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Simulation of Moment, Cumulant, Kurtosis and the Characteristics Function of Dagum Distribution 264 IJCSNS International Journal of Computer Science and Network Security, VOL.17 No.8, August 2017 Simulation of Moment, Cumulant, Kurtosis and the Characteristics Function of Dagum Distribution Dian Kurniasari1*,Yucky Anggun Anggrainy1, Warsono1 , Warsito2 and Mustofa Usman1 1Department of Mathematics, Faculty of Mathematics and Sciences, Universitas Lampung, Indonesia 2Department of Physics, Faculty of Mathematics and Sciences, Universitas Lampung, Indonesia Abstract distribution is a special case of Generalized Beta II The Dagum distribution is a special case of Generalized Beta II distribution. Dagum (1977, 1980) distribution has been distribution with the parameter q=1, has 3 parameters, namely a used in studies of income and wage distribution as well as and p as shape parameters and b as a scale parameter. In this wealth distribution. In this context its features have been study some properties of the distribution: moment, cumulant, extensively analyzed by many authors, for excellent Kurtosis and the characteristics function of the Dagum distribution will be discussed. The moment and cumulant will be survey on the genesis and on empirical applications see [4]. discussed through Moment Generating Function. The His proposals enable the development of statistical characteristics function will be discussed through the expectation of the definition of characteristics function. Simulation studies distribution used to empirical income and wealth data that will be presented. Some behaviors of the distribution due to the could accommodate both heavy tails in empirical income variation values of the parameters are discussed. and wealth distributions, and also permit interior mode [5]. Key words: A random variable X is said to have a Dagum distribution Dagum distribution, moment, cumulant, kurtosis, characteristics function. with the parameters a, b and p, if and only if the density function of X is as follows [4]: 1. Introduction ap−1 ap x , x > 0, a,b,p >0 Nowadays the development of the statistical theory is very p+1 a f (x | a,b,p) = ap x advance, especially the theory of distribution. One of the b 1+ distributions that used a lot in practice is Dagum a distribution. This distribution is a special case of 0 Otherwise. Generalized Beta II Distribution (GB 2) with the parameter at GB2 is q=1. This distribution has three parameters, namely (a, p) as shape parameters and b as a where ( , ) are shape parameters and b is a scale scale parameter. There are many studies in the theory of parameter. The mean and variance of Dagum distribution distribution discuss the characteristics function from a are as follows: distribution. Through the characteristics function we can discuss the mean, variance and standard deviation, also we p + 1 can discuss the moment, cumulant and characteristics ( ) = 1 1 (p) function. Moment is one of the importance characteristics Г � a� Г � − a� of a distribution; there are two kinds of moment, moment with respect to origin and moment with respect to mean. and the variance Г Moment with respect to the mean can be used to see the (p) p + 1 p + 1 skewness and kurtosis which can explain the behavior of = 2 2 1 1 (p) 2 the distribution of the data through graph. Moment with 2 Г Г � a� Г � − a� − Г � a� Г � − a� 2 respect to the origin is the moment which can be used to Г evaluate the mean. Cumulant is one of the importance 2. Method of analysis characteristics alternative of moment of a distribution, in other words the moment can define cumulant. Besides 2.1 Moment Generating Function(Mgf) moment and cumulant, the characteristics function also very importance. The study of Dagum distribution will be Moment generating function of Dagum distribution is discussed in this paper. The Dagum distribution was introduced by Camilo Dagum 1970’s [1, 2, 3], this Manuscript received August 5, 2017 Manuscript revised August 20, 2017 IJCSNS International Journal of Computer Science and Network Security, VOL.17 No.8, August 2017 265 p + 1 then ( ) is the central moment of order k. The ( ) = n n moment with respect to mean is defined by , and [7, 9] ! (p) − Г � a� Г � − a� � =0 Г = ( ) 2.2 Characteristics Function Then the first central moment −of the Dagum distribution A characteristic function is unique for a distribution and is: the characteristic function can be used to find the moment = = 0 of a random variable. If the random variable X and the , characteristic function denoted by Φ for all , then [6] The second central Moment′ is: ′ 1 1 − 1 2 2 1 1 (p) p+ 1 2 p+ 2 1 ( ) ( ) 2 a a a a c t = E e ℛ = = , Г Г Г −2(−)Г Г − � � � �p � � � � The function c(t) generally itxis complex, where: ′ ′ 2 E(e ) = E{cos tx +Φi sin tx} The third2 Central2 − Moment1 � is: = Г 3 + 2� itx Φ ( ) ( ) ′ ′ ′ ′3 = 2 3 3 2 32 13 1 +1 2 1 Г p Г�p+( �)Г�1− � 3Г p Г�p+ �Г�1−( �)Г�p+�Г�−1− � 3 a a a a a a c(t) 3 3 � Г p − Г p p + 1 3 1 3 1 , ..., 2Г �p+ (�Г)�1− � =Φ ∞ ( 1) 2k 2k a a 22 !2 (p) 3 Г � a � Г � − a � and Гthep -r central� moment is � − ( ) =0 pГ + 1 ( ) 2+1 2+1 = − r−i r−i 1 . + ∞ 1 2+1 2+1 ( ) ( ) 2 + 1! (p) Г�p+ a �Г�1− a � Г p+1 Г�1−a� Г � a � Г � − a � =0 Г p Г p � − 1 ∑ � � �− � where i is complex=0 number , i= .Г 2.5 Cumulant 2.3. The-r moment with respect√− to origin Cumulant of a random variable X is defined by [8, 9] : If X is a random variable and r is a positive integer, then ( ) = log( ( )) the-r moment of the random variable X with respect to the origin in defined as [7] : where ( ) is moment generating function of a distribution, and its coefficient of the Taylor series is = ( ) cumulant, namely [9]: The-r moment with respect′ to the origin can be found from the characteristics function as follows [8]: log( ( )) = ! d C(t) By using Taylor series we have: = ( i) for t = 0 rdt ′ r 1 r r log( ( )) = + ( ) C(t) µ − � � 2 where is characteristic function. 1 ′ 2 ′ ′ 2 + (2 ı 3 2+− )ı+ 3 So that we found the first moment is: 3 ′ ′ ′ ′ Then we have: ı − ı2 3 ⋯ = 1( ) 1 , kı = , Г�p+a�Г�1−a� Second moment is: ′ Г p k₂ = ( 1 ), ′ 2 = k₃ = (2′₂ − 3ı + ), and so on. 2( ) 2 , ..., and Г�p+a�Г�1−a� ′ the-r moment is: 2 Cumulant also can be found from moment with respect to Г p ′ı − ı ′₂ ′₃ p + 1 origin. = r r (p) Г � a� Г � − a� 1 = −1 1 2.4 The-r moment with respectГ to mean ′ − ′ The first cumulant of the− �Dagu� m distribution� − is: Let k be a positive integer and C is a constant, and then =1 − ( ) is the moment of order k. If = ( ) = , k1 = '1 − 266 IJCSNS International Journal of Computer Science and Network Security, VOL.17 No.8, August 2017 2 p + 1 k2 = '2 ̶ ' 1 = 1 1 ( ) (p) p + 1 p + 1 a p a Г � � Г � − � = 2 2 1 1 The second cumulant is: (p) 2 2 Г 2 Г Г � a� Г � − a� − Г � a� Г � − a� � 2 � Г The third cumulant is: 3 k3 = '3 ̶ 3 '1 '2 + 2 ' 1 (p) p + 1 3 (p) p + 1 p + 1 2 p + 1 = 3 3 1 1 2 2 + 1 1 2 (p) (p) 3 (p)3 3 Г Г � a� Г � − a� Г Г � a� Г � − a� Г � a� Г � − a� Г � a� Г � − a� � 3 − 3 3 � Г Г Г ,...., and the-r cumulant is: (p) p + 1 1 (p) p + 1 r r −1 − r−n r−1 (p) 1 r−n (p) Г Г � a� Г � − a� − Г Г � a � Г � − a � � r − � � � r−n � Г =1 − Г E(X ) Skew(X) = = 2.6 Skewness k 3 − µ µ 3 3 2 A distribution is symmetric if theσ graph of2 the distribution Skewness is the degree of asymmetric of a distribution. A µ symmetric distribution is a distribution which has the is symmetric around the central point. same mean, median and mode values. When the mean, If the ( ) = 0 , then the distribution is symmetric at the median and mode of a distribution are not the same, it will central point or the mean; if Skew(X) > 0 then the imply that the distribution is not symmetric. This property distribution is skew to the right, and if Skew(X) < 0 then can be seen in the skewness of a distribution. When the the distribution is skew to the left. The skewness of the moment with respect to the mean has been found, the Dagum distribution is skewness can be defined as follows [10]: ( ) ( ) ( ) Skew(X) = = + / /2 ( ) / 3 3/ 1 1 / ( ) / 3 / 1 3 1 µ₃ Г p Г�p+a�Г�1−a�−3 p Г p �p+a�Г�1−a� +2Г �p+a�Г �1−a� 3 2 3 2 3 2 2 3 2 2 3 1 3 1 3 2 +36 2(p) 2p +3 2 12 3 p1+ 3 1 3 p + 1 µ₂ Г p Г �p+a�Г �1−a�−Г �p+a�Г �1−a� Г p Г �p+a�Г �1−a�−Г �p+a�Г �1−a� 2 2 2 1 2 1 4 1 4 1 2.7 Kurtosis Г Г � a� Г � −+ a�(Гp)� p a+� Г � −1 a� − Г � a� Г � − a� 4 4 1 4 1 Kurtosis is a measurement to see the sharpness of a peak Г Г � a� Г � − a� of a distribution.
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