# Inference on a new lifetime distribution for a parallel

## Transcript Of Inference on a new lifetime distribution for a parallel

Inference on a new lifetime distribution for a parallel-series system under

progressively type-II censored samples

Alaa H. Abdel-Hamid1, Atef F. Hashem1, Co¸skun Ku¸s2

1 Department of Mathematics and Computer Science, University of Beni-Suef, Beni-Suef, Egypt (e-mail: hamid−[email protected], dr−[email protected])

2 Department of Statistics, University of Selcuk, Selcuk, Turkey (e-mail: [email protected])

Abstract. In this paper we introduce a new lifetime distribution with increasing, decreasing or upside-down bathtub shaped hazard rates, called doubly Poisson exponential distribution. One of the motivations of the new distribution is that it may represent the lifetime of units connected in a parallel-series system. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods for the involved parameters are considered. The methods are maximum likelihood, moments, least squares, weighted least squares and Bayes (using linear-exponential and general entropy loss functions) estimations. Bayes estimates for the parameters are obtained using Markov chain Monte Carlo algorithm. The performance of these methods is compared through an extensive numerical simulation, based on mean of mean squared errors and mean of relative absolute biases. Two real data sets are used to compare the new distribution with other ﬁve distributions. The comparison shows that the former distribution is better to ﬁt the data than the other ﬁve distributions.

Keywords: parallel-series system, progressive type-II censoring, exponential distribution, maximum likelihood, moments, least squares, weighted least squares and Bayes estimations, simulation.

Bayesian estimation on the exponentiated Pareto distribution under type II censoring

Hanaa Abu-Zinadah

Department of Statistics, Faculty of Science - AL Faisaliah Campus, King Abdulaziz University, Jeddah, Saudi Arabia (e-mail: [email protected])

Abstract. In this paper, Bayes estimates of the two shape parameters, reliability and failure rate functions of the exponentiated Pareto lifetime model are derived from complete and type II censored samples. When the Bayesian approach is concerned, conjugate priors for either the one or the two shape parameters cases are considered. An approximation form due to Lindley (1980) is used for obtaining the Bayes estimates under the squared error loss and LINEX loss functions. The root-mean square errors of the estimates are computed. Comparisons are made between the Bayes estimators.

Keywords: Bayes estimators, exponentiated Pareto distribution, LINEX loss function, squared error loss function.

References

1. Ahmad K.E., Fakhry M.E., Jaheen Z.F. (1997). Empirical Bayes estimation of P (Y < X) and characterizations of Burr-type X model. J. Statist. Plan. Infer., 64, 297–308.

2. AL-Hussaini E.K., Jaheen Z.F. (1992). Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model. J. Statist. Comput. Simul., 41, 31–40.

3. AL-Hussaini E.K., Jaheen Z.F. (1994). Approximate Bayes estimator applied to the Burr model. Commun. Statist.—Theory Meth., 23, 99–121.

4. Basu A.P., Ebrahimi N. (1991). Bayesian approach to life testing and reliability estimation using asymmetric loss function. J. Stat.Plan. Infer., 29, 21–31.

5. Calabria R., Pulcini G. (1996). Point estimation under asymmetric loss functions for left-truncated exponential samples. Commun. Statist.—Theory Meth., 25, 585–600.

6. Gupta R.C., Gupta R.D., Gupta P.L. (1998). Modeling failure time data by Lehman lternatives. Commun. Statist.—Theory Meth., 27, 887–904.

7. Howlader H.A., Weiss G. (1989). Bayesian estimators of the reliability of the logistic distribution. Commun. Statist.—Theory Meth., 18, 245–259.

8. Ismail M.A. (2001). Bayesian analysis of hybrid censored life tests using asymmetric loss functions. Annual Conference on Statistics and Computer Modeling in Human and Social Sciences, 13, 42–63.

9. Lindley D.V. (1980). Approximate Bayesian methods. Trabajos de Estadistica, 31, 223–245.

10. Nassar M.M., Eissa F.H. (2004). Bayesian estimation for the exponentiated Weibull model. Commun. Statist.—Theory Meth., 33, 2343–2362.

11. Pandey B.N. (1997). Testimator of the scale parameter of the exponential distribution using LINEX loss function. Commun. Statist.—Theory Meth., 26, 2191– 2202.

12. Pandey B.N., Rai O. (1992). Bayesian estimation of mean and square of mean of normal distribution using LINEX loss function. Commun. Statist.—Theory Meth., 21, 3369–3391.

13. Pandey B.N., Singh B.P., Mishra C.S. (1996). Bayes estimation of shape parameter of classical Pareto distribution under LINEX loss function. Commun. Statist.—Theory Meth., 25, 3125–3145.

14. Sadooghi-Alvandi S.M., Parsian A. (1992). Estimation of the binomial parameter n using a LINEX loss function. Commun. Statist.–Theory Meth., 21, 1427–1439.

15. Singh U., Gupta P.K., Upadhyay S.K. (2005). Estimation of parameters for exponentaited Weibull family under type-II censoring scheme. Computational Statistics and Data Analysis, 48, 509–523.

16. Sinha S.K., Sloan J.A. (1988). Bayes estimation of the parameters and reliability function of the 3-parameter Weibull distribution. IEEE Trans. Reliability, 37, 364–369.

17. Soliman A.A. (2000). Comparison of LINEX and quadratic Bayes estimators for the Rayleigh distribution. Commun. Statist.—Theory Meth., 29, 95–107.

18. Soliman A.A. (2001). LINEX and quadratic approximate Bayes estimators applied to the Pareto model. Commun. Statist.—Simulat. Comput., 30, 63–77.

19. Tsionas E.G. (2003). Pareto regression: A Bayesian analysis. Commun. Statist.— Theory Meth., 32, 1213–1225.

20. Varian H.R. (1975). A Bayesian approach to real estate assessment. Amsterdam: North Holland, 195-208.

21. Zellner A. (1986). Bayesian estimation and prediction using asymmetric loss functions. J. Amer. Statist. Assoc., 81, 446–451.

Pointwise estimation of bivariate Pickands dependence function: A Bernstein copula approach

Alireza Ahmadabadi, Burcu Hudaverdi Ucer

Department of Statistics, Dokuz Eylul University, Izmir, Turkiye (e-mail: [email protected], [email protected])

Abstract. Modeling dependence structures of joint extreme events has been interest of various applicational areas, such as, environmental science, insurance, ﬁnance, etc. Pickands dependence function that characterizes the extreme-value copula is widely used to model these extreme events. In this study, Bernstein copula approximation is used to estimate Pickands dependence function. Pointwise estimation procedure with a visual tool is proposed for investigating the extreme-value dependence structure. The performance of the estimator is presented with a simulation study. Test results mainly show that the estimator has a good performance in detecting the tail behavior.

Keywords: extreme-value, Bernstein copula, Pickands dependence function.

References

1. Cap´era`a P., Foug`eres A.L., Genest C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.

2. Deheuvels P. (1991). On the limiting behavior of the pickands estimator for bivariate extreme-value distributions. Statist. Probab. Lett. 12, 429–439

3. Einmahl J.H.J., de Haan L., Li, D. (2006). Weighted approximations to tail copula processeswith application to testing the bivariate extreme value condition. The Annals of Statistics, 34, 1987–2014

4. Falk M., Reiss R.-D. (2005). On Pickands coordinates in arbitrary dimensions. Journal of Multivariate Analysis, 92, 426–453.

5. Galambos J. (1987). The asymptotic theory of extreme order statistics, second edn. Robert E. Krieger Publishing Co. Inc., Melbourne, FL

6. Gudendorf G., Segers J. (2012). Nonparametric estimation of multivariate extremevalue copulas. Journal of Statistical Planning and Inference, 142, 3073–3085

7. Hall P., Tajvidi N. (2000). Distribution and dependence-function estimation for bivariate extremevalue distributions. Bernoulli, 6, 835–844.

8. Pickands J. (1981). Multivariate extreme value distribution. In Proceedings of the 43rd Session of the International Statistical Institute, 859–878.

9. Sancetta A., Satchell S. (2004). The Bernstein copula and its application to modeling and approximations of multivariate distributions. Econometric Theory, 20, 535–562.

10. Tiago de Olivera J. (1989). Intrinsic estimation of the dependence structure for bivariate extremes. Statist. Probab. Letters, 8, 213-218.

11. Zhang D., Wells M.T., Peng L. (2008) Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Multivariate Anal., 99, 577–588

Joint signatures

Narayanaswamy Balakrishnan

Department of Mathematics and Statistics, McMaster University, Canada (e-mail: [email protected])

Abstract. In this talk, I will introduce the notion of joint signatures of two systems and present some of their properties including mixture representations for joint distributions of lifetimes of the two systems. I will then use this representation to develop some statistical inferential methods for characteristics of both systems and components based on system lifetime data. I will present some examples to illustrate the results developed. Finally, I will conclude the talk by mentioning some further issues that are worth of further study.

Asymptotic behavior of the joint record values, with applications

H. M. Barakat1, M. A. Abd Elgawad2

1 Department of Mathematics, University of Zagazig, Zagazig, Egypt (e-mail: [email protected])

2 Department of Mathematics and Statistics, University of Central China Normal University, Wuhan 430079, China (e-mail: mohamed−[email protected])

Abstract. The class of limit distribution functions of the joint upper record values, as well as the joint of lower record values, is fully characterized. Suﬃcient conditions for the weak convergence are obtained. As an application of this result, the suﬃcient conditions for the weak convergence of the record quasi-range, record quasi-midrange, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit distribution functions of these statistics are derived.

Keywords: weak convergence, record values, joint record values, record functions.

References

1. Barakat H.M., Abd Elgawad M.A. Asymptotic behavior of the joint record values, with applications. (submitted)

Limit distributions of generalized order statistics in a stationary Gaussian sequence

H. M. Barakat, E. M. Nigm, E. O. Abo Zaid

Department of Mathematics, University of Zagazig, Zagazig, Egypt (e-mail: [email protected], s−[email protected], [email protected])

Abstract. In this paper we study the limit distributions of extreme, intermediate and central m-generalized order statistics (gos), as well as m-dual generalized order statistics (dgos), of a stationary Gaussian sequence under equi-correlated set up. Moreover, the result of extremes is extended to a wide subclass of gos, as well as dgos, (which contains the most important models of ordered random variables), when the parameters γ1,n, γ2,n, ..., γn,n are assumed to be pairwise diﬀerent.

Keywords: Gaussian sequences, generalized order statistics, dual generalized order statistics.

References

1. Barakat H.M., Nigm E.M., Abo Zaid E.O. Asymptotic behavior of the joint record values, with applications. (submitted)

The estimations under power normalization for the tail index, with comparison

H. M. Barakat1, E. M. Nigm1, O. M. Khaled2, H. A. Alaswed3

1 Department of Mathematics, University of Zagazig, Zagazig, Egypt (e-mail: [email protected], s−[email protected])

2 Department of Mathematics, University of Port-Said, Port-Said, Egypt (e-mail: [email protected])

3 Department of Statistics, University of Sebha, Libya (e-mail: [email protected])

Abstract. The objective of this research is to suggest two classes of moment and moment ratio estimators under power normalization for the tail index. Moreover, for quantitative comparison of the suggested estimators and other estimators, we use a mean square error criterion. The problem of weighting between the linear and power models to describe the given extreme data is challenging. For this purpose, we suggest the coeﬃcient variation criterion. A simulation study is conducted, to assess and compare the accuracy of the suggested estimators and other estimators, as well as the suggested statistical criterions. The suggested estimators and other estimators, as well as the suggested criterions are used to analyze a real data sets. All computations in this work are performed by R-package.

Keywords: power normalization, generalized Pareto distributions, Hill estimators, moment estimator, moment ratio estimator.

References

1. Barakat H.M., Nigm E.M., Khaled O.M., Alaswedz H.A. The estimations under power normalization for the tail index, with comparison. (submitted)

On some new models of multivariate record values

I˙smihan Bayramoglu

Department of Mathematics, Izmir University of Economics, Turkey (e-mail: [email protected])

Abstract. The theory of record values in multivariate sequences of random variables is considered. One of the considered models is based on coordinatewise ordering of multivariate observations. The distributional properties of record values in a new scheme of multivariate records is presented. Some examples with well known bivariate distributions as underlying distributions of original sample are given and graphical illustrations are provided. In the second record model, we consider the N-ordering scheme for random vectors and deﬁne new records according to this ordering. The distributional theory of bivariate records and record times is given. Examples and graphical illustrations are provided. Challenging unsolved problems on record theory of multivariate random sequences are discussed.

Keywords: order statistics, record values, multivariate orderings.

References

1. Ahsanullah M. (1995). Record Statistics. Nova Science Publishers Inc., New York. 2. Arnold B.C., Balakrishnan N., Nagaraja H.N. (1998). Records. John Wiley and

Sons, New York. 3. Arnold B.C., Castillo E., Sarabia J.S. (2009a). Multivariate order statistics via

multivariate concomitants. Journal of Multivariate Analysis, 100, 946–951. 4. Arnold B.C., Castillo E., Sarabia J.S. (2009b). On multivariate order statistics.

Application to ranked set sampling. Computational Statistics and Data Analysis, 53, 4555–4569. 5. Bairamov I. (2006). Progressive type-II censored order statistics for multivariate observations. Journal of Multivariate Analysis, 97, 797–809. 6. Bairamov I.G., Gebizlioglu O.L. (1998). On the ordering of random vectors in a norm sense. Journal of Applied Statistical Sciences, 6, 77–86. 7. Barnet V. (1976). The ordering of multivariate data. Journal of Royal Statistical Society, A, 139, 318–343.

The role of record values in reliability⋆

F´elix Belzunce

Department of Statistics and Operations Research, University of Murcia, Murcia, Spain (e-mail: [email protected])

Abstract. The notion of record values was introduced by Chandler (1952). Since then, a great number of results have been provided for record values. The record values have other interpretations mainly in reliability, such as failures times of minimal repair policies and the relevation transform. Under this interpretations two main areas of research have been developed along the years. One is to study aging properties of record values, such as IFR, NBU or ILR (see Pellerey, Shaked and Zinn, 2000) and and the other one is the comparison of record values arising from diﬀerent parent populations (see Belzunce, Lillo, Ruiz and Shaked, 2001). The purpose of this talk is to provide, ﬁrst, a historical review of the diﬀerent interpretations of record values in reliability and a review of some of the main results about aging properties and comparison of record values. Next I will present some new questions about record values, that can be addressed from the point of view of reliability. Mainly I will discuss some new results about the comparison of a minimal repair process with a renewal process and some new results about the role of relevation in allocation of redundant components. This talk is also intended to be a tribute to Moshe Shaked, who made great contributions on the topics of this talk.

Keywords: record values, minimal repair process, relevation transform.

References

1. Belzunce F., Lillo R., Ruiz J.M., Shaked M. (2001). Stochastic comparisons of nonhomogeneous processes. Probability in the Engineering and Informational Sciences, 15, 199–224.

2. Chandler K.N. (1952). The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220–228.

3. Pellerey F., Shaked M., Zinn J. (2000). Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences, 14, 353– 373.

⋆ This work has been supported by the Ministerio de Econom´ıa y Competitividad under grant MTM2012-34023-FEDER

progressively type-II censored samples

Alaa H. Abdel-Hamid1, Atef F. Hashem1, Co¸skun Ku¸s2

1 Department of Mathematics and Computer Science, University of Beni-Suef, Beni-Suef, Egypt (e-mail: hamid−[email protected], dr−[email protected])

2 Department of Statistics, University of Selcuk, Selcuk, Turkey (e-mail: [email protected])

Abstract. In this paper we introduce a new lifetime distribution with increasing, decreasing or upside-down bathtub shaped hazard rates, called doubly Poisson exponential distribution. One of the motivations of the new distribution is that it may represent the lifetime of units connected in a parallel-series system. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods for the involved parameters are considered. The methods are maximum likelihood, moments, least squares, weighted least squares and Bayes (using linear-exponential and general entropy loss functions) estimations. Bayes estimates for the parameters are obtained using Markov chain Monte Carlo algorithm. The performance of these methods is compared through an extensive numerical simulation, based on mean of mean squared errors and mean of relative absolute biases. Two real data sets are used to compare the new distribution with other ﬁve distributions. The comparison shows that the former distribution is better to ﬁt the data than the other ﬁve distributions.

Keywords: parallel-series system, progressive type-II censoring, exponential distribution, maximum likelihood, moments, least squares, weighted least squares and Bayes estimations, simulation.

Bayesian estimation on the exponentiated Pareto distribution under type II censoring

Hanaa Abu-Zinadah

Department of Statistics, Faculty of Science - AL Faisaliah Campus, King Abdulaziz University, Jeddah, Saudi Arabia (e-mail: [email protected])

Abstract. In this paper, Bayes estimates of the two shape parameters, reliability and failure rate functions of the exponentiated Pareto lifetime model are derived from complete and type II censored samples. When the Bayesian approach is concerned, conjugate priors for either the one or the two shape parameters cases are considered. An approximation form due to Lindley (1980) is used for obtaining the Bayes estimates under the squared error loss and LINEX loss functions. The root-mean square errors of the estimates are computed. Comparisons are made between the Bayes estimators.

Keywords: Bayes estimators, exponentiated Pareto distribution, LINEX loss function, squared error loss function.

References

1. Ahmad K.E., Fakhry M.E., Jaheen Z.F. (1997). Empirical Bayes estimation of P (Y < X) and characterizations of Burr-type X model. J. Statist. Plan. Infer., 64, 297–308.

2. AL-Hussaini E.K., Jaheen Z.F. (1992). Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model. J. Statist. Comput. Simul., 41, 31–40.

3. AL-Hussaini E.K., Jaheen Z.F. (1994). Approximate Bayes estimator applied to the Burr model. Commun. Statist.—Theory Meth., 23, 99–121.

4. Basu A.P., Ebrahimi N. (1991). Bayesian approach to life testing and reliability estimation using asymmetric loss function. J. Stat.Plan. Infer., 29, 21–31.

5. Calabria R., Pulcini G. (1996). Point estimation under asymmetric loss functions for left-truncated exponential samples. Commun. Statist.—Theory Meth., 25, 585–600.

6. Gupta R.C., Gupta R.D., Gupta P.L. (1998). Modeling failure time data by Lehman lternatives. Commun. Statist.—Theory Meth., 27, 887–904.

7. Howlader H.A., Weiss G. (1989). Bayesian estimators of the reliability of the logistic distribution. Commun. Statist.—Theory Meth., 18, 245–259.

8. Ismail M.A. (2001). Bayesian analysis of hybrid censored life tests using asymmetric loss functions. Annual Conference on Statistics and Computer Modeling in Human and Social Sciences, 13, 42–63.

9. Lindley D.V. (1980). Approximate Bayesian methods. Trabajos de Estadistica, 31, 223–245.

10. Nassar M.M., Eissa F.H. (2004). Bayesian estimation for the exponentiated Weibull model. Commun. Statist.—Theory Meth., 33, 2343–2362.

11. Pandey B.N. (1997). Testimator of the scale parameter of the exponential distribution using LINEX loss function. Commun. Statist.—Theory Meth., 26, 2191– 2202.

12. Pandey B.N., Rai O. (1992). Bayesian estimation of mean and square of mean of normal distribution using LINEX loss function. Commun. Statist.—Theory Meth., 21, 3369–3391.

13. Pandey B.N., Singh B.P., Mishra C.S. (1996). Bayes estimation of shape parameter of classical Pareto distribution under LINEX loss function. Commun. Statist.—Theory Meth., 25, 3125–3145.

14. Sadooghi-Alvandi S.M., Parsian A. (1992). Estimation of the binomial parameter n using a LINEX loss function. Commun. Statist.–Theory Meth., 21, 1427–1439.

15. Singh U., Gupta P.K., Upadhyay S.K. (2005). Estimation of parameters for exponentaited Weibull family under type-II censoring scheme. Computational Statistics and Data Analysis, 48, 509–523.

16. Sinha S.K., Sloan J.A. (1988). Bayes estimation of the parameters and reliability function of the 3-parameter Weibull distribution. IEEE Trans. Reliability, 37, 364–369.

17. Soliman A.A. (2000). Comparison of LINEX and quadratic Bayes estimators for the Rayleigh distribution. Commun. Statist.—Theory Meth., 29, 95–107.

18. Soliman A.A. (2001). LINEX and quadratic approximate Bayes estimators applied to the Pareto model. Commun. Statist.—Simulat. Comput., 30, 63–77.

19. Tsionas E.G. (2003). Pareto regression: A Bayesian analysis. Commun. Statist.— Theory Meth., 32, 1213–1225.

20. Varian H.R. (1975). A Bayesian approach to real estate assessment. Amsterdam: North Holland, 195-208.

21. Zellner A. (1986). Bayesian estimation and prediction using asymmetric loss functions. J. Amer. Statist. Assoc., 81, 446–451.

Pointwise estimation of bivariate Pickands dependence function: A Bernstein copula approach

Alireza Ahmadabadi, Burcu Hudaverdi Ucer

Department of Statistics, Dokuz Eylul University, Izmir, Turkiye (e-mail: [email protected], [email protected])

Abstract. Modeling dependence structures of joint extreme events has been interest of various applicational areas, such as, environmental science, insurance, ﬁnance, etc. Pickands dependence function that characterizes the extreme-value copula is widely used to model these extreme events. In this study, Bernstein copula approximation is used to estimate Pickands dependence function. Pointwise estimation procedure with a visual tool is proposed for investigating the extreme-value dependence structure. The performance of the estimator is presented with a simulation study. Test results mainly show that the estimator has a good performance in detecting the tail behavior.

Keywords: extreme-value, Bernstein copula, Pickands dependence function.

References

1. Cap´era`a P., Foug`eres A.L., Genest C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.

2. Deheuvels P. (1991). On the limiting behavior of the pickands estimator for bivariate extreme-value distributions. Statist. Probab. Lett. 12, 429–439

3. Einmahl J.H.J., de Haan L., Li, D. (2006). Weighted approximations to tail copula processeswith application to testing the bivariate extreme value condition. The Annals of Statistics, 34, 1987–2014

4. Falk M., Reiss R.-D. (2005). On Pickands coordinates in arbitrary dimensions. Journal of Multivariate Analysis, 92, 426–453.

5. Galambos J. (1987). The asymptotic theory of extreme order statistics, second edn. Robert E. Krieger Publishing Co. Inc., Melbourne, FL

6. Gudendorf G., Segers J. (2012). Nonparametric estimation of multivariate extremevalue copulas. Journal of Statistical Planning and Inference, 142, 3073–3085

7. Hall P., Tajvidi N. (2000). Distribution and dependence-function estimation for bivariate extremevalue distributions. Bernoulli, 6, 835–844.

8. Pickands J. (1981). Multivariate extreme value distribution. In Proceedings of the 43rd Session of the International Statistical Institute, 859–878.

9. Sancetta A., Satchell S. (2004). The Bernstein copula and its application to modeling and approximations of multivariate distributions. Econometric Theory, 20, 535–562.

10. Tiago de Olivera J. (1989). Intrinsic estimation of the dependence structure for bivariate extremes. Statist. Probab. Letters, 8, 213-218.

11. Zhang D., Wells M.T., Peng L. (2008) Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Multivariate Anal., 99, 577–588

Joint signatures

Narayanaswamy Balakrishnan

Department of Mathematics and Statistics, McMaster University, Canada (e-mail: [email protected])

Abstract. In this talk, I will introduce the notion of joint signatures of two systems and present some of their properties including mixture representations for joint distributions of lifetimes of the two systems. I will then use this representation to develop some statistical inferential methods for characteristics of both systems and components based on system lifetime data. I will present some examples to illustrate the results developed. Finally, I will conclude the talk by mentioning some further issues that are worth of further study.

Asymptotic behavior of the joint record values, with applications

H. M. Barakat1, M. A. Abd Elgawad2

1 Department of Mathematics, University of Zagazig, Zagazig, Egypt (e-mail: [email protected])

2 Department of Mathematics and Statistics, University of Central China Normal University, Wuhan 430079, China (e-mail: mohamed−[email protected])

Abstract. The class of limit distribution functions of the joint upper record values, as well as the joint of lower record values, is fully characterized. Suﬃcient conditions for the weak convergence are obtained. As an application of this result, the suﬃcient conditions for the weak convergence of the record quasi-range, record quasi-midrange, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit distribution functions of these statistics are derived.

Keywords: weak convergence, record values, joint record values, record functions.

References

1. Barakat H.M., Abd Elgawad M.A. Asymptotic behavior of the joint record values, with applications. (submitted)

Limit distributions of generalized order statistics in a stationary Gaussian sequence

H. M. Barakat, E. M. Nigm, E. O. Abo Zaid

Department of Mathematics, University of Zagazig, Zagazig, Egypt (e-mail: [email protected], s−[email protected], [email protected])

Abstract. In this paper we study the limit distributions of extreme, intermediate and central m-generalized order statistics (gos), as well as m-dual generalized order statistics (dgos), of a stationary Gaussian sequence under equi-correlated set up. Moreover, the result of extremes is extended to a wide subclass of gos, as well as dgos, (which contains the most important models of ordered random variables), when the parameters γ1,n, γ2,n, ..., γn,n are assumed to be pairwise diﬀerent.

Keywords: Gaussian sequences, generalized order statistics, dual generalized order statistics.

References

1. Barakat H.M., Nigm E.M., Abo Zaid E.O. Asymptotic behavior of the joint record values, with applications. (submitted)

The estimations under power normalization for the tail index, with comparison

H. M. Barakat1, E. M. Nigm1, O. M. Khaled2, H. A. Alaswed3

1 Department of Mathematics, University of Zagazig, Zagazig, Egypt (e-mail: [email protected], s−[email protected])

2 Department of Mathematics, University of Port-Said, Port-Said, Egypt (e-mail: [email protected])

3 Department of Statistics, University of Sebha, Libya (e-mail: [email protected])

Abstract. The objective of this research is to suggest two classes of moment and moment ratio estimators under power normalization for the tail index. Moreover, for quantitative comparison of the suggested estimators and other estimators, we use a mean square error criterion. The problem of weighting between the linear and power models to describe the given extreme data is challenging. For this purpose, we suggest the coeﬃcient variation criterion. A simulation study is conducted, to assess and compare the accuracy of the suggested estimators and other estimators, as well as the suggested statistical criterions. The suggested estimators and other estimators, as well as the suggested criterions are used to analyze a real data sets. All computations in this work are performed by R-package.

Keywords: power normalization, generalized Pareto distributions, Hill estimators, moment estimator, moment ratio estimator.

References

1. Barakat H.M., Nigm E.M., Khaled O.M., Alaswedz H.A. The estimations under power normalization for the tail index, with comparison. (submitted)

On some new models of multivariate record values

I˙smihan Bayramoglu

Department of Mathematics, Izmir University of Economics, Turkey (e-mail: [email protected])

Abstract. The theory of record values in multivariate sequences of random variables is considered. One of the considered models is based on coordinatewise ordering of multivariate observations. The distributional properties of record values in a new scheme of multivariate records is presented. Some examples with well known bivariate distributions as underlying distributions of original sample are given and graphical illustrations are provided. In the second record model, we consider the N-ordering scheme for random vectors and deﬁne new records according to this ordering. The distributional theory of bivariate records and record times is given. Examples and graphical illustrations are provided. Challenging unsolved problems on record theory of multivariate random sequences are discussed.

Keywords: order statistics, record values, multivariate orderings.

References

1. Ahsanullah M. (1995). Record Statistics. Nova Science Publishers Inc., New York. 2. Arnold B.C., Balakrishnan N., Nagaraja H.N. (1998). Records. John Wiley and

Sons, New York. 3. Arnold B.C., Castillo E., Sarabia J.S. (2009a). Multivariate order statistics via

multivariate concomitants. Journal of Multivariate Analysis, 100, 946–951. 4. Arnold B.C., Castillo E., Sarabia J.S. (2009b). On multivariate order statistics.

Application to ranked set sampling. Computational Statistics and Data Analysis, 53, 4555–4569. 5. Bairamov I. (2006). Progressive type-II censored order statistics for multivariate observations. Journal of Multivariate Analysis, 97, 797–809. 6. Bairamov I.G., Gebizlioglu O.L. (1998). On the ordering of random vectors in a norm sense. Journal of Applied Statistical Sciences, 6, 77–86. 7. Barnet V. (1976). The ordering of multivariate data. Journal of Royal Statistical Society, A, 139, 318–343.

The role of record values in reliability⋆

F´elix Belzunce

Department of Statistics and Operations Research, University of Murcia, Murcia, Spain (e-mail: [email protected])

Abstract. The notion of record values was introduced by Chandler (1952). Since then, a great number of results have been provided for record values. The record values have other interpretations mainly in reliability, such as failures times of minimal repair policies and the relevation transform. Under this interpretations two main areas of research have been developed along the years. One is to study aging properties of record values, such as IFR, NBU or ILR (see Pellerey, Shaked and Zinn, 2000) and and the other one is the comparison of record values arising from diﬀerent parent populations (see Belzunce, Lillo, Ruiz and Shaked, 2001). The purpose of this talk is to provide, ﬁrst, a historical review of the diﬀerent interpretations of record values in reliability and a review of some of the main results about aging properties and comparison of record values. Next I will present some new questions about record values, that can be addressed from the point of view of reliability. Mainly I will discuss some new results about the comparison of a minimal repair process with a renewal process and some new results about the role of relevation in allocation of redundant components. This talk is also intended to be a tribute to Moshe Shaked, who made great contributions on the topics of this talk.

Keywords: record values, minimal repair process, relevation transform.

References

1. Belzunce F., Lillo R., Ruiz J.M., Shaked M. (2001). Stochastic comparisons of nonhomogeneous processes. Probability in the Engineering and Informational Sciences, 15, 199–224.

2. Chandler K.N. (1952). The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220–228.

3. Pellerey F., Shaked M., Zinn J. (2000). Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences, 14, 353– 373.

⋆ This work has been supported by the Ministerio de Econom´ıa y Competitividad under grant MTM2012-34023-FEDER