×

Some new approaches to multivariate probability distributions. (English) Zbl 0633.62044

Authors’ abstract: We extend and generalize to the multivariate set-up our earlier investigations related to expected remaining life functions and general hazard measures including representations and stability theorems for arbitrary probability distributions in terms of these concepts. The univariate case has been discussed in detail by the authors in Adv. Appl. Probab. 12, 903-921 (1980; Zbl 0454.62085).
Reviewer: K.S.Miller

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62N05 Reliability and life testing

Citations:

Zbl 0454.62085
Full Text: DOI

References:

[1] Basu, A. P., Bivariate failure rate, J. Amer. Statist. Assoc., 66, 103-104 (1971) · Zbl 0218.62118
[2] Bertoluzza, C.; Forte, B., Mutual dependence of random variables and maximum discretized entropy, Ann. Probab., 13, 630-637 (1985) · Zbl 0563.60023
[3] Billingsley, P., (Convergence of Probability Measures (1968), Wiley: Wiley New York) · Zbl 0172.21201
[4] Block, H. W., Monotone failure rate for multivariate distributions, Naval Research Logistics Quarterly, 24, 627-637 (1977) · Zbl 0383.62069
[5] Burrill, C. W., (Measure, Integration and Probability (1972), McGraw-Hill: McGraw-Hill New York) · Zbl 0248.28001
[6] Cox, D. R., Regression models and life tables, J. Roy. Statist. Soc. Ser. B, 34, 187-220 (1972) · Zbl 0243.62041
[7] Davies, P. L.; Shanbhag, D. N., A generalization of a theorem of Deny with applications in characterization theory, Quart. J. Math. (Oxford) (1987), in press · Zbl 0617.60016
[8] Galambos, J.; Kotz, S., (Characterization of Probability Distributions (1978), Springer-Verlag: Springer-Verlag Heidelberg/New York), Lecture Notes in Mathematics, No. 675 · Zbl 0381.62011
[9] Glänzel, W.; Telcs, A.; Schubert, A., Characterization by truncated moments and its application to Pearson-type distributions, Z. Wahrsch. Verw. Gebiete, 66, No. 2, 173-183 (1984) · Zbl 0523.62016
[10] Gupta, R. C., On the characterization of distributions by conditional expectations, Commun. in Statist., 4, 99-103 (1975) · Zbl 0299.62011
[11] Hall, W. J.; Wellner, J. A., Mean residual life, (Csörgö, M.; Dawson, D. A.; Rao, J. N.K; Saleh, A. K.Md. E., Statistics and Related Topics (1981), North-Holland: North-Holland Amsterdam), 169-184 · Zbl 0481.62078
[12] Hamdan, M. A., On a characterization by conditional expectations, Technometrics, 14, 497-499 (1972) · Zbl 0234.62006
[13] Hollander, M.; Proschan, F., Nonparametric Concepts and Methods in Reliability, (Krishnaiah, P. R.; Sen, P. K., Handbook of Statistics, Vol. 4 (1984), North-Holland: North-Holland Amsterdam), 613-655 · Zbl 0596.62099
[14] Johnson, N. L.; Kotz, S., (Distributions in Statistics: Continuous Multivariate Distributions (1972), Wiley: Wiley New York) · Zbl 0248.62021
[15] Johnson, N. L.; Kotz, S., A vector multivariate hazard rate, J. Multivariate Anal., 5, 498 (1975) · Zbl 0323.60014
[16] Johnson, N. L.; Kotz, S., On some generalized Farlie-Gumbel-Morgenstern distributions, Commun. in Statist., 4, 415-428 (1975) · Zbl 0342.62006
[17] Kendall, D. G., Extreme points methods in stochastic analysis, Z. Wahrsch. Verw. Gebiete, 1, 295-300 (1963) · Zbl 0113.11904
[18] Kotlarski, I. I., On a characterization of some probability distributions by conditional expectation, Sankhy \(a\), Ser. A, 34, 461-467 (1972) · Zbl 0275.60019
[19] Kotz, S.; Shanbnag, D. N., Some new approaches to probability distributions, Advan. Appl. Probab., 12, 903-921 (1980) · Zbl 0454.62085
[20] Krishnaiah, P. R., On generalized gamma type distributions and their applications in reliability, (Tsokos, C. P.; Shimi, I. N., Theory and Applications in Reliability, Vol. 1 (1977), Academic Press: Academic Press New York) · Zbl 0415.62035
[21] Lau, Ka-Sing; Rao, C. R., Integrated Cauchy functional equation and characterizations of the exponential law, Sankhy \(a\) A, 44, 72-90 (1982) · Zbl 0584.62019
[22] Marshall, A. W., Some comments on the hazard gradient, Stochastic Process Appl., 3, 293-300 (1975) · Zbl 0329.62040
[23] Parthasarathy, K. R., (Probability Measures on Metric Spaces (1967), Academic Press: Academic Press New York) · Zbl 0153.19101
[24] Phelps, R. R., (Lectures on Choquet’s Theorem (1966), Van Nostrand: Van Nostrand Princeton) · Zbl 0135.36203
[25] Prakasa Rao, B. L.S, On a property of bivariate distributions, J. Multivariate Anal., 4, 106-113 (1974) · Zbl 0275.60020
[26] Prohorov, Yu. V., Convergence of random processes and limit theorems in probability theory, Theoret. Probab. Appl., 1, 177-238 (1956) · Zbl 0075.29001
[27] Puri, P. S.; Rubin, H., On a characterization of the family of distributions with constant multivariate failure rates, Ann. Probab., 2, 738-740 (1974) · Zbl 0286.60007
[28] Rao, C. R.; Shanbhag, D. N., Recent results on characterization of probability distributions: a unified approach through extensions of Deny’s theorem, Advan. Appl. Probab., 18, 660-678 (1986) · Zbl 0607.62005
[29] Seneta, E., (Non-negative Matrices and Markov Chains (1981), Springer: Springer New York) · Zbl 0471.60001
[30] Shanbhag, D. N., Characterizations for exponential and geometric distributions, J. Amer. Statist. Assoc., 65, 1256-1259 (1970) · Zbl 0224.62007
[31] Shanbhag, D. N., Some refinements in distribution theory, Stankhy \(a\), Ser. A, 41, 252-262 (1979) · Zbl 0477.60017
[32] Shanbhag, D. N.; Bhaskara Rao, M., A note on characterizations of probability distributions based on conditional expected values, Sankhy \(a\), Ser. A, 37, 297-300 (1975) · Zbl 0355.62011
[33] Zahedi, H., Some new classes of multivariate survival distribution functions, J. Statist. Plann. Inference, 11, 171-188 (1985) · Zbl 0587.62177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.