# Monotone Likelihood Ratio Property of the Poisson Signal with Three Sources of Errors in the Parameter

• Kim, Joo-Hwan (Department of Computer Science & Statistics, Dongguk University)
• Published : 1998.08.01

#### Abstract

When a neutral particle beam(NPB) aimed at the object and receive a small number of neutron signals at the detector, it follows approximately Poisson distribution. Under the four assumptions in the presence of errors and uncertainties for the Poisson parameters, an exact probability distribution of neutral particles have been derived. The probability distribution for the neutron signals received by a detector averaged over the three sources of errors is expressed as a four-dimensional integral of certain data. Two of the four integrals can be evaluated analytically and thereby the integral is reduced to a two-dimensional integral. The monotone likelihood ratio(MLR) property of the distribution is proved by using the Cauchy mean value theorem for the univariate distribution and multivariate distribution. Its MLR property can be used to find a criteria for the hypothesis testing problem related to the distribution.

#### References

1. Reviews of Modern Physics v.59 Report to The American Physical Society of the Study Group on Science and Technology of Directed Energy Weapons
2. National Bureau of Standards, Applied Mathematics Series #55 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Abramowitz, M.;Stegun, I. A.
3. Technometrics v.29 Fitting the Student-t Distribution to Grouped Data, with Application to a Particle Scattering Experiment Beckman, R. J.;Johnson, M.
4. Discrimination with Neutral Particle Beams and Protons, LA-UR 87-3140 Beyer, W. A.;Qualls, C. R.
5. Journal of Applied Statistics v.19 Algorithm AS 32 : The Incomplete Gamma Integral Bhattacharjee, G. P.
6. Advanced Calculus(Third Ed.) Buck, R. C.
7. An Introduction to Probability Theory and Its Applications: Volume II(Third Ed.) Feller, W.
8. Distributions in Statistics : Continuous Univariate Distributions I Johnson, N. L.;Kotz, S.
9. `95 춘계공동학술대회논문집(II) Exact Poisson Distribution in the Use of NPB with Aiming Errors Kim, J. H.
10. The Korean Communications in Statistics v.2 no.2 Properties of the Poisson-power Function Distribution Kim, J. H.
11. The Korean Communications in Statistics v.3 no.1 Error Rate for the Limiting Poisson-power Function Distribution Kim, J. H.
12. The Korean Communications in Statistics v.4 no.1 The Minimum Dwell Time Algorithm for the Poisson Distribution and the Poisson-power Function Distribution Kim, J. H.
13. Annals of Mathematical Statistics v.27 The Theory of Decision Procedures for Distributions with Monotone Likelihood Ratio Karlin, K.;Rubin, H.
14. Aiming Error and Its Effect on Discrimination, LA-UR 87-20q Wehner,T. R.