A  new  test  of  Poissonity  based  on  a  characteristic  property  of  Poisson  distributions  is 
proposed

Gupta , Arjun K.

Székely, Gábor J.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

,PERIODICA POLl'TECHNICA SER CIVIL ENG, VOL, 36, NO, 4, pp, 365-36d (1992j GEYSERS AND TESTS 1 Gabor J. SZEKELY and Arjun K. GUPTA* Department of Mathematics, Faculty of Civil Engineering Technical LT niversity, H-1521 Budapest "Department of Mathematics and Statistics, BGSU, USA Received: December 1. 1992 Abstract 'Dieu! La regarderai-je?' (E. R.: CdB Quatrieme acte, scene V) A new test of Poissonity based on a characteristic property of Poisson distributions is proposed. Keywords: Poisson distribution, exponential distribution, characterization of distribu-tions. Prelude In 1875 a young professor of our University, Gyula Konig (also Rector between 1891 and 1894) gave an interesting lecture on the determination of the period cjJ of a periodic event if only very rough observations are available, say, the signs of cos(kcjJ) k = 0, 1, .... E.g. if we can only observe periodic eruptions of a geyser occurring at night ) or in the daytime (+) then from this sequence of + and signs we can reconstruct the value of cjJ( mod 7i). If Wn denotes the changes of signs in this sequence of + and -of length n, then et = lim Wn. 1i 71-00 n This result was published in E:ONIG (1876). Variations Variation 1 Konig's idea can be used to find a solution of the algebraic equation f ( ) 1! ,,- I 0 X =aox +alx +···+a,,=. 1 I{c,;earch supported hy HUllgariall ;\atiollal FOUlldatioll for Scil'lltific l\C'sC'arch. Grallt )\!2. 1'1O.'i 366 G. J. SZeKELY and A. K. GUPTA If we compute and Wn denotes the changes of signs in the sequence 0:0, ... O:n, then one of the roots (zeros) of our equation is where and ::vo = r ( cos w + i sin w) , w l' Wn -= Im-11" n~oo n T = lim 10:,,1-;,1 n--cx:: (in fact Xo is a root with the smallest absolute value r). VaTiation 2 If the period <P is random and the random periods cPll cPl., ... between consec-utive eruptions of a geyser are independent, identically distributed random variables with unknown distribution function then we might want to determine F from rough observations of n = 1,2, .. ,. (5" can be considered a random arithmetic progression where <P1, <P'2, ... are not necessarily only identically distributed. The case rP'l. = ... = cP would correspond to Konig's sequence SA: = Suppose that a rough observation of 511 is the integer part of Sll denoted by [Sit] (if the unit is one day, then we can only observe the dates of eruptions). By the law of large numbers the expectation of F is n-:x n n-x n with probability one. A much deeper result of BAH.TFAJ (1966) and EHI)()S R(;i\\{j (1970) shows that not only the expectation of F but F itself can also be recon-structed from the sequence [5,,] (see also CS()H(;6·R(;vJ~sZ (1981)). GEYSERS ASD TESTS 367 Rondo Try to test the hypothesis that the random number of eruptions (of a geyser) per day is a Poisson random variable. In other words if VI, V2, ... , Vn denote the random number of eruptions on the first, second, ... ,n-th day, then we want to test the hypothesis that VI, V2, ... ,Vn are (independent) Poisson random variables, P(V; = k) = k! i = 1,2, ... ,n; k = 0,1,2, .... is the unknovvn expected number of eruptions per day). Testing Poissonity in a powerful way is not an easy problem. Neither IS easy to get rid of the unknovm parameter A (here A is neither location nor scale). In fact, one can prove (see PE;\A-RoHATGI-SZEEELY (1992)) there is no (nonconstant) function of 1/1, ... ,1/n whose distribution function does not depend on A. Approach the problem in the following way. If <PI, <P2,'" are inde-pendent, exponentially distributed random variables with density function Ae - AX for x ;:::: 0 (and 0 otherwise) and 5" = <P i, then in the sequence of increasing integer parts [51],[52],[5;1], ... the number of O's, l's, 2's, etc. is a sequence VI, V'l., ... of independent Poisson random variables. By Vari-ation 2 above, this sequence uniquely determines the distribution F of cp's, therefore F must be exponential, v is Poisson if and only if <P is exponential. Thus, testing for Poissonity of v is equivalent to testing for exponentiality of <p. In the density of <p, however, the parameter A is a scale parameter and, thus, the problem of testing is easier and well known (see e.g. SHAPlRO and \VILE (1962), LILLIEFORS (1969), BAHTIIOLO\1E\\' (19.57). :-'IOHA\' (1951). EpSTEI\' (1960), STOHMEH (1962)), There is only one more missing link: how can we reconstruct the se-quence <PI, <P:I., .,. from VI, V'2", .. Take independent, uniformly distributed samples of size Vi on the interval [i - 1, i] i 1,2, . , . , n. If the (in-creasingly) ordered joint sample is ~l, ~2," . ,~'" where m = L;~l Vi, then <Pi = ~i+l - ~i, i = 1,2, ... is a sequence of independent, exponentially distributed random variables with parameter A, i.e. {<Pi} and {cf;i} are identically distributed therefore {<Pi} can play the role of {<Pi} in the com-putations above. Acknowledgements Tht' ~1I(hors ,up thallkflll to G. TlIslIRdi for all illt('i'p:'till~ disclIssion 011 this topic. 368 G. J. 3ZEKELY and A. K. GUPTA Referen.ces BARTHOLOMEW, D. J. (1957): Testing for Departure from the Exponential Distribution. Biometrika, Vo!. 44, pp. 253-256. BARTFAI, P. (1966): Die Bestimmung der zu einem wiederkehrenden Prozess gehorenden Verteilungsfunktion aus den mit Fehlern behafteten Daten einer einzigen Realisa-tion. Studia Sci. Math. Hung. Vo!. 1, pp. 161-168. CSORGO, M. - REVESZ, P. (1981): Strong Approximations in Probability and Statistics, Akademiai Kiad6, Budapest. EpSTEIN, B. (1960): Tests for the Validity of the Assumption that the Underlying Distri-bution of Life is Exponential. Teclmometrics, Vo!. 2, pp. 83-10l. ERDOS, P. - RENYI, A. (1970): On a New Law of Large Numbers, JOUT. Analyse M athe.matique Vo!. 22, pp. 103-11l. LILLlEFORS, H. W. (1969): On the Kolmogorov-Smirnov Test for the Exponential Distri-bution with Mean Unknown. J. AmcT. Statisl. Assoc., Vo!. 64, pp. 387-389. KONIG, Gy. (1876): On a General Solution ... , Miiegyetemi Lapok, Vol. 1, pp. 20-28. II2-II6. 2vloRAN, P. A. P. (19.51): The Random Divisioll of all Inten'al 11. J. Hog. St.al-isl. Soc. SeT. E, Vo!. 13, pp. 147-1·50. PE~A, E. A. - ROHATGI, V. K. SZEKELY, G. j.(1992): 011 the NOIl-existence of Ancillary Statistics, PTobability and Statistics LettcTs Vo!. 1·5. pp. :5.57-360. SHAPIRO, S. S. WILK, M. B. (1972): An Analysis of Variance Test for the Exponential Distribution, Technometrics, Vo!. 14, pp. 35.5-370. STOR)"1ER, H. (1962): On a Test for the Exponential Distribution, Metrika, Vo!. .5, pp. 128-137. 

GEYSERS AND TESTS

https://pp.bme.hu/ci/article/download/3854/2959

GEYSERS AND TESTS

Abstract

Similar works

Full text

Available Versions

Periodica Polytechnica (Budapest University of Technology and Economics)