A GENERALIZED LOGISTIC DISTRIBUTION

Transcription

1 A GENERALIZED LOGISTIC DISTRIBUTION SARALEES NADARAJAH AND SAMUEL KOTZ Received 11 October 2004 A generalized logistic distribution is proposed, based on the fact that the difference of two independent Gumbel-distributed random variables has the standard logistic distribution. 1. Introduction If X 1 and X 2 are independent Gumbel-distributed random variables with the common cdf Fx = exp { exp x }, 1.1 then it is well known that the difference Z = X 1 X 2 has the standard logistic distribution with the pdf f Z z = expz { 1+expz } for <z<. The properties of this distribution and its generalizations have been studied by several authors. Of particular eminence are the numerous papers on this topic by Professor N. Balakrishnan and his colleagues; see, for example, Balakrishnan [1, 2, 3], Balakrishnan and Aggarwala [4], Balakrishnan et al. [5, 7, 12], Balakrishnan and Chan [6], Balakrishnan and Joshi [8], Balakrishnan and Kocherlakota [9], Balakrishnan and Leung [10], BalakrishnanandMalik [11], Balakrishnanand Puthenpura [13], Balakrishnan and Sandhu [14], and Balakrishnan and Wong [15]. In this short note, we construct a new generalization of 1.2 bytakingx i, i = 1,2, to have the general Gumbel distribution with the cdf { F i x = exp exp x µ } i 1.3 for <x<, <µ i <, and > 0. This distribution which is also known as the extreme-value distribution of type I has received special attention in the probabilisticstatistical literature and in various applications in the second half of the twentieth century. Copyright 2005 International Mathematics and Mathematical Sciences 2005: DOI: /IJMMS

2 3170 A generalized logistic distribution A recent book by Kotz and Nadarajah [16], which describes this distribution, lists over fifty applications ranging from accelerated life testing through to earthquakes, floods, horse racing, rainfall, queues in supermarkets, sea currents, wind speeds, and track race records to mention just a few. 2. The generalization The pdf corresponding to 1.3is f i x = 1 exp x µ { i exp exp and thus the pdf of Z = X 1 X 2 can be written as x µ } i, 2.1 f Z z = f 1 x f 2 x zdx = 1 exp x µ 1 σ 1 σ 1 exp x z µ 2 { exp exp x µ } 1 σ 1 { exp exp x z µ 2 } dx. 2.2 Setting y = exp x/σ 1, 2.2 can be expressed as where I denotes the integral f Z z = 1 µ1 exp + µ 2 + z I µ 1,µ 2,σ 1,, 2.3 σ 1 I µ 1,µ 2,σ 1, = 0 [ { y σ1/σ2 exp exp µ1 σ 1 y +exp µ2 + z y σ1/σ2 }] dy. 2.4 We refer to 2.3 asthegeneralized logistic distribution. The integral term in 2.4 isdifficult to calculate. However, for some particular choices of µ 1,µ 2,σ 1,, one can obtain the following explicit expressions. i If σ 1 = = σ, then by standard integration one can obtain 1 { exp µ1 /σ +exp µ 2 + z /σ } If, in addition, µ 1 = µ 2 = µ, then the above reduces to exp 2µ/σ { 1+expz/σ }

3 ii If σ 1 = 2, then one can show by [17, equation ] that α πα exp 8 αβ 2 4 { 2erfc αβ + αβ 2 erfc 2 S. Nadarajah and S. Kotz 3171 } αβ α2 β 2 4, 2.7 where α = exp{µ 2 + z/ }, β = exp{µ 1 /2 }, anderfc, denotes the complementary error function defined by erfcx = 1 2 x exp t 2 dt. 2.8 π 0 iii If 0 <σ 1 / = p/q < 1wherep 1andq 1 are co-prime integers, then one can show by [17, equation ] that q 1 j=0 α j p1 + j Γ 1+ j! q p+1 F q 1, p,1+ β 1+p1+j/q p1 + j q ; q,1+ j; 1q p p α q q q β p, 2.9 where α = exp{µ 2 + z/ }, β = expµ 1 /σ 1, k,a denotes the sequence k,a = a k, a +1 k,..., a + k 1, 2.10 k mf n denotes the generalized hypergeometric function defined by mf n α1,...,α m ;β 1,...,β n ;x = k=0 α1 k α m k β1 k β x k n k k!, 2.11 and c k = cc +1 c + k 1 denotes the ascending factorial. iv If σ 1 / = p/q > 1wherep 1andq 1 are coprime integers, then one can showagain by [17, equation ]that p 1 j=0 q β j Γ 1+ pj! q+1 F p 1, q1 + j p q,1+ α 1+q1+j/p q1 + j p ; p,1+ j; 1p q q β p p p α q, 2.12 where α = exp{µ 2 + z/ } and β = expµ 1 /σ 1.

4 3172 A generalized logistic distribution 0.3 f x x σ 1 / = 0.2 σ 1 / = 1 σ 1 / = 2 σ 1 / = 5 σ 1 / = 10 Figure 2.1. The generalized logistic pdf 2.3forσ 1 / = 0.2,1,2,5,10, = 1, µ 1 = 0, and µ 2 = f x x σ 1 / = 0.2 σ 1 / = 1 σ 1 / = 2 σ 1 / = 5 σ 1 / = 10 Figure 2.2. The generalized logistic pdf 2.3forσ 1 / = 0.2,1,2,5,10, = 1, µ 1 = 1, and µ 2 = 0. Figures 2.1 and 2.2 illustrate possible shapes of the pdf 2.3 for selected values of µ 1,µ 2,σ 1,. The magnitude of σ 1 / clearly controls the shape of the pdf. In fact, if µ 1 = 0, then f Z z 1 { } µ2 + z µ2 + z exp exp exp 2.13

5 S. Nadarajah and S. Kotz 3173 as σ 1 / 0. Also, f z 0foreveryz 0, asσ 1 /. On the other hand, if µ 1 0, then f z 0foreveryz 0, irrespective of whether σ 1 / 0orσ 1 /. 3. Applications The standard logistic distributiongiven by 1.2 has important uses in describing growth and as a substitute for the normal distribution. It has also attracted interesting applications in the modeling of the dependence of chronic obstructive respiratory disease prevalence on smoking and age, degrees of pneumoconiosis in coal miners, geological issues, hemolytic uremic syndrome data for children, physiochemical phenomenon, psychological issues, survival time of diagnosed leukemia patients, and weight gain data. The main feature of the generalized logistic distribution in 2.3 is that new parameters are introduced to control both location and scale. Thus, 2.3 allowsforagreaterdegreeofflexibility and we can expect this to be useful in many more practical situations. References [1] N. Balakrishnan, Order statistics from the half logistic distribution, J. Statist. Comput. Simulation , no. 4, [2], Approximate maximum likelihood estimation for a generalized logistic distribution, J. Statist. Plann. Inference , no. 2, [3] N. Balakrishnan ed., Handbook of the Logistic Distribution, Statistics: Textbooks and Monographs, vol. 123, Marcel Dekker, New York, [4] N. Balakrishnan and R. Aggarwala, Relationships for moments of order statistics from the righttruncated generalized half logistic distribution, Ann. Inst. Statist. Math , no. 3, [5] N. Balakrishnan, M. Ahsanullah, and P. S. Chan, On the logistic record values and associated inference, J. Appl. Statist. Sci , no. 3, [6] N. Balakrishnan and P. S. Chan, Estimation for the scaled half logistic distribution under type II censoring, Comput. Statist. Data Anal , no. 2, [7] N. Balakrishnan, S. S. Gupta, and S. Panchapakesan, Estimation of the mean and standard deviation of the logistic distribution based on multiply type-ii censored samples, Statistics , no. 1-2, [8] N. Balakrishnan and P. C. Joshi, Means, variances and covariances of order statistics from symmetrically truncated logistic distribution, J. Statist. Res , no. 1-2, [9] N. Balakrishnan and S. Kocherlakota, On the moments of order statistics from the doubly truncated logistic distribution, J. Statist. Plann. Inference , no. 1, [10] N. Balakrishnan and M. Y. Leung, Means, variances and covariances of order statistics, BLUEs for the type I generalized logistic distribution, and some applications, Comm. Statist. Simulation Comput , no. 1, [11] N. Balakrishnan and H. J. Malik, Moments of order statistics from truncated log-logistic distribution, J. Statist. Plann. Inference , no. 2, [12] N. Balakrishnan, H. J. Malik, and S. Puthenpura, Best linear unbiased estimation of location and scale parameters of the log-logistic distribution, Comm. Statist. Theory Methods , no. 12, [13] N. Balakrishnan and S. Puthenpura, Best linear unbiased estimators of location and scale parameters of the half logistic distribution, J. Statist. Comput. Simulation , no. 3-4,

6 3174 A generalized logistic distribution [14] N. Balakrishnan and R. A. Sandhu, Recurrence relations for single and product moments of order statistics from a generalized half logistic distribution with applications to inference, J.Statist. Comput. Simulation , no. 4, [15] N. Balakrishnan and K. H. T. Wong, Best linear unbiased estimation of location and scale parameters of the half-logistic distribution based on type II censored samples, Amer. J. Math. Management Sci , no. 1-2, [16] S. Kotz and S. Nadarajah, Extreme Value Distributions. Theory and Applications, Imperial College Press, London, [17] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions, Gordon & Breach Science, New York, [18], Integrals and Series. Vol. 2. Special Functions, Gordon & Breach Science, New York, [19], Integrals and Series. Vol. 3. More Special Functions, Gordon & Breach Science, New York, Saralees Nadarajah: Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA address: snadaraj@chuma1.cas.usf.edu Samuel Kotz: Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052, USA address: kotz@seas.gwu.edu

7 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization