### On a Generalization of the Positive Exponential Family of Distributions and the Estimation of Reliability Characteristics

#### Abstract

A generalization of positive exponential family of distributions developed by Liang (2008) is taken into consideration. Its properties are studied. Two measures of reliability are discussed. Uniformly minimum variance unbiased estimators (UMVUES), maximum likelihood estimators (MLES) and method of moment estimators (MMES) are developed for the reliability functions. The performances of three types of estimators are compared through Monte Carlo simulation. Real life data sets are also analyzed.

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A. AWAD, M. GHARRAF (1986). Estimation of P(Y < X) in the Burr case: A comparative study. Communications in Statistics - Simulation and Computation, 15, no. 2, pp. 389–403.

A. BAKLIZI (2008a). Estimation of P(X < Y) using record values in the one and two parameter exponential distributions. Communications in Statistics - Theory and Methods, 37, no. 5, pp. 692–698.

A. BAKLIZI (2008b). Likelihood and Bayesian estimation of P(X < Y) using lower record values from the generalized exponential distribution. Computational Statistics and Data Analysis, 52, no. 7, pp. 3468–3473.

D. BARTHOLOMEW (1957). A problem in life testing. Journal of the American Statistical Association, 52, no. 279, pp. 350–355.

D. BARTHOLOMEW (1963). The sampling distribution of an estimate arising in life testing. Technometrics, 5, no. 3, pp. 361–374.

A. BASU (1964). Estimates of reliability for some distributions useful in life testing. Technometrics, 6, no. 2, pp. 215–219.

A. CHAO (1982). On comparing estimators of P(Y < X) in the exponential case. IEEE Transactions on Reliability, 31, no. 4, pp. 389–392.

A. CHATURVEDI, N. KUMAR, K. KUMAR (2018). Statistical inference for the reliability functions of a family of lifetime distributions based on progressive type II right censoring. Statistica, 78, no. 1, pp. 81–101.

A. CHATURVEDI, T. KUMARI (2016). Estimation and testing procedures for the reliability functions of a general class of distributions. Communications in Statistics - Theory and Methods, 46, no. 22, pp. 11370–11382.

A. CHATURVEDI, A. MALHOTRA (2017). Estimation and testing procedures for the reliability functions of a family of lifetime distributions based on records. International Journal of System Assurance Engineering and Management, 8, no. 2, pp. 836–848.

A. CHATURVEDI, A. MALHOTRA (2018). Estimation of P(X > Y) for the positive exponential family of distributions. Statistica, 78, no. 2, pp. 149–167.

A. CHATURVEDI, A. PATHAK (2012). Estimation of the reliability functions for exponentiated Weibull distribution. Journal of Statistics and Applications, 7, no. 3-4, pp. 113–120.

A. CHATURVEDI, A. PATHAK (2013). Estimation of the reliability function for four parameter exponentiated generalized Lomax distribution. International Journal of Scientific and Engineering Research, 5, no. 1, pp. 1171–1180.

A.CHATURVEDI, U. RANI (1997). Estimation procedures for a family of density functions representing various life-testing models. Metrika, 46, no. 1, pp. 213–219.

A. CHATURVEDI, U. RANI (1998). Classical and Bayesian reliability estimation of the generalized Maxwell failure distribution. Journal of Statistical Research, 32, no. 1, pp. 113–120.

A. CHATURVEDI, K. SINGH (2006). Bayesian estimation procedures for a family of lifetime distributions under squared-error and entropy losses. Metron, 64, no. 2, pp. 179–198.

A. CHATURVEDI, K. SINGH (2008). A family of lifetime distributions and related estimation and testing procedures for the reliability function. Journal of Applied Statistical Science, 16, no. 2, pp. 35–50.

A. CHATURVEDI, K. SURINDER (1999). Further remarks on estimating the reliability function of exponential distribution under type I and type II censorings. Brazilian Journal of Probability and Statistics, 13, no. 1, pp. 29–39.

A. CHATURVEDI, S. TOMER (2002). Classical and Bayesian reliability estimation of the negative binomial distribution. Journal of Applied Statistical Science, 11, no. 1, pp. 33–43.

A. CHATURVEDI, S. TOMER (2003). UMVU estimation of the reliability function of the generalized life distributions. Statistical Papers, 44, no. 3, pp. 301–313.

A. CHATURVEDI, S. VYAS (2017). Estimation and testing procedures for the reliability functions of three parameter Burr distribution under censorings. Statistica, 77, no. 3, pp. 207–235.

K. CONSTANTINE, M. KARSON, S. TSE (1986). Estimation of P(Y < X) in the gamma case. Communications in Statistics - Simulation and Computation, 15, no. 2, pp. 365–388

D. DAVIS (1952). An analysis of some failure data. Journal of the American Statistical Association, 47, no. 258, pp. 113–150.

S. ERYILMAZ (2008a). Consecutive k-out-of-n: G system in stress-strength setup. Communications in Statistics - Simulation and Computation, 37, no. 3, pp. 579–589.

S. ERYILMAZ (2008b). Multivariate stress-strength reliability model and its evaluation for coherent structures. Journal of Multivariate Analysis, 99, no. 9, pp. 1878–1887.

S. ERYILMAZ (2010). On system reliability in stress-strength setup. Statistics and Probability Letters, 80, no. 9-10, pp. 834–839.

S. ERYILMAZ (2011). A new perspective to stress-strength models. Annals of the Institute of Statistical Mathematics, 63, no. 1, pp. 101–115.

M.HUANG, Z.WANG (2012). Inference about the reliability parameter with gamma stress and strength. Journal of Statistical Planning and Inference, 142, no. 4, pp. 848–854.

N. JOHNSON, S. KOTZ (1970). Continuous Univariate Distributions - I. Wiley, New York.

G. KELLY, J. KELLY,W. SCHUCANY (1976). Efficient estimation of P(Y < X) in the exponential case. Technometrics, 18, no. 3, pp. 359–360.

S. KOTZ, M. PENSKY (2003). The Stress-Strength Model and Its Generalizations: Theory and Applications. World Scientific, Singapore.

K. KRISHNAMOORTHY, Y. LIN (2010). Confidence limits for stress-strength reliability involving Weibull models. Journal of Statistical Planning and Inference, 140, no. 7, pp. 1754–1764.

K. KRISHNAMOORTHY, Y. LIN, Y. XIA (2009). Confidence limits and prediction limits for a Weibull distribution based on the generalized variable approach. Journal of Statistical Planning and Inference, 139, no. 8, pp. 2675–2684.

K. KRISHNAMOORTHY, S. MUKHERJEE, H. GUO (2007). Inference on reliability in two parameter exponential stress-strength model. Metrika, 65, no. 3, p. 261.

K. KUMAR, R. GARG, H. KRISHNA (2017). Nakagami distribution as a reliability model under progressive censoring. International Journal of System Assurance Engineering and Management, 8, no. 1, pp. 109–122.

D. KUNDU, M. RAQAB (2009). Estimation of R=P(Y < X) for three-parameter Weibull distribution. Statistics and Probability Letters, 79, no. 17, pp. 1839–1846.

T. LIANG (2008). Empirical Bayes estimation of reliability in a positive exponential family. Communications in Statistics - Theory and Methods, 37, no. 13, pp. 2052–2065.

M. NAKAGAMI (1960). The m-distribution- A general formula of intensity distribution of rapid fading. In W. HOMAN (ed.), Statistical Methods in Radio Wave Propagation, Pergamon Press, Oxford, pp. 3–36.

J. PATEL, C. KAPADIYA, D. OWEN (1976). Handbook of Statistical Distributions. Dekker, New York.

E. PUGH (1963). The best estimate of reliability in the exponential case. Operations Research, 11, no. 1, pp. 57–61.

S.REZAEI, R. TAHMASBI, M.MAHMOODI (2010). Estimation of P(Y < X) for generalized Pareto distribution. Journal of Statistical Planning and Inference, 140, no. 2, pp. 480–494.

V.ROHATGI,A. SALEH (2012). An Introduction to Probability and Statistics. John Wiley and Sons, U.K.

B. SARACOGLU, M. KAYA (2007). Maximum likelihood estimation and confidence intervals of system reliability for Gompertz distribution in stress-strength models. Selcuk Journal of Applied Mathematics, 8, no. 2, pp. 25–36.

Y. SATHE, S. SHAH (1981). On estimating P(Y < X) for the exponential distribution. Communications in Statistics - Theory and Methods, 10, no. 1, pp. 39–47.

H. SCHAFFT, T. STATON, J. MANDEL, J. SHOTT (1987). Reproducibility of electromigration measurements. IEEE Transactions on Electron Devices, 34, no. 3, pp. 673–681.

S. SINHA (1986). Reliability and Life Testing. Wiley Eastern, New Delhi.

H. TONG (1974). A note on the estimation of P(Y < X) in the exponential case. Technometrics, 16, no. 4, p. 625.

H. TONG (1975). Letter to the editor. Technometrics, 17, no. 3, p. 393.

R. TYAGI, S. BHATTACHARYA (1989). A note on the MVU estimation of reliability for the Maxwell failure distribution. Estadistica, 41, no. 137, pp. 73–79.

V.VODA (1978). Concerning an application of the Weibull distribution to reliability aspects in metrology. Metrologia Aplicata, 25, pp. 123–124.

Z. XIA, J. YU, L. CHENG, L. LIU, W. WANG (2009). Study on the breaking strength of jute fibres using modified Weibull distribution. Composites, Part A, 40, no. 1, pp.54–59.

DOI: 10.6092/issn.1973-2201/8638