Poisson-Mishra Distribution
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1 International Journal o Mathematics and Statistics Invention (IJMSI E-ISSN: 77 P-ISSN: - 79 Volume Issue March. 7 PP-- Poisson-Mishra Distribution Binod Kumar Sah Department o Statistics, R.R.M. Campus, Janapur, Tribhuvan University, Nepal Abstract: A one parameter Poisson-Mishra distribution has been obtained by compounding Poisson distribution with Mishra distribution o B.K.Sah(. The irst our moments about origin have been obtained. The maimum lielihood method and the method o moments have been discussed or estimating its parameter. The distribution has been itted to some data-sets to test its goodness o it. It has been ound that this distribution gives better it to all the discrete data sets which are used by Sanaran (97 and others to test goodness o it o Poisson-Lindley distribution. Keywords: Mishra distribution, Poisson-Lindley distribution, miture, moments, estimation o parameters, goodness o it. I. INTRODUCTION Lindley (98 introduced a one-parameter continuous distribution given by its probability density unction,, e ;>, >... (. This distribution is nown as Lindley distribution and its cumulative distribution unction has been obtained as F, e ; >, > (. Sanaran (97 obtained a Poisson-Lindley distribution given by its probability mass unction P ; =,,, (. assuming that the parameter o a Poisson distribution has Lindley distribution which can symbolically be epressed as Poisson Lindley. (. He discussed that this distribution has applications in errors and accidents. The irst our moments about origin o this distribution have been obtained as ( ( (. ( ( ( ( (. ( ( ( ( ( ( (.7 ( ( ( ( ( ( ( ( (.8 And its variance as (.9 Ghitany et al (9 discussed the estimation methods or the one parameter Poisson-Lindley distribution (. and its applications. In this paper, a one parameter Poisson-Mishra distribution has been obtained by compounding Poisson distribution with Mishra distribution II. MISHRA DISTRIBUTION A continuous random variable X is said to ollow Mishra distribution with parameter i it assumes only nonnegative value and its probability density unction (pd is given by Page
2 ( e ( ; ;, (. ( The r th moment about origin o this distribution is obtained as r r! ( r ( r ( r r ( (. Moment Generating Function M ( t Moment generating unction o Mishra distribution can be obtained as t M ( t e ( d ( t ( t. ( ( t... (. Distribution Function Distribution unction o this distribution unction can be obtained as F ( ( ( e d ( e ( (. III. POISSON-MISHRA DISTRIBUTION A Poisson-Mishra distribution can be obtained by miing Poisson distribution with the Mishra distribution (.. Suppose that the parameter o Poisson distribution ollows Mishra distribution (..Thus, the one parameter Mishra miture o Poisson distribution can be obtained as PMD (; = = e ( e d! ( ( ( ( (. ( ( ;,,,... ; ( (. We name this distribution as Poisson-Mishra distribution (PMD. The epression (. is the probability mass unction o PMD. IV. MOMENTS The r th moment about origin o the PMD (. can be obtained as r r e r E E ( X / = ( d (!... (. Obviously, the epression under bracet is the r th moment about origin o the Poisson distribution. So, the irst our moments about origin o the PMD can be obtained as ( e d ( (... (. = ( Taing r= in (. and using the second moment about origin o the Poisson distribution, the second moment about origin o the PMD is obtained ( ( e d ( ( (... (. ( ( Similarly, taing r= and in (. and using the respective moments o the Poisson distribution, we get inally, ater a little simpliication, the third and ourth moments about origin o the one parameter PMD Page
3 ( ( ( (. ( ( ( ( ( ( ( ( ( ( ( ( Probability Generating Function: The probability generating unction o the one parameter PMD (. can be obtained as P X ( t E ( t ( t e ( e d ( (. ( t ( t (.7 ( ( Moment Generating Function: The moment generating unction o the one parameter PMD (. can be obtained as M X ( t t ( e e ( e d ( ( e t ( e t t ( ( e (.8 V. ESTIMATION OF PARAMETER Here, at irst, the method o moments has been used to estimate the parameter o one parameter PMD. An estimate o the parameter o this distribution can be obtained by solving the epression (. o as ( ( ( (. Replacing the population moment by the respective sample moment and solving the epression (. by using Regula-Falsi method, we get an estimate o. The Maimum Lielihood Estimates: Let,,..., n be the value o observations o a random sample o size n rom the two-parameter PMD and let be the observed requency in the sample corresponding to X= (=,,..., such that t / n,where is the largest observed value having non-zero requency. Let the lielihood unction L o the two- parameter PMD is given by L. ( ( and the log lielihood unction is obtained as ( ( ( ( ( (. log L n log ( log( log ( ( ( ( ( (. The lielihood equation is obtained as log L n n ( ( ( ( ( ( ( ( (( ( Solving the epression (., we get an estimate o (. VI. GOODNESS OF FIT The one parameter Poisson-Mishra distribution (PMD has been itted to a number o discrete data-sets to which earlier one parameter Poisson-Lindley distribution (PLDhas been itted and it is ound that to all these data-sets, the PMD provides better its than the one parameter PLD o Sanaran (97 and the two-parameter PLD o Rama Shaner and Mishra (. Here the ittings o the PMD to three data-sets have been presented in the 7 Page
4 ollowing tables. The irst data is the Student's historic data Hemocytometer counts o yeast cell, used by Borah (98 or itting the Gegenbauer distribution and the second is due to Kemp and Kemp (9 regarding the distribution o mistaes in copying groups o random digits and the third is due to Beall(9 regarding the distribution o Pyraustanublilalis in 97. Table I Hemocytometer Counts o Yeast Cell Number o Yeast Observed requency Epected requency o Epected requency o Epected requency o Cell per square PLD two-parameter PLD PMD Total d. P-value Table-II Distribution o mistaes in copying groups o random digits Number o errors per group Observed requency Epected requency o PLD Epected requency o two-parameter PLD Epected requency o PMD Total 9. μ P-value Table-III Distribution o Pyraustanublilalis in 97 Number o errors per group Observed requency Epected requency o PLD Epected requency o two-parameter PLD Epected requency o PMD Total 8 Page
5 P-value The Kolmogorov - Smirnov Test (K-S Test: The K-S test is a simple non parametric method or testing whether there is a signiicant dierence between the observed requency distribution and a theoretical requency distribution. The K-S test is thereore another measure o the goodness o it o a theoretical requency distribution. For more validity o the Poisson-Mishra distribution, we may apply K-S test to test whether the PMD is a good it to the above mentioned data-sets or not. We have obtained the absolute deviation o epected relative cumulative requency o the PMD and observed relative cumulative requency o the above mentioned data-sets and are placed in the table, and respectively. Some notations used or applying K-S test are as ollows. F e = Epected Relative Cumulative Frequency o the Poisson-Mishra distribution. F o = Observed Relative Cumulative Frequency D n (Ma = the maimum absolute deviation o F e and F o D n (Tabulated = the critical value o the K-S test statistic at % level o signiicance Calculation o the absolute deviation o F e and F o : Table IV Hemocytometer counts o Yeast Cell Number o Yeast Cell per square Observed requency Epected requency o PMD F e F o F e- F o The maimum absolute deviation o F e and F o = D n(calculated =. at =.The Critical value or D n at % level o signiicance can be computed by D. 8 n (Tabulated. n Table-V Distribution o mistaes in copying groups o random digits Number o errors per group Observed requency Epected requency o PMD F e F o F e- F o Total The maimum absolute deviation o F e and F o = D n(calculated =. at =.The Critical value or D n at % level. o signiicance can be computed by D. n(tabulated n Table-VI Distribution o Pyraustanublilalis in 97 Number o errors per group Observed requency Epected requency o PMD F e F o F e- F o Total Page
6 The maimum absolute deviation o F e and F o = D n(calculated =.8 at =. The Critical value or D n at % level o signiicance can be computed by D n(tabulated..8 n Poisson-Mishra Distribution By comparing the calculated and tabulated value o K-S test statistic, we may conclude that the Poisson-Mishra distribution is a good it or the data-set. VII. CONCLUSION It has been observed that the Poisson-Mishra distribution gives better it to all the discrete data-sets than the one parameter PLD o Sanaran and the two-parameter PLD o Rama Shaner and Mishra because o p-value. Hence, it provides a better alternative to the one parameter PLD o Sanaran (97 and two-parameter PLD o Rama Shaner and Mishra (. Perhaps, it is the best alternative to all the eisting Quasi Poisson-Lindley distributions. ACKNOWLEDGEMENTS The author epresses their gratitude to the reeree or his valuable comments and suggestions which improved the quality o the research article. The hearty than goes to Proessor A. Mishra or his tremendous guidance and valuable suggestions. REFERENCES []. Beall.G. (9: The it and signiicance o contagious distribution when applied to observations on larval insects, Ecology,,- 7. []. Borah, M. (98: The Gegenbauer distribution revisited: Some recurrence relation or moments, cumulants, etc., estimation o parameters and its goodness o it, Journal o Indian Society o Agricultural Statistics, Vol., pp []. Ghitany, M. E, and Al-Mutairi, D.K. (9: Estimation methods or the discrete Poisson-Lindley distribution. J.Stat.Compt. Simul.79 (; -9. []. Kemp, C.D. and Kemp, A.W. (9: Some properties o the Hermite distribution, Biometria, Vol., pp.8-9. []. Lindley, D.V. (98: Fiducial distribution and Bayes theorem, Journal o Royal Statistical Society, Ser.B, Vol., pp.-7. []. Sah, Binod Kumar, (: Mishra distribution, International Journal o Mathematics and Statistics Inventions(IJMSI, Vol., Issue 8, pp.-7 [7]. Sanaran, M. (97: The discrete Poisson Lindley distribution, Biometrics,,-9. [8]. Shaner, R. and Mishra, A. (: A two-parameter Poisson-Lindley distribution, Statistics in Transition New Series, Vol., No., pp.-. Page
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