The bivariate generalized linear failure rate distribution and its multivariate extension. (English) Zbl 1247.62137
Summary: The two-parameter linear failure rate distribution has been used quite successfully to analyze life time data. Recently, a new three-parameter distribution, known as the generalized linear failure rate distribution, has been introduced by exponentiating the linear failure rate distribution. The generalized linear failure rate distribution is a very flexible life time distribution, and the probability density function of the generalized linear failure rate distribution can take different shapes. Its hazard function also can be increasing, decreasing and bathtub shaped. The main aim of this paper is to introduce a bivariate generalized linear failure rate distribution, whose marginals are generalized linear failure rate distributions. It is obtained using the same approach as was adopted to obtain the Marshall-Olkin bivariate exponential distribution. Different properties of this new distribution are established. The bivariate generalized linear failure rate distribution has five parameters and the maximum likelihood estimators are obtained using the EM algorithm. A data set is analyzed for illustrative purposes. Finally, some generalizations to the multivariate case are proposed.
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62N05 | Reliability and life testing |
Keywords:
Marshall-Olkin copula; maximum likelihood estimator; failure rate; EM algorithm; Fisher information matrixReferences:
| [1] | Aarset, M. V., How to identify a bathtub failure rate?, IEEE Transactions on Reliability, 36, 106-108 (1987) · Zbl 0625.62092 |
| [2] | Akaike, H., Fitting autoregressive model for regression, Annals of the Institute of Statistical Mathematics, 21, 243-247 (1969) · Zbl 0202.17301 |
| [3] | Bain, L. J., Analysis of the linear failure rate distribution, Technometrics, 15, 875-887 (1974) |
| [4] | Barlow, R. E.; Proschan, F., Statistical Theory of Reliability and Life Testing, Probability Models (1981), Silver Spring: Silver Spring Maryland |
| [5] | Blomqvist, N., On a measure of dependence between two random variables, Annals of Mathematical Statistics, 21, 593-600 (1950) · Zbl 0040.22403 |
| [6] | Domma, F., Some properties of the bivariate Burr type III distribution, Statistics, 44, 2, 203-215 (2009) · Zbl 1291.60027 |
| [7] | Franco, M.; Vivo, J.-M., A multivariate extension of Sarhan and Balakrishnan’s bivariate distribution and its aging and dependence properties, Journal of Multivariate Analysis, 101, 3, 494-499 (2009) |
| [8] | Joe, H., Multivariate Model and Dependence Concept (1997), Chapman and Hall: Chapman and Hall London · Zbl 0990.62517 |
| [9] | Kim, G.; Silvapulle, M. J.; Silvapulle, P., Comparison of semiparametric and parametric methods for estimating copulas, Computational Statistics and Data Analysis, 51, 2836-2850 (2006) · Zbl 1161.62364 |
| [10] | Kundu, D.; Gupta, R. D., Bivariate generalized exponential distribution, Journal of Multivariate Analysis, 100, 581-593 (2009) · Zbl 1169.62046 |
| [11] | Lehmann, E. L., Some concepts of dependence, Annals of Mathematical Statistics, 37, 1137-1153 (1966) · Zbl 0146.40601 |
| [12] | Lin, C. T.; Wu, S. J.S.; Balakrishnan, N., Parameter estimation for the linear hazard rate distribution based on records and inter-record times, Communications in Statistics—Theory and Methods, 32, 729-748 (2003) · Zbl 1048.62026 |
| [13] | Lin, C. T.; Wu, S. J.S.; Balakrishnan, N., Monte Carlo methods for Bayesian inference on the linear hazard rate distribution, Communications in Statistics—Theory and Methods, 35, 575-590 (2006) · Zbl 1093.62035 |
| [14] | Marshall, A. W.; Olkin, I., A multivariate exponential model, Journal of the American Statistical Association, 62, 30-44 (1967) · Zbl 0147.38106 |
| [15] | Meintanis, S. G., Test of fit for Marshall-Olkin distributions with applications, Journal of Statistical Planning and Inference, 137, 3954-3963 (2007) · Zbl 1122.62054 |
| [16] | Mudholkar, G. S.; Srivastava, D. K.; Freimer, M., The exponentiated Weibull family: a reanalysis of the bus motor failure data, Technometrics, 37, 436-445 (1995) · Zbl 0900.62531 |
| [17] | Nelsen, R. B., An Introduction to Copulas (1999), Springer: Springer New York · Zbl 0909.62052 |
| [18] | Pandey, A.; Singh, A.; Zimmer, W. J., Bayes estimation of the linear hazard model, IEEE Transactions on Reliability, 42, 636-640 (1993) · Zbl 0797.62094 |
| [19] | Sarhan, A.; Balakrishnan, N., A new class of bivariate distributions and its mixture, Journal of Multivariate Analysis, 98, 1508-1527 (2007) · Zbl 1116.62060 |
| [20] | Sarhan, A.; Kundu, D., Generalized linear failure rate distribution, Communications in Statistics—Theory and Methods, 38, 642-660 (2009) · Zbl 1165.62008 |
| [21] | Schwarz, G., Estimating the dimension of a model, Annals of Statistics, 6, 461-464 (1978) · Zbl 0379.62005 |
| [22] | Sen, A.; Bhattacharyya, G. K., Inference procedures for linear failure rate model, Journal of Statistical Planning and Inference, 46, 59-76 (1995) · Zbl 0822.62087 |
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