The distributional properties of half logistic distribution and Type I generalized logistic distribution were studied, bringing out the L-moments (up to order four) of each of these. Skewness and Kurtosis were obtained

Adesakin T. J.

Olagunju M. O.

Osuntuyi A. A.

English

Archivio istituzionale della ricerca - Università degli Studi di Venezia Ca' Foscari

IfeJoumslofSclencevol. 10, no. 1 (2008) THE DISTRIBUTIONAL PROPERTIES OF THE FAMILY OF LOGISTIC DISTRIBUTIONS* T.J. ADESAKIN,A.A. OSUNTUYI+ and M.A. OLAGUNJU Department of Mathematics, Obafemi Awolowo University, lle-Ife, Nigeria. (Received: August 2007; Accepted: May 2008) Abstract 245 The distributional properties ofhalflogistic distribution and Type I generalized logistic distribution were studied, bringing out the L-moments (up to order four) of each of these. Skewness and Kurtosis were obtained. Key words: Logistic distribution, L-moments • 1. Introduction Family oflogistic distribution like standard logistic, halflog_istic, Type I generalized logistic, etc., were studied in Ph.D thesis ofOiapade (2006); he obtained the propet1ies of the family, using moment generating flinction and characteristic functions but did not consider the L-moment of the family of the distribution studied in this paper. Also, the L-skewness and L-kurtosis of the family of'+t.: :.: :"tribution are obtained. The standard probability density function of the logist;~ random variable xis given by: e.r ft(x)= ( r , -00 <X < 00 1 + e·r -(I. I) The cumulative distribution function (c.d.f) is given as l .r Ft (x) = ( _ )'-oo < x < oo . · 1 + e .r (1.2) The quartile (inverse distribution) function is given as: x(F) = ln(_f_),o ~ F ~ 1 1-F and the L-moment of a given distribution as proposed by Hosking (1990) can be expressed as : I L, = Jx(F)P,_1 (F)dF, 0 r-1 (r -lJ(r + k -lJ Where P,_1 (F) = L ( -Iy-k-t F k, k=O k k called Legendre polynomial of order (r -1) , Hosking ( 1990). The L-moment of the logistic distribution can be expressed as; 1 1 = J(ln F)P,_, (F)dF- J(ln(l- F))P,_1 (F)dF 0 0 = c[!F' QnF}iF - IF' (ln(l- F))dF} + corresponding author( email: babapassat@yahoo.com) * Presented in part at the First Faculty of Science Conf~rence, Obafemi Awolowo University, Ile-Ife, July 3-5, 2007. 246 Adesakin eta/.: . Distributional properties of th~ ~mily,of logls~!€:·dJ~rlbutions_ ., Where c = f.(-UHH:(:r """lJ"(r -+:k -1JFk. k=~ k k . . The following identities of integ_t'als wi11 help us to sili1plify the above -equations I ('t) J k 1 " . )d -l ox n,~ x·= (k+1)2; I ~ 1 ( ii) fxk ln(l- x)dx ="-:Li . ( . r o It=<~ n n + k + 1 lim ,; 1 Therefot:e, -if k = 0, £ ( ) = 1; j --+- 00 na<l 1:l n + 1. . lim J .. 1 j .lim :i l 11 i'f k = o; . I ( ) = ~; if k;:: 2, . _ 2: . c . ) = - ; J ~ oo ,=1 n n + 2 .4 J ~ .co u=l n n + 3 . 18 the first four t.-momentofthe lqgistic distribution can be expressed as: L1 =0 L" =1 L3, =0 _ L4 = 4.2 According to Hoskip.g t 1990), the L-skewness and L-kmtosis can be obtained b~ using L, . 2 Tr=-,1> · . L'l Therefore for the logistic djstrjbution, L-skewness = TJ: = 0: .and 'L:-kUrtosis = r 4 = 4.2 If the .location (i) and ·scale (co) are included ·in the quartile function of legistic distribution as: x(F) = §+ cvln(. ~).o· ::; F ~ 1 . 1- .F ,· The condition .given by Bickel ~d Lehn1artn (1976) stated ·suppqse tl1cere has bee~r defined a partial ordering, with F<G, meaning !that G possesses the attribute under consideration more strongly than li't then.. the;firsh~onclition.required ofa·mea:sure 8 of this. attribute is that ,$(F) ~. B(G ), whenever F <G. A secQnd condition characterizes. the behavior of 9(F) (which we shall also denote by 8(x) when X is a random variable with. distribution F). under linear transfonnatiort.. Thus, a meaSilre of location .·should satisfy &(aX-+" I?)= a 8{X) + b, for all a, b; and .a measure of scale .9(aX +b)= Jaf8(X), fo11 all a*· 0' and all b-. Ftillowing this, we _obta,in out L-m-oment offue lqgjstic distribution as: L1 = D Ll = CO L3, = 0· L~ =4.2c~ The Property of'the Half logistic Distribution, Qne ofthe probabi.l ity c;l istributio.ns· which 's a memper of the t:am ily ofthe Jog}~tic -di~tribution i_s half logistic distribution. · fts probab.i1 ity density function can· be exp.Tessed as: L1 = 0 Adesaldn et aL: Disbibutional properties of the family of logistic distributions 247 2e}' /y(y)= .· 2 ,0 < .l' < o?. (1 + e}' ) Tlte cwnulative distribution t\mction is · ) e7 -1 Fy{ v = ( ro < v < o?. ·· 1 + er -The inverse (quartile) distribution function of the halflogistic distJ.ibution can be expressed as: 1+F v(F)= bt--. . · 1-F Tlte L-moment of the ltalflogistic distribution can be expressed as: L = J1 1~(l + F )P. (F)dF T 0 .u\ l - F ) T-1 . I l = J (In(l + F )~-1 (F)dF- I (bt(l- F ))P,._I (F)dF 0 () = {[ F' In( I+ F }iF- [ F' (In!. I- F ))rTF J The identity of integrals below will give clues on solving the L-moment, of the half logistic disb.ibution: · 1 QJ ( ~r+' 1l (i) J x.t llt(l +x)dx = L - x : 0 1l~l 11 1 = 1 (ii) J.-rk ln(l- .T)d-r =-' < r 0 !:fup+k+l Therefot-e, .we have the equation L, above to be L=c) +) I ( Qll (-1}"+1 Qll 1 \ ' !:i' n(n + k + 1) !:lu(n + k + 1) 1 c' ('"' 1 CD 1 J ;:: . 1 + k t; ( 211- 1) ~ (211 + k) . lim ) ( 1 1 \ If k=O, . L ( ) --( )J= lo~. 2: J ~ Cl) 11• 1 2n - 1 2n If k = L lim f( 1 - . 1 I= t: . j ~ 00 ;:: ( 211- r) ( 2u + 1)) If k - 2. )' . - . - log. - + - . ~- lim J( l 1 J- ., 1 · j-+CI);;:t (2n-1) (2n+2) 2 If k = 3, '\' l . - 1 = 4 . lim J( J j ~ a)ttl(2n-l) (2n+3} 3 The L-moment of the halflogistic dill'hibution can be expre.sz:eed 3$; L1 == 2log1 2, L1 = -2log. 2 + 2 = 2(1-log. :!) • J l \ L3 =:!log. 2- 6+ 4f log. 2 +-I= 6log. 2- 4~ ' 2; J l \ 4 1 L4 = 2log. 2+12- 2(5)j log. 2+-1+- = 15- - 22log. 2 ' 2) 3 3 , The L-skewnesss and L-kurtosis of the = Let 6 2-4 = 2 2+ pb { I )=b .!. =II, 1-Fb \. b.> <X<CI),b>O. I x= u=O 2-1 5--3 is are can I kll-\. the =l.u=l. of distributions I b l J Adesakin eta/.: Distributional properties of the family of logistic distributions I ( _!_J I "' 1 Also, Jxk ln 1- x h dx = b Jukh+h-l ln(l - u )du = -b I ( )' o . 0 11=1 n bk + b + n "' 1 "' ( 1 1 J ~ n(bk+ b+ n)' = ~ n(bk+b) - b(k + nXn+bk+b) 1 00 (1 1 J = I - --:----(bk + b )n=1 n (n + bk +b) · If k = 0 f (_!_ - 1 J = f (_!_) ' 11=1 • n (n +b) 11=1 n . If k = 1 , b = only value, I -- =I -· .,., ( 1 1 J 2h ( 1) II= I n (n + 2b) 11=1 n If k = 2 f (_!_- l J = f (_!_) , n= 1 n (n + 3b) 11=1 n . lfk=3 f (_!_- 1 J=f(_!_) , n =1 n (n + 4b) 11=1 n . We can assume for any value of k , f (_!_ -( 1 )J =(I:''(_!_)· ll=l n n + bk + b 11= 1 n But k and b must be positive integers so the L-moment of the type I generalized logistic Distribution can be expressed as ( 1 1 (~h( 1 )J ~ b(k + tY + b(k + 1) b -;; 249 The value of b is obtained, using the maximum likelihood method of estimation, and it is assumed to be approximated to the nearest integer. lf b = I, the distribution gives the standard logistic distribution. REFERENCES Berkson , J ., 1994. Application of the logistic distribution function to bi-assay. Journal of American Statistical Association, 39, 357-365. Bickel, P.J. and Lehmann, I 976. Descriptive Statistics for nonparametric models III Diepersion. Annals of Statistics, 4, I I 39-1158. George, E.O. and Ojo, M.O., 1980. On a generalization of the logistic distribution. Annals of Statistics Mathematics, 32, 2, A, 16 I-169. Hosking, J.R.M., 1989. L-moment: Analysis and Estimation of Distributions using Linear Combination of Order Statistics. Hosking, J.R.M., 1989. Some Ttheoretical Results concerning L-moments. Hosking, J.R.M., 1994. The four parameter kappa distribution. IBM Journal Development, 38(3), 251-258. Karvanan, J., 2007. Characterizing the generalized lambda distribution by L-moments. National Public Health Institute. Olapade, A.K., 2006. The Theories and Appl ications of Family of Logistic Distribution., Obafemi Awolowo University, Ile-Ife (unpublished Ph.D Thesis). Olapade, A.K. and Ojo, M.O., 2006. On logistic and halflogistic distribution , (In press). 

The distributional properties of the family of logistic distributions

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The distributional properties of the family of logistic distributions

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