphandra wilsonii rizz. Master thesis, Universidade Federal de Minas Gerais. ##Glänzell, W. (1990), Some consequences of a characterization theorem based on truncated moments. Statistics, 21(4), 613-618.##Gupta, R. D. and Kundu, D. (2001), Exponentiated exponential family: an alternative to gamma and Weibull. Biometrical Journal, 43, 117-130.##Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998), Modeling failure time data by Lehmann alternatives.Communications in Statistics - Theory and Methods, 27, 887-904.##Hamedani, G. G. (2013), On certain generalized gamma convolution distributions II (No. 484). Technical Report.##Hamedani, G. G., Altun, E., Korkmaz, M. C., Yousof, H. M. and Butt, N. S. (2018), A new extended G family of continuous distributions with mathematical properties, characterizations and regression modeling. Pakistan Journal of Statistics and Operation##Research, 14(3), 737-758.##Hashimoto, E. M, Ortega, E. M. M., Cordeiro, G. M. and Pascoa, M. A. R. (2015), The McDonald Extended Weibull Distribution. Journal of Statistical Theory and Practice, 9(3), 608-632.##Korkmaz, M. C. and Genc, A. I. (2017), A new generalized two-sided class of distributions with an emphasis on two- sided generalized normal distribution. Communications in Statistics-Simulation and Computation, 46(2), 1441-1460.##Korkmaz, M. C., Yousof, H. M. and Hamedani, G. G. (2018a), The Exponential Lindley Odd Log-Logistic-G Family: Properties, Characterizations and Applications. Journal of Statistical Theory and Applications, 17(3), 554-571.##Korkmaz, M. C., Yousof H. M., Hamedani, G. G. and Ali M. M. (2018b), The Marshall-Olkin Generalized G Poisson Family Of Distributions. Pakistan Journal of Statistics, 34(3), 251-267.##Korkmaz, M. C., Alizadeh, M., Yousof, H. M. and Butt, N. S. (2018c), The generalized oddWeibull generated family of distributions: statistical properties and applications. Pakistan Journal of Statistics and Operation Research, 14(3), 541-556.##Korkmaz, M. C. (2019a), A new family of the continuous distributions: the extended Weibull-G family. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68(1), 248-270.##Korkmaz, M. C., Cordeiro, G. M., Yousof, H. M., Pescim, R. R., Afify, A. Z., and Nadarajah, S. (2019b), The Weibull Marshall-Olkin family: Regression model and application to censored data. Communications in Statistics-Theory and Methods Accepted, DOI:10.1080/03610926.2018.1490430.##Lindley, D. V. (1958), Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B, 20, 102-107.##Marshall, A.W. and Olkin, I. (1997), A new methods for adding a parameter to a family of distributions with application to the Exponential andWeibull families. Biometrika, 84, 641-652.##Mudholkar, G. S. and Srivastava, D. K. (1993), Exponentiated Weibull family for analysing bathtub failure rate data. IEEE Transactions on Reliability, 42, 299-302.##Nadarajah, S., Cordeiro, G. M., and Ortega, E. M. M., (2014), The Zografos-Balakrishnan-G Family of Distributions: Mathematical Properties and Applications. Communications in Statistics - Theory and Methods, 44, 186-215.##Nofal, Z. M., Afify, A. Z., Yousof, H. M. and Cordeiro, G. M. (2017), The generalized transmuted-G family of distributions. Communications in Statistics - Theory and Methods, 46, 4119-4136.##Pogany, T. K., Saboor, A. and Provost, S., (2015), The Marshall Olkin Exponential Weibull Distribution. Hacettepe Journal of Mathematics and Statistics, 44(6), 1579-1594.##Silva, R. B., Bourguignon, M., Dias, C. R. B. and Cordeiro, G. M. (2013), The compound class of extendedWeibull power series distributions. Computational Statistics andData Analysis, 58, 352-367.##Swain, J. J., Venkatraman, S., andWilson, J. R., (1988), Least- squares estimation of distribution functions in johnson’s translation system. Journal of Statistical Computation and Simulation, 29, 271- 297.##Yousof, H. M., Afify, A. Z., Alizadeh, M., Butt, N. S., Hamedani, G. G. and Ali, M. M. (2015), The transmuted exponentiated generalized-G family of distributions. Pakistan Journal of Statistics and Operation Research, 11, 441-464.##Yousof, H. M., Afify, A. Z., Hamedani, G. G. and Aryal, G. (2016), the Burr X generator of distributions for lifetime data. Journal of Statistical Theory and Applications, 16, 288-305.##Yousof, H. M., Majumder, M., Jahanshahi, S. M. A., Ali, M. M. and Hamedani, G. G. (2018), A new Weibull class of distributions: theory, characterizations, and applications. Journal of Statistical Research of Iran, 23, 13-31.## ##]
Parameter Estimation of Some Archimedean Copulas Based on Minimum Cramér-von-Mises Distance
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2
The purpose of this paper is to introduce a new estimation method for estimating the Archimedean copula dependence parameter in the non-parametric setting. The estimation of the dependence parameter has been selected as the value that minimizes the Cramér-von-Mises distance which measures the distance between Empirical Bernstein Kendall distribution function and true Kendall distribution function. A Monte Carlo study is performed to measure the performance of the new estimator and compared to conventional estimation methods. In terms of estimation performance, simulation results show that the proposed Minumum Cramér-von-Mises estimation method has a good performance for low dependence and small sample size when compared
with the other estimation methods. The new minimum distance estimation of the dependence parameter is applied to model the dependence of two real data sets as illustrations.
163
183
selim orhun
susam
Selim Orhun
Susam
Department of Econometrics, Munzur University, Turkey.
orhunsusam@munzur.edu.tr
Cramér-von-Mises
Archimedean copula
Parameter estimation
Bernstein polynomials
[Biau, G. and Wegkamp, M. (2005), Minimum distance estimation of copula densities. Statistics & Probability letters, 73, 105–114.##Deheuvels, P. (1978). Caracterisation complete des lois extremes multivariees et de la convergence des types extremes. Publications de l’Institut de Statistique de l’Universite´ de Paris 3, 1–36.##Duchesne, T., Rioux, J. and Luong, A. (1997). Minimum crame´r-von mises distance methods for complete and grouped data. Communications in statistics - theory and methods 26, 401–420. ##Fermanian, J.D., Radulovic, D. and Wegkamp, M. (2004). Weak convergence of empirical copulaprocesses. Bernoulli 10, 847–860. ##Genest, C. and Mackay, R.J. (1986). Copules archimediennes et families de lois bidimensionelles dont les marges sont donnees. Canadian journal of statistics 14, 145–159. ##Genest, C. and Rivest, L. (1993). Statistical inference procedures for bivariate archimedean copulas. Journal of the American Statistical Association 88, 1034–1043. ##Genest, C., Molina, J. and Lallena, J. (1995), De l’impossibilite de construire des lois a marges multidimensionnelles donnees a partir de copules. Comptes rendus de l’Acadmie des Sciences, 320, 723–726. ##Genest, C., Rémillard, B. and Beaudoin, D. (2008), Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 199–213.##Joe, H. (1978). Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis 94, 401–419. ##Joe, H. and Xu, J. (1996). The estimation method of inference functions for margins for multivariate models.Technical Report 166, UBC, Department of Statistics. ##Joe, H. (1997). Multivariate Models and Dependence Concepts. London, England: Chapman and Hall.##Joe, H. (2005), Asymptotic eciency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis, 94, 401–419.##Kim, G., Silvapulle, M. and Silvapul, P. (2007). Comparison of semiparametric and parametric methods for estimating copulas. Communications in Statistics. Simulation and Computation 51, 2836–2850.##Kojadinovic, I. and Yan, J. (2010). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics 47, 52–63.##Leblanc, A. (2012). On estimating distribution functions using bernstein polynomials. Annals of the Institute of Statistical Mathematics 64, 919–943.##Mendes, B., De Melo B. and Nelsen, R. (2007), Robust fits for copula models. Communications in statistics: Simulation and computition, 36(5), 997–1017.##Michiels, F., Koch, I. and De Schepper, A. (2012), How to improve the fit of Archimedean copulas by means of transforms. Statistical Papers, 53, 345-355.##Najafabadi, A., Farid-Rohani, M. and Qazvini, M. (2013), A GLM-Based Method to Estimate a Copula’s Parameter(s). Journal of the Iranian statistical society, 12(2), 321-334. ## Nelsen, R.B. (2006). An introduction to copulas. New York, USA: Springer.##Oakes, D. (1994). Multivariate survival distributions. Journal of Nonparameric Statistics 3, 343–354.##Sklar, A. (1959). Fonctions de re´partition a´ n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universite´ de Paris 8, 229–231.##Susam, S.O. and Ucer, B.U. (2018). Testing independence for archimedean copula based on bernstein estimate of kendall distribution function. Journal of Statistical Computation and Simulation 88, 2589–2599.##Tsukahara H. (2005), Semiparametric estimation in copula models. Canadian Journal of Statistics, 33(3), 357–375.##Weib, G. (2011). Copula parameter estimation by maximum-likelihood and minimum-distance estimators: a simulation study. Computitional Statistics 26, 31–54.## ##]
Bivariate Extension of Past Entropy
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2
Di Crescenzo and Longobardi (2002) has been proposed a measure of uncertainty related to past life namely past entropy. The present paper addresses the question of extending this concept to bivariate set-up and study some properties of the proposed measure. It is shown that the proposed measure uniquely determines the distribution function. Characterizations for some bivariate lifetime models are obtained using the proposed measure. Further, we define new classes of life distributions based on this measure and properties of the new classes are also discussed. We also proposed a non-parametric kernel estimator for the proposed measure and illustrated performance of the estimator using a numerical data.
185
208
Rajesh
G
Ganapathi
Rajesh
Department of Statistics, CUSAT, Cochin-22, India.
rajeshg75@yahoo.com
Abdul
Sathar E I
Enchakudiyil Ibrahim
Abdul-Sathar
Department of Statistics, University of Kerala, Thiruvananthapuram - 695 581, India.
sathare@gmail.com
Reshmi
Reshmi
Krishnan Vijayalekshmi Ammal
Reshmi
Department of Statistics, University of Kerala, Thiruvananthapuram - 695 581, India.
reshmikv81@gmail.com
Bivariate Reversed Hazard Rate
Bivariate Mean Inactivity Time
Non-parametric Kernel Estimation
Past Entropy.
[Abdul-Sathar, E. I., Rajesh, G., and Nair, K. R. M. (2010), Bivariate geometric vitality function and some characterization results. Calcutta Statistical Association Bulletin, 62(3-4), 207–228.##Asadi, M. and Zohrevand, Y. (2007). On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference, 137(6), 1931–1941.##Balakrishnan, N. and Lai, C. D. (2009). Continuous bivariate distributions. Springer Science & Business Media.##Belzunce, F., Navarro, J., Ruiz, J. M., and Aguila, Y. D. (2004). Some results on residual entropy function. Metrika, 59(2), 147–161.##Bismi, N. G. (2005). Bivarite burr distributions. PhD thesis, Cochin University of Science and Technology.##Di Crescenzo, A. and Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. Journal of Applied Probability, 39(2), 434–440.##Ebrahimi, N. (1996). How to measure uncertainty in the residual life time distribution. Sankhy a: The Indian Journal of Statistics, Series A, 48–56.##Ebrahimi, N., Kirmani, S., and Soofi, E. S. (2007). Multivariate dynamic information. Journal of Multivariate Analysis, 98(2), 328–349.##Kim, H. and Kvam, P. H. (2004). Reliability estimation based on system data with an unknown load share rule. Lifetime Data Analysis, 10(1), 83–94.##Kundu, A. and Kundu, C. (2017). Bivariate extension of (dynamic) cumulative past entropy. Communications in Statistics-Theory and Methods, 46(9), 4163–4180.##Kundu, A. and Kundu, C. (2018). Bivariate extension of generalized cumulative past entropy. Communications in Statistics-Theory and Methods, 47(8), 1962–1977.##Nair, K. R. M. and Rajesh, G. (2000). Geometric vitality function and its applications to reliability. IAPQR TRANSACTIONS, 25(1), 1–8.##Nair, N. U. and Asha, G. (2008). Some characterizations based on bivariate reversed mean residual life. ProbStat Forum, 1, 1–14.##Nanda, A. K. and Paul, P. (2006). Some properties of past entropy and their applications. Metrika, 64(1), 47–61.##Rajesh, G., Abdul-Sathar, E. I., Nair, K. R. M., and Reshmi, K. V. (2014a). Bivariate extension of dynamic cumulative residual entropy. Statistical Methodology, 16, 72–82.##Rajesh, G., Abdul-Sathar, E. I., Reshmi, K. V., and Nair, K. R. M. (2014b). Bivariate generalized cumulative residual entropy. Sankhya A, 76(1), 101–122.##Rajesh, G., Sathar, A. E. I., and Nair, K. R. M. (2009). Bivariate extension of residual entropy and some characterization results. Journal of Indian Statistical Association, 47, 91–107.##Rao, M., Chen, Y., Vemuri, B. C., and Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE transactions on Information Theory, 50(6), 1220–1228.##Roy, D. (2002). Acharacterization of model approach for generating bivariate life distributions using reversed hazard rates. Journal of the Japan Statistical Society, 32(2), 239–245.##Sathar, A. E. I., Nair, K. R. M., and Rajesh, G. (2009). Generalized bivariate residual entropy function and some characterization results. South African Statistics Journal, 44, 1–18.##Shaked, M. and Shanthikumar, J. G. (2007). Stochastic orders. Springer Science & Business Media.##Shannon, C. E. (1948). A mathematical theory of communication. Bell system technical journal, 27(3), 379–423.##Sunoj, S. and Linu, M. (2012). Dynamic cumulative residual renyi’s entropy. Statistics, 46(1), 41–56.## ##]
The Weighted Exponentiated Family of Distributions: Properties, Applications and Characterizations
2
2
In this paper a new method of introducing an additional parameter to a continuous distribution is proposed, which leads to a new class of distributions,
called the weighted exponentiated family. A special sub-model is discussed. General expressions for some of the mathematical properties of this class such as the moments, quantile function, generating function and order statistics are derived; and certain characterizations are also discussed. To estimate the model parameters, the method of maximum likelihood is applied. A simulation study is carried out to assess the finite sample behavior of the maximum likelihood estimators. Finally, the usefulness of the proposed method via two applications to real data sets is illustrated.
209
228
Zubair
Ahmad
Zubair
Ahmad
Department of Statistics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan.
z.ferry21@gmail.com
Gholamhossien
Hamedani
Gholamhossien
Hamedani
Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53201-1881, USA.
gholamhoss.hamedani@marquette.edu
Mohammed
Elgarhy
Mohammed
Elgarhy
Valley High Institute for Management Finance and Information Systems, Obour, Qaliubia, Egypt.
m_elgarhy85@yahoo.com
Exponentiated Family
Weighted Family
Exponential Distribution
Moments
Order Statistics
Characterizations
Maximum Likelihood Estimation.
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