Vol 4 | Issue 4 | 2014 | 183-185. Asian Journal of Pharmaceutical Science & Technology e-ISSN: 2248 – 9185 Print ISSN: 2248 – 9177 www.ajpst.com FIVE-PARAMETER TYPE I GENERALIZED LOGISTIC DISTRIBUTION A.K. Olapade1, R.O. Iyiola*2 and T.J. Adesakin1 1 Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria. Department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Nigeria. 2 ABSTRACT In this paper, we consider further generalization of the four-parameter type I generalized logistic distributions in Olapade (2004) to a five-parameter type I generalized logistic distribution. Some theorems that relate the five-parameter type I generalized logistic to other distributions are established. A possible application of one of the theorems is included 2000 Mathematics Subject Classifications, Primary 62E15, Secondary 62E10. Key words: Generalized logistic distribution, Exponential distribution, Gamma distribution, Gumbel distribution, Pareto distribution, Homogeneous differential equation. INTRODUCTION The probability density function of a random variable that has logistic distribution is (1.1) and the corresponding cumulative distribution function is given by (1.2) The importance of the logistic distribution has already been felt in many areas of human endeavour. Verhulst [1] used it in economic and demographic studies. Berkson [2-4] used the distribution extensively in analyzing bio-assay and quantal response data. George and Ojo [5], Ojo [6], Ojo [7] are few of many publications on logistic distribution. The simplicity of the logistic distribution and its importance as a growth curve has made it one of the many important statistical distributions. The shape of the logistic distribution that is similar to that of the normal distribution makes it simpler and also profitable on suitable occasions to replace the normal by the logistic distribution with negligible errors in the respective theories. Balakrishman and Leung [8] show the probability density function of a random variable X that has type I generalized logistic distribution. It is given by The corresponding cumulative distribution function is (1.4) and the characteristic function of X is (1.5) The means, variances and covariances of order statistics from the type I generalized logistic distribution have been tabulated for some values of b in Balakrishman and Leung [9]. Five-Parameter Type I Generalized Logistic Distribution Olapade [10] presented a four-parameter generalized logistic distribution called extended type I generalized logistic distribution. In this research, we take a step forward by defining a suitable random variable which has a five-parameter generalized logistic distribution as shown below. Theorem 2.1: suppose a continuously distributed random variable X has a Gumbel distribution with probability density function (2.1) (1.3) Corresponding Author: R.O. Iyiola E-mail: [email protected] 183 | P a g e Vol 4 | Issue 4 | 2014 | 183-185. Assuming that has a gamma distribution with probability density function parameter p. β and λ if and only if Y has an exponential probability distribution with parameter p. Proof: If Y has exponential distribution with parameter p, then the probability density function of Y is (3.1) (2.2) We obtain the probability density function of the compound distribution using equations (2.1) and (2.2) as Then (2.3) If we introduce the location parameter µ and scale parameter σ in the equation (2.3) we have (3.2) which is the five-parameter type I generalized logistic density function. Conversely, if X is a five-parameter type I generalized logistic random variable, then . Therefore (2.4) We refer to the probability density function in equation (2.4) as the five-parameter type I generalized logistic distribution. For the rest of this paper, we shall assume that without loss of generality. The probability density function of the five-parameter type I generalized logistic distribution then becomes as shown in equation (2.3) and the corresponding cumulative distribution function is implies that implies that (3.3) and (3.4) Since this is the probability density function of an exponential random variable Y with parameter p, the proof is complete. Theorem 3.2: Suppose Y1 and Y2 are independently distributed random variables. If Y1 has the gamma distribution with probability density (3.5) (2.5) When we have the ordinary logistic distribution and when , we have the type I generalized logistic of Balakrishman and Leung [7]. For the generalized logistic distribution given in the equation (2.3), we obtain its characteristic we obtain its characteristic function as (2.6) and its analogous moment generating function as (2.7) This characteristic function and the cumulative distribution function in the equation (2.5) are important tools in proving some theorems that characterize the generalized logistic distribution as we shall see in the next section. and Y2 has the exponential distribution with probability density (3.6) Then the random variable X = InY1 – InY2 has a fiveparameter type I generalized logistic distribution with parameters p, λ and β. Proof: Let Y1 and Y2 be independently distributed random variables with probability density functions h1 and h2 respectively given above. The characteristic function of is obtained as (3.7) Similarly, the characteristic function of –InY2 is given by (3.8) Therefore, Some theorems that relate the Five-Parameter type I generalized logistic to some other distributions: We state some theorems and prove them in this section. Theorem 3.1: Let Y be a continuously distributed random variable with probability density function . Then the random variable X = has a five- parameter type I generalized logistic distribution with (3.9) Since the characteristic function of the five-parameter type I generalized logistic distribution given in the equation (2.3) is the product of the equations (3.7) and (3.8), the theorem follows. Theorem 3.3: Let Y be a continuously distributed random variable with probability density function . Then 184 | P a g e Vol 4 | Issue 4 | 2014 | 183-185. the random variable is a five-parameter type I generalized logistic random variable if and only if Y follows a generalized Pareto distribution with parameters λ, p, and β, which are positive real numbers. Proof: If Y has the generalized Pareto distribution with parameters λ, p and β then (3.10) (see McDonald and Xu [11]. Then implies that transformation is Therefore, and the Jacobian of the if the random variable X follows a five-parameter type I generalized logistic, it is easily shown that the F above satisfies the equation (3.13) Conversely, let us assume that F satisfies the equation (3.13). Separating the variables in the equation (3.13) and integrating, we have (3.15) where k is a constant. Obviously from the equation (3.14) (3.16) (3.11) Which is the five-parameter type I generalized logistic density function. Conversely, if X is a five-parameter type I generalized logistic random variable with probability distribution function shown in the equation (2.3), then (3.12) = where prime denotes differentiation, F denotes F(x) and F1 denotes F1(x) Proof: Since (3.13) Since the equation (3.12) is the cumulative distribution function for the generalized Pareto distribution given in the equation (3.9), the proof is complete. Theorem 3.4: The random variable X is a five-parameter type I generalized logistic with probability distribution function F given in the equation (2.5) if and only if F satisfies the homogenous differential equation (3.14) The value of k that makes F a distribution function is Possible Application of Theorem 3.4: From the equation (3.13), we have (3.17) Thus, the importance of the theorem (3.13) lies in the linearizing transformation (3.16). The transformation (3.16) which we call “five-parameter type I generalized logit transform” can be regarded as another generalization of Berkson’s logit transform in Berkson (1944) for the ordinary logistic model. Therefore, in the analysis of bioassay and quantal response data, if model (2.3) is used, what Berkson’s logit transform does for the ordinary logistic can be done for the five-parameter type I generalized logistic model (2.3) by the transformation (3.16). REFERENCES 1. Verhulst PF. Recherches mathematiques sur la loi d’accresioement de la population. Academic Royale Secience et Metres Bruxelee, Series, 2, 18, 1845, 1-38. 2. Berkson J. Application of the logistic function to bioassay. Journal of the American Statistical Association, 39, 1944, 357365. 3. Berkson J. Why I prefer logits to probits. Biometrics, 7, 1951, 327-339. 4. Berkson, J. A statistically precise and relatively simple method of estimating the bio-assay and quantal response, based on the logistic function. Journal of the American Statistical Association, 48, 1953, 565-599. 5. George EO and Ojo MO. On a generalization of the logistic distribution. Annals of Statistical Mathematics, 32(2)A, 1980, 161-169. 6. Ojo MO. Some Relationships between the generalized logistic and other distributions. Statistical, LVII No. 4, 1997, 573-579. 7. Ojo MO. Approximations to the distribution of the sum of the generalized logistic random variable. Kragujevac Journal of Mathematics, 24, 2002, 135-145. 8. Balakrishman N and Leung MY. Order statistics from the Type I Generalized Logistic Distribution. Communications in Statistics – Simulation and Computation, 17(1), 1988a, 25-50. 9. Balakrishman N and Leung MY. Means, variances and covariances of order statistics, BLUE’s for the Type I generalized logistic distribution and some applications. Communications in Statistics – Simulation and Computation, 17(1), 1988b, 5184. 10. Olapade AK. On Extended Type I generalized logistic distribution. International Journal of Mathematics and Mathematical Sciences, 57, 2004, 3069-3074. 11. McDonald BJ and Xu JY. A generalization of the beta distribution with applications. J Econometrics, 66, 1995, 133-152. 185 | P a g e
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