Linking the Negative Binomial and Logarithmic Series Distributions via their Associated Series

Transcription

1 Revista Colombiana e Estaística Diciembre 2008, volumen 31, no. 2, pp. 311 a 319 Linking the Negative Binomial an Logarithmic Series Distributions via their Associate Series Relacionano las istribuciones binomial negativa y logarítmica vía sus series asociaas Mauricio Sainle a Departamento e Estaística, Faculta e Ciencias, Universia Nacional e Colombia, Bogotá, Colombia Abstract The negative binomial istribution is associate to the series obtaine by taking erivatives of the logarithmic series. Conversely, the logarithmic series istribution is associate to the series foun by integrating the series associate to the negative binomial istribution. The parameter of the number of failures of the negative binomial istribution is the number of erivatives neee to obtain the negative binomial series from the logarithmic series. The reasoning in this article coul be use as an alternative metho to prove that the probability mass function of the negative binomial istribution sums to one. Finally, an interpretation of the logarithmic series istribution is given by using the presente reasoning. Key wors: Convergent series, Logarithmic series istribution, Negative binomial istribution, Power series istributions. Resumen La istribución binomial negativa está asociaa a la serie obtenia e erivar la serie logarítmica. Recíprocamente, la istribución logarítmica está asociaa a la serie obtenia e integrar la serie asociaa a la istribución binomial negativa. El parámetro el número e fallas e la istribución binomial negativa es el número e erivaas necesarias para obtener la serie binomial negativa e la serie logarítmica. El razonamiento presentao puee emplearse como un métoo alternativo para probar que la función e masa e probabilia e la istribución binomial negativa suma uno. Finalmente, se presenta una interpretación e la istribución logarítmica usano el razonamiento planteao. Palabras clave: istribución binomial negativa, istribución e series e potencias, istribución logarítmica, series convergentes. a Stuent. msainleg@unal.eu.co 311

2 312 Mauricio Sainle 1. Introuction In statistical literature, many pages have been eicate fining relations between probability istributions. For example, Casella & Berger (2002) present an interesting iagram that illustrates the relationships between several common istributions through transformations or limits. Recently, an extension of this iagram has been presente by Leemis & McQueston (2008). Their paper shows the relationships among 76 univariate probability istributions: 19 iscrete an 57 continuous. The relations between some common iscrete istributions with infinite support are well known since the work of Quenouille (1949), who showe the links between the logarithmic series, Poisson, an negative binomial istributions. Also Anscombe (1950) carries out several investigations in orer to establish relations between the negative binomial an logarithmic series istributions by fining their sampling properties. He also presents the metho for parameter estimation an some tests for eparture from these istributions. Power series have been use to fin properties of iscrete istributions. For example, Samaniego (1992) uses the geometric series an relate equalities to fin the moments of the geometric istribution. In another way, Casella & Berger (2002, p. 74) use the metho of taking erivatives of the series (1 π)πn to fin the mean of the geometric istribution. Following these type of reasonings, this paper uses the iea first presente by Noack (1950) of associating a convergent power series to a probability mass function (pmf) of some infinite iscrete ranom variable. Though probability istributions moel real situations, their mathematical expressions are sometimes connecte. In this paper, integrals an erivatives of the series associate to the negative binomial an logarithmic series istributions are use to fin the relationship between these istributions. In Section 3 the parameter of the number of failures of the negative binomial istribution is foun to be the number of erivatives of the logarithmic series neee to fin the negative binomial series. The work in Section 3 can be also use to prove that the pmf of the negative binomial istribution sums to one. In Section 4 the above parameter is foun to be relate to the number of integrals taken from the series associate to a negative binomial istribution with a major parameter. The relationship between the logarithmic series istribution an the negative binomial is not as explicit as the relation between the geometric istribution an the negative binomial istribution. However, many authors have showe the existent relationship through ifferent ways (Fisher et al. 1943, Anscombe 1950, Or 1967). The metho explaine here provies a simple way to fin this relationship, which is presente in Section 4. At the en of this article it is shown a iagram inspire by the work of Leemis & McQueston (2008) that summarizes the ieas presente.

3 Linking the Negative Binomial an Logarithmic Series Distributions From Convergent Power Series to Probability Mass Functions Suppose that you have a convergent power series, i.e. a n x n = L(x) n=l with a n epening only on n, an x a real number in the convergence interval ( r, r). Then, it is possible to establish an associate pmf of some infinitely iscrete ranom variable as follows P(n x) = { anx n L(x), if n = l, l + 1,...; 0, otherwise. (1) provie that a n x n /L(x) 0 for all n. See the papers of Noack (1950) an Khatri (1959) for a general iscussion about this istribution an its properties with l = 0, an the paper of Patil (1962) in which he introuces the generalize power series istribution when n varies within an arbitrary non-null subset of non-negative integers. Example 1. It is well known that the series xn /n! converges to e x, for all x. If x > 0, then e x x n /n! > 0 for n = 0, 1, 2,..., an the pmf of the Poisson istribution is obtaine. In this paper we work with convergent series for x (0, 1) which enotes a non-trivial probability of success in a Bernoulli trial. To follow a classical notation let us replace x by π. For the evelopments presente in the following sections, take into account that if a power series converges for all π (0, 1), then, also for all π (0, 1), π a n π n = a n π πn = a n nπ n 1 an a n π n π = a n π n π = a n n + 1 πn+1 + c This is obtaine from general theorems about power series. See, e.g., Apostol (1988, p. 529) an Casella & Berger (2002, p. 74) for results about convergent series.

4 314 Mauricio Sainle 3. From the Logarithmic Series to the Negative Binomial Distribution The logarithmic series pmf is given by P(n π) = with π (0, 1). Its associate power series is π n, n = 1, 2,... n log(1 π) π + π2 2 + π3 3 + π πn + = log(1 π) (2) n Taking erivatives of this series with respect to π, the geometric series is foun. 1 + π + π 2 + π π n + = 1 1 π Multiplying both sies by (1 π), it is foun that (3) (1 π) + (1 π)π + (1 π)π (1 π)π n + = 1 where the generic term of this series, (1 π)π n, is the pmf of the well known geometric istribution, where n = 0, 1, 2,... represents the number of successes before the first failure in a sequence of inepenent Bernoulli trials with parameter π. Now let us take erivatives at both sies of (3) 1 + 2π + 3π nπ n 1 + (n + 1)π n + = 1 (1 π) 2 (4) an following (1), i.e. in this case by multiplying by (1 π) 2, one fins a series which sums to 1 with generic term ( ) ( ) n n 1 (n + 1)(1 π) 2 π n = (1 π) 2 π n = (1 π) 2 π n which is the pmf of the negative binomial istribution with parameters 2 an π, for n = 0, 1, 2,.... Again, let us take erivatives of (4) π + 4 3π (n + 1)nπ n 1 + (n + 2)(n + 1)π n + = 2 (1 π) 3 an the associate pmf is foun to be, again as in (1), ( ) ( ) (n + 2)(n + 1) n n 1 (1 π) 3 π n = (1 π) 3 π n = (1 π) 3 π n for n = 0, 1, 2,... which is the pmf of the negative binomial istribution with parameters 3 an π.

5 Linking the Negative Binomial an Logarithmic Series Distributions 315 Following this iea, taking k erivatives at each sie of (2), one fins the series from which the negative binomial istribution with parameters k an π can be obtaine. Given the previous work, to emonstrate this result (by mathematical inuction), we nee to assume that the result is true for k an emonstrate that it is true for k + 1, as follows: Suppose that by taking k erivatives of (2) we fin the series (k 1)! + k! 1! (k + 1)! π + π ! (k + n 1)! π n + = n! (k 1)! (1 π) k (5) which is associate to the negative binomial istribution with parameters k an π, following (1) (k + n 1)! (1 π) k π n = (k 1)!n! ( k + n 1 k 1 ) (1 π) k π n for n = 0, 1, 2,.... Let us take erivatives of (5) with respect to π k! + (k + 1)! π + 1! (k + 2)! π ! (k + n)! π n + = n! Following (1) we obtain ( ) k + n (1 π) k+1 π n, for n = 0, 1, 2,... k k! (1 π) k+1 which is the pmf of the negative binomial istribution with parameters k + 1 an π, an the result is emonstrate. As state above, taking the generic term of the series an iviing it by the sum of the series, we fin the associate pmf. If instea of this if we take all the series an ivie it by its sum, we will fin a proof that the associate pmf sums to one. This coul be useful if we start, for instance, from the wiely known convergence of the geometric series, following the steps of this section we fin the series associate to the negative binomial istribution an iviing by its sum, we emonstrate that the pmf of this istribution sums to one. Casella & Berger (2002, p. 95) mentions that the traitional proof of the fact that the negative binomial pmf sums to one utilizes an extension of the binomial theorem that inclues negative exponents. This usual proof leas to the use of binomial coefficients with negative integers, which is not taught in regular calculus courses an which many people fin ifficult to hanle. Thus, the above results can be use to present a more peagogical way to prove that the pmf of the negative binomial istribution sums to one. 4. From the Negative Binomial to the Logarithmic Series Distribution In the previous section we foun that by taking erivatives of the series associate to the negative binomial istribution with parameters k an π, the series

6 316 Mauricio Sainle associate to the negative binomial istribution with parameters k + 1 an π is obtaine. Conversely, taking the series associate to the negative binomial with parameters k an π an integrating it, it can be foun the series associate to the negative binomial with parameters k 1 an π, taking the constant of integration to be equal to (k 2)! in the sie of the series an zero in the sie of the sum. This means that if we are stuying the probability of getting n successes before k failures, an if we integrate the associate series, we will obtain a series with an associate pmf for stuying the probability of getting n successes before k 1 failures. The geometric pmf, which is equivalent to the negative binomial pmf with parameters 1 an π, has associate the geometric series, an from the above iea, it is natural to think of integrating it to fin a way of moeling the probability of getting n successes, taking the parameter of the number of failures as zero. The constant of integration (k 2)! = (1 2)! = ( 1)! is not efine, so for convenience it is taken as zero in this case, an the following series is obtaine π + π2 2 + π πn + = log(1 π) n This series has the associate pmf π n, for n = 1, 2,... n log(1 π) which is the logarithmic series pmf. The interpretation of the above istribution is not simple. In fact, this istribution is a limit case of the negative binomial istribution, an it is foun by Fisher et al. (1943) through a ifferent reasoning. It is also presente by Anscombe (1950) as a limiting process of the negative binomial istribution by consiering a sample of N Bernoulli trials (N 0), letting N ten to infinity an k to zero in terms of the Gamma function, as follows Γ(k + n) n!γ(k) (1 π)k π n, for n = 0, 1, 2,... Or (1967), by graphical methos, foun in a natural way that the logarithmic series istribution occurs as the limit of the negative binomial, when the parameter of the number of failures tens to zero. 5. Conclusions The parameter of the number of failures of the negative binomial istribution is the number of erivatives neee to obtain the negative binomial series from the logarithmic series. Thus the negative binomial pmf with parameters k an π is foun by

7 Linking the Negative Binomial an Logarithmic Series Distributions 317 P NB (n k, π) = { k π k ( (n + k) 1 π n+k)/ k π k ( log(1 π) ), if n = 0, 1, 2,...; 0, otherwise. π n = log(1 π) n n=1 π π π n = 1 1 π π π (n + 1)π n 1 = (1 π) 2 π π (n + 2)(n + 1)π n 2 = (1 π) 3 π... π. (k + n 2)! π n (k 2)! = n! (1 π) k 1 π π (k + n 1)! π n (k 1)! = n! (1 π) k anπ n =L(π) LogS(π) G(π) NB(2, π) NB(3, π) NB(k 1, π) NB(k, π) Figure 1: Relationships between logarithmic series istribution LogS(π), geometric istribution G(π) an negative binomial istribution NB(n, π).

8 318 Mauricio Sainle Conversely, this iea can be applie by replacing erivatives by i integrals with the analogous terms of the series associate to the negative binomial with parameters k an π. For i = k k we can fin the negative binomial with parameters k an π, for 1 < k < k. For i = k 1 we can fin the geometric istribution, an finally, for i = k we can fin the logarithmic series istribution. Do not forget to take into account the appropriate integration constants. The logarithmic series istribution is interprete by using the presente metho as a istribution for moeling the probability of getting n successes letting the number of failures of the negative binomial experiment ten to zero, as shown by Or (1967). Figure 1 shows the relationships between the ifferent pmf of the above mentione istributions. The link to go from one istribution to another one follows the next algorithm. First, choose a given pmf. Secon, fin the associate series to such pmf. Thir, either ifferentiate or integrate that series as many times as neee to obtain the series associate to the pmf wante. Fourth, ivie the general term of the obtaine series by its summation. For instance, the first erivative of the geometric series, associate to the geometric istribution, prouces the series associate to the negative binomial istribution with parameters 2 an π. While integrating once an taking the integration constant as zero, prouces the logarithmic series, associate to the logarithmic series istribution. Similarly, if one woul like to obtain the negative binomial pmf with parameters 3 an π from the geometric series, thus one shoul ifferentiate it twice an finally ivie the general term of the series foun by the summation. Finally, if you o not follow the fourth step in the algorithm above, an instea of this you ivie the series obtaine in the thir step by its sum, you will fin a way to prove that the pmf, of the istribution associate to the series, sums to one. Acknowlegments The author woul like to thank A. Villamarín, S. Granaa an R. Herrera for reaing the preliminary versions of this paper an F. H. Nieto for his suggestions. The author also appreciates the very helpful comments offere by the referees an the eitor. [ Recibio: marzo e 2008 Aceptao: octubre e 2008 ] References Anscombe, F. J. (1950), Sampling Theory of the Negative Binomial an Logarithmic Series Distributions, Biometrika 37(3/4), Apostol, T. M. (1988), Calculus, secon en, Reverté.

9 Linking the Negative Binomial an Logarithmic Series Distributions 319 Casella, G. & Berger, R. L. (2002), Statistical Inference, secon en, Duxbury Thomson Learning, Pacific Grove, Unite States. Fisher, R. A., Corbet, A. S. & Williams, C. B. (1943), The Relation between the Number of Species an the Number of Iniviuals in a Ranom Sample of an Animal Population, The Journal of Animal Ecology 12(1), Khatri, C. G. (1959), On Certain Properties of Power-Series Distributions, Biometrika 46(3/4), Leemis, L. M. & McQueston, J. T. (2008), Univariate Distribution Relationships, The American Statistician 62(1), Noack, A. (1950), A Class of Ranom Variables with Discrete Distributions, The Annals of Mathematical Statistics 21(1), Or, J. K. (1967), Graphical Methos for a Class of Discrete Distributions, Journal of the Royal Statistical Society 130(2), Patil, G. P. (1962), On Homogeneity an Combine Estimation for the Generalize Power Series Distribution an Certain Applications, Biometrics 18(3), Quenouille, M. H. (1949), A Relation between the Logarithmic, Poisson, an Negative Binomial Series, Biometrics 5(2), Samaniego, F. J. (1992), Elementary Derivations of Geometric Moments, The American Statistician 46(2),