# Inverse Maxwell Distribution as a Survival Modal, Genesis and Parameter Estimation

Author Affiliations

^{1}Department of Mathematics and Statistics, DDU Gorakhpur University, Gorakhpur, INDIA^{2}Department of Mathematics and Statistics, DDU Gorakhpur University, Gorakhpur, INDIA

*Res. J. Mathematical & Statistical Sci.,* **Volume 2, Issue (7),** Pages 23-28, July,12 **(2014)**

## Abstract

If a random variable follows a particular distribution then the distribution of the inverse of that random variable is called inverted distribution. Some authors have discussed both continuous and discrete inverted distribution and its applications to various disciplines eg. In social sciences to geological, engineering, environmental and medical sciences etc. In this paper we have derived the probability density function of Inverse Maxwell distribution and studied its properties and its suitability as a survival model has been discussed by obtaining its survival and hazard functions, these are plotted on a graph paper and their properties have been discussed. The maximum likelihood estimator, an moment equation estimator of the parameter have been obtained

## References

- Stephan F.F., The expected value and variance of thereciprocal and other negative powers of a positiveBernoullian variate, Ann. Math. Statist, 16, 50-61, (1945)
- Grab E.L. and Savage I.R., Tables of the expected value of1/X for positive Bernoulli and Poisson variables, J. Amer.Statist. Assoc., 49, 169-177 (1954)
- Mendenhall W. and Lehmann E.H., Jr., An approximationto the negative moments of the positive binomial useful inlife testing, Technometrics 2, 227-41 (1960)
- Govindarajulu Z., The reciprocal of the decapitatednegative binomial variable, J. Amer. Statist. Assoc., 57,906-913 (1962)
- Govindarajulu Z., Recurrence relations for the inversemoments of the positive binomial variable, J. Amer.Statist. Assoc., 58, 467-473 (1963)
- Tiku M.L., A note on the negative moments of a truncatedPoisson variate, J. Amer. Statist. Assoc., 59, 1220-4.(1964)
- Vijsokouska E.S., Reliability of tools used in semiautomaticlathes, Russian Engineering Journal, XLVI(6),46-50 (1966)
- Stancu D.D., On the moments of negative order of thepositive Bernoulli and Poisson variables, StudiaUniversitatis Babes Bolyai Series, Mathematics andPhysics, 1, 29-31 (1968)
- Skibusky M., A characterization of hyper-geometricdistributions, J.Amer. Statist. Assoc., 65, 926-929 (1970)
- Kabe D.G., Inverse moments of discrete distributions, TheCanadian Journal of Statistics, 4(1), 133-141.(1976)
- Shahnbhag, D.N. and Busawa, T.V., On a characterizationproperty of the multinomial distribution, Ann. Math.Statist, 42, 2200 (1971)
- Chao M.T. and Strawderman W.E., Negative moments ofpositive random variables, J. Amer. Statist. Assoc. 67,429-431 (1972)
- Gupta R.C., Modified power series distribution and someof its applications, Sankhya B., 36, 288-298 (1974)
- Gupta R.C., Estimating the probability of winning (losing)in a gambler’s ruin problem with applications, Journal ofStatistical Planning and Inference, 9, 55-62 (1984)
- Lepage Y., Negative factorial moments of positiverandom variables, Industrial Mathematics, 28, 95-100(1978)
- Kumar A. and Consul P.C., Negative moments of amodified power series distribution and bias of themaximum likelihood estimator, Communications inStatistics. A., 8, 151-166 (1979)
- Cressie N. and Borkent M., The moment generatingfunction have its moments, Journal of Statistics Planningand Inference, 13, 337-344 (1986)
- Cressie N., Davis A.S. Folks J.L. and Policello G.E., Themoment generating function and negative integermoments, American Statistician., 35(3), 148-150 (1981)
- Ahmad M. and Sheikh A.K., Reliability computation forBernstein distribution strength and stress, Submitted to the10th Pak. Statistical Conference, Islamabad, Pakistan(1981)
- Ahmad M. and Sheikh A.K., Some estimation problem ofa class of inverted distributions, 44th session of theInternational Statistical Institute. Spain. 1-6, Ahmad: Onthe Theory of Inversion 51 (1983)
- Ahmad M. and Sheikh A.K., Bernstein Reliability Model:Derivation and Estimation of Parameters, ReliabilityEngineering, 8, 131-148 (1984)
- Ahmad M. and Sheikh A.K., Some remarks on the hazardfunctions of the inverted distribution, ReliabilityEngineering, 11, 1-6 . (1987)
- Jones M.C., Inverse factorial moments, Statistics andProbability Letters, 6, 37-42 (1987)
- 24. Roohi A., Negative Moments of Non-Negative RandomVariables. Un-published M. Phil dissertation, Universityof Lahore, Lahore, 535 (2002)
- Jones M.C. and Zhigljavsky Anatoly A. Approximatingthe negative moments of the Poisson distribution,Statistics and Probability Letters, 66(2), 171-182 (2004)
- Rempala G.A., Asymptotic factorial powers expansion forbinomial and negative binomial reciprocals, Proc. Amer.Math. Society, 132(1), 261-273. (2004)
- Ahmad M. and Roohi A., Characterization of the HyperPoissonProbability Function Using First Order NegativeMoments. Proceedings of the Inter- national Conferenceon Distribution Theory, Order Statistics and Inference inHonour of Barry C. Arnold, held at University ofCantabria, Santander Spain June 16-18, (2004a)
- Ahmad M. and Roohi A., On sums of somehypergeometric series functions-I, Pak. J. Statist., 20(1),193-197 (2004b)
- Ahmad M. and Roohi A., On sum of somehypergeometric series function-II. Pak. J. Statist., 21(3),245-248 (2004c)
- Ahmad M. and Roohi A., Characterization of the PoissonProbability Function, Pak. J. Statist., 20(2), 301-304(2004d)