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Here We Provide the Mathematics of the Discrete Pareto Distribution By I. DISCRETE PARETO DISTRIBUTION Now suppose that N dependent random variates xi of the Here we provide the mathematics of the discrete Pareto discrete Pareto distribution are given. Using (2) and (3), ρ^, distribution by first expressing some fundamental blocks that the estimator of ρ, which is obtained by the MLE method the distribution is built upon. satisfies: The Lerch Transcendent Function ([1], [2], [3]) is N 0 1 X −ζ (^ρ + 1) defined as: ln(x ) = : (6) N i ζ (^ρ + 1) 1 i=1 X zj Φ(z; s; a) = ; (1) (a + j)s [7], [6] provide more information for the solution of (6) j=0 as well as an alternative poorer estimation for the parameter jzj < 1; a 6= 0; −1; −2; ··· : ρ in (5). The Hurwitz Zeta Function ζ(s; a) is a special case of II. YULE,WARING AND BETA-NEGATIVE BINOMIAL (1) which is defined as ζ(s; a) = Φ(1; s; a) where <(s) > 1, DISTRIBUTIONS a 6= 0; −1; −2; ··· 1. The Riemann Zeta Function ζ(s) is a special case of Here we express the mathematics that describe the Yule, the Hurwitz Zeta function, ζ(s) = ζ(s; 1), which is defined Waring and Beta-Negative Binomial distributions. But first by: the required concepts and terms, as well as fundamental 1 X 1 mathematics, are provided. ζ (s) = ; <(s) > 1 : (2) js Suppose that there is a finite population and consider just j=1 two outcomes: Success with the probability of p and Failure The derivate of the ζ(s) is given by: with the probability of 1 − p, i.e. a Bernoulli trial, then the 1 1 probability of k successes in n draws without replacement 0 X ln (j) X ln (j) ζ (s) = − = − ; <(s) > 1 : (3) is classically called Hypergeometric distribution. When the js js j=1 j=2 same is done with replacement it is called the Binomial distribution. In the first case the draws are dependent, while Based on (1) the General Lerch Distribution is defined in the later case they are independent. The distribution of the which has the PDF of ([4], [5]): number of failures before n successes happen is called the x z Negative Binomial distribution. This distribution is usually P [X = x] = s ; (4) (a + x) Φ(z; s; a) denoted by NB(n; p) in which p is the probability of success x = 0; 1; 2; ··· : in our Bernoulli trial. The Gamma Function is defined as: The general Lerch distribution in special cases yields some Z 1 other distributions such as Lotka, Estoup, Good, etc ([6], Γ(a) = ta−1e−tdt; <(a) > 0 : [7], [4], [5]). 0 2 The Discrete Pareto Distribution which is also one When a is a positive integer we have: Γ(a) = (a − 1)! . a = 1; s > 1; z = 1 special case of (4) happens when The Beta Function is defined by: and is defined as: Z 1 x−s x−(ρ+1) a−1 b−1 P [X = x] = = ; (5) B (a; b) = t (1 − t) dt ζ (s) ζ (ρ + 1) 0 Γ(a)Γ(b) x = 1; 2; ··· : = ; <(a) > 0; <(b) > 0 : Γ(a + b) where ρ = s − 1 > 0 and ρ 2 R 3. Zoring¨ and Altmann [4] have provided the moments of The Ascending Factorial is denoted by x(n) and is the general Lerch distribution as well as estimation of the defined as: x(n) = x (x + 1) (x + 2) ··· (x + n − 1) = Γ(x+n) (0) parameters of (4). More specifically, when ρ > 1 and r < ρ, Γ(x) ; x 2 R; n 2 N and x = 1. Simi- the rth moment of (5) about zero, Raw or Crude Moment, larly the Descending Factorial is defined by: x(n) = 0 Γ(x+1) is denoted by µr and is given by ([7], [6]): x (x − 1) (x − 2) ··· (x − n + 1) = Γ(x−n+1) ; x 2 ; n 2 and x = 1. Additionally we have x = 0 ζ (ρ − r + 1) R N (0) (n) µ = : n (n) r ζ (ρ + 1) (−1) (−x) . There is also another symbol for the ascending factorial The moment is infinite for r ≥ ρ. that is used in the related literature. It is called the Pochham- 1The <(x) symbol represents the real part of x. mer Symbol and is denoted by (x)n. The Pochhammer 2Also referred to as the Zipf distribution or the Riemann Zeta distribution. symbol should be handled with caution since it is similar to 3 (n) R indicates the set of all real numbers. the symbol of descending factorial, so we have x = (x)n. The Stirling Number of the First Kind is defined by we set n = 1. The crude moments of it are also obtained by ([5]): setting n = 1 in (10). 1 d n The Beta-Negative Binomial Distribution is an exten- s (m; n) = x ; sion to the Yule and Waring distributions which is obtained n! dx n (m) x=0 from the Negative Binomial distribution NB (n; p) when m; n 2 f0; 1; 2; · · · g : the probability of success in our Bernoulli trials, p, changes In particular when n > m or n = 0; m 6= 0 or m = between different experiments and is distributed according to 0; n 6= 0 or n = m = 0 then s(m; n) = 0. The following continuous univariate Beta distribution i.e. p ∼ Beta (α; β). relation also holds: Beta distribution is defined on the interval of (0; 1) with m two shape parameters of α and β and is denoted by X n BNB (α; β; n) ([8], [6]). x(m) = s (m; n) x ; x 2 R; m 2 f0; 1; 2; · · · g : (7) n=0 The PDF of the Beta-Negative Binomial distribution is defined as: The rth Descending Factorial Moment of a random 0 h X! i α(n)β(x)n(x) variable X is defined by µ[r] = E X(n) = E (X−r)! P [X = x] = ([6]). Since we have (7) it follows that: x!(α + β)(n) (α + β + n)(x) r Γ(α + n)Γ(β + x)Γ(α + β)Γ(n + x) 0 X 0 = ; µ[r] = s (r; j) µj : Γ(α)Γ(β)Γ(n)Γ(α + β + n + x) Γ (1 + x) j=0 x = 0; 1;2; ··· ; α > 0; β > 0; n > 0 and α; β; n 2 R : Particularly we have: Setting n = 1 and changing β by n and α by b we obtain 0 the Waring distribution. Similarly by setting n = 1, β = 1 µ[1] = µ, 0 0 and changing α by b we get the Yule distribution ([5]). µ[2] = µ2 − µ, The factorial moment of the Beta-Negative Binomial 0 0 0 distribution BNB (α; β; n), in which α, β and n can be µ[3] = µ3 − 3µ2 + 2µ, 0 0 0 0 any positive real numbers, is given by ([9]): µ[4] = µ4 − 6µ3 + 11µ2 − 6µ : (r) (r) 0 n β The Yule Distribution has the PDF of ([5]): µ[r] = ; 0 ≤ r < α (α − 1)(r) P [X = x] = b B (b + 1; x + 1) Γ(α − r)Γ(β + r)Γ(n + r) b x! = ; 0 ≤ r < α : = ; (8) Γ(α)Γ(β)Γ(n) (b + 1)(x+1) x = 0; 1; 2; ··· ; b > 0 and b 2 : R REFERENCES The PDF of the Waring Distribution is given by ([5], [1] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, [6]): Higher transcendental functions Vol.1. McGraw-Hill, 1955, B (n + x; b + 1) vol. 1. P [X = x] = B (n; b) [2] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series b n(x) (9) and Products, 7th Edition. Academic Press, 2007. = ; (x+1) (b + n) [3] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, x = 0; 1; 2; ··· ; b > 0; n ≥ 0 and b; n 2 R : NIST Handbook of Mathematical Functions. NIST (National Institute of Standards and Technology) and Cambridge Uni- The first three crude moments of the Waring distribution are versity Press, 2010. given by: [4] P. Zornig¨ and G. Altmann, “Unified representation of zipf 0 n distributions,” Computational Statistics and Data Analysis, µ1 = − ; b > 1 1 − b vol. 19, no. 4, pp. 461 – 473, 1995. 0 n 2n(1 + n) µ = − + ; b > 2 2 1 − b (1 − b)(2 − b) [5] G. Wimmer and G. Altmann, Thesaurus of Univariate Discrete Probability Distributions, 1st ed. Stamm, 1999. 0 n 6n(1 + n) 6n(1 + n)(2 + n) µ = − + − ; b > 3 : 3 1 − b (1 − b)(2 − b) (1 − b)(2 − b)(3 − b) [6] N. L. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete (10) Distributions, 3rd ed. Wiley Interscience, 2005. According to (8), (9), it is easy to see that the Yule [7] ——, Univariate Discrete Distributions, 2nd ed. Wiley distribution is a special case of the Waring distribution when Interscience, 1992. [8] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, 2nd ed. Wiley, 1994, vol. 2. [9] J. O. Irwin, “The generalized waring distribution. Part I,” Journal of the Royal Statistical Society. Series A (General), vol. 138, no. 1, pp. pp. 18–31, 1975. [Online]. Available: http://www.jstor.org/stable/2345247.
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