On the Moments , Cumulants , and Characteristic Function of the Log-Logistic Distribution

This research examine about the moments, cumulants, and characteristic function of the log-logistic distribution. Therefore, the purposes of this article are (1) finding moments of the log-logistic distribution by using moment generating function and by definition of expected values of the log-logistic random variable and (2) finding the cumulants and characteristic function of the log-logistic distribution. Log-logistic distribution has two parameters: the shape parameter α and β as a parameter scale. Moments of the log-logistic distribution can be determined by using the moment generating function or the definition of expected value. Cumulants determined by the moments that have been found previously. Furthermore, skewness and kurtosis can be determined from the log-logistic distribution. While the characteristic function is the expected value of , which I as an imaginary number. Keywords Log-logistic Distribution, Moments, Cumulants, Characteristics Function. Abstrak Penelitian ini mengkaji tentang momen, kumulan, dan fungsi karakteristik dari distribusi log-logistik. Oleh karenanya tujuan dari tulisan ini adalah (1) menentukan moment ke-r distribusi log-logistik berdasarkan fungsi pembangkit momen, dan membuktikannya berdasarkan definisi nilai harapan dari distribusi log-logstik, dan (2) menentukan fungsi karakteristik distribusi log-logistik. Distribusi log-logistik memiliki dua parameter yaitu α sebagai parameter bentuk dan β sebagai parameter skala. Momen dari distribusi log-logistik dapat ditentukan dengan menggunakan fungsi pembangkit momen atau definisi dari nilai harapan. Kumulan ditentukan dengan momen yang telah ditemukan sebelumnya. Selanjutnya, dapat ditentukan skewness dan kurtosis dari distribusi log-logistik. Sedangkan fungsi karakteristik adalah nilai harapan , dimana I sebagai angka imajiner. Kata Kunci Distribusi Log-logistik, Momen, Kumulan, FungsiKarakteristik.


I. INTRODUCTION 1
log-logistic distribution is a probability distribution of random variable that the logaritm has logistic distribution.Log-logistic distribution with two parameters, α as the form parameter and β as the scale parameter.According to [3] a random variable X is said to be a log-logistic distribution with the form parameter α and the scale parameter β, are denotated X L L (α,β), if Probability Density Function (PDF) is given by : where, and .The log-logistic distribution has the same form with log-normal distribution but has heavier tail.Probability Density Function of log-logistic distribution In statistics, the log-logistic distribution is continuous probability distribution for non-negative random variable.For instances, death value caused by cancer diagnose or treatment, and also used in hydrology for rate of flow water model and rainfall, in economics as a simple model of wealth or incomedistribution.
Generally, the main focus in the research of the loglogistic distribution propertiesis in the areas of the expected value, variance, and quantile of this distribution.In this article we extend to derive properties of the log-logistic distribution in terms of the r-moment, cumulants, and characteristic function mathematically.
Therefore, the purposes of this article are (1) Finding moments of the log-logistic distribution (α,β) by using Moment Generating Function (MGF) and proved by definition, that is through the expected value of the loglogistic random variable.(2) Finding the cumulants and characteristic function of the log-logistic distribution (α,β).
The second section of this article discuss about used methods.The Moment Generating Function (MGF) to find moment, cumulants, and characteristic function of the log-logistic distribution is determined in In the section 3.1.In the section 3.2 along with skewness function and kurtosis function, we generate the moments, cumulants, and characteristic function of the log-logistic distribution.In the section 3.3 graphically, we discuss the Probability Density Function (PDF), skewness and kurtosis function of the Log-logistic distribution.Finally, the last section is the conclusion of the article.

II. METHOD
This section discusses basic theories in mathematically deriving the moment, cumulants, and characteristic function of the log-logistic distribution (α,β).

Moment Generating Function A.
Moment generating function is used to determine rmoment of the X log-logistic random variable.Moment generating fnction is denoted ( ).The moment generating function for the log-logistic distribution is given as follows [5]: A Average and variance actually are special kind of other measurements called moments.To generate these moments, we need to differentiate the moment generating function.The moments of the log-logistic distribution then can be retrieved as follows [9]: The first moment, The second moment, The third moment, The fourth moment, The r-moment, Expectation of the Random Variable C.
Let X be a continuous random variable having a probability density function f(x) such that we have certain absolute convergence [5]; namely, The expectation of a random variable is Cumulants D.
The other characteristics or properties of distributionscan be determined by their cumulantss.On calculating to determine the cumulants, we use moments that have been determined before [6].The cumulantss are defined as follows: (10) (13) Characteristic Function E.
The characteristic function is one of important features in the probability and distribution concept.Similar to the moment generating function, the characteristic function could be used to calculate moments of the X random variable.The characteristic function can be defined as follows [8]: