A Note on Logistic Mixture Distributions

Biostatistics and Biometrics Open Access Journal ISSN: 2573-2633

Satheesh Kumar C1* and Manju L2 1Department of Statistics, University of Kerala, India 2Department of Community Medicine, Sree Gokulam Medical College, India Submission: May 25, 2017; Published: August 23, 2017 *Corresponding author: Satheesh Kumar C, Department of Statistics, University of Kerala, India; Email:

Abstract Here we consider a wide class of logistic distributions which are obtained by mixing well known type I and type II logistic distributions. We investigate some important properties of the distribution and illustrated the usefulness of the model with the help of a real life dataset.

Keywords: Mixture distributions; Model selection; Type I logistic distribution; Type II logistic distribution

e−β x Introduction = − Fx3 ( ) 1.β (6) + −x The logistic function has wide applications in several areas of (1 e ) research such as demographic studies to estimate the growth of If Z follows , then follows . is human population [1], as a growth model in biology [2], bioassay LDI YZ= − LDII LDI problems [3-8], survival data [9], public health [10], income negatively skewed for 01<<α and positively skewed for α >1. is positively skewed for and negatively skewed for distributions [10] etc. For a detailed account of the properties LDII β <1 and applications of the logistic model see Balakrishnan [12]. β >1. Both these classes of distributions have applications A continuous random variable X is said to follow the standard introduce a new class of distributions which is a convex mixture logistic distribution (LD) if its probability density function (PDF) in several areas of scientific research. Through this paper we of the and the and examine its important properties. is of the following form. LDI LDII

−x e (1) fx1 ( ) = 2 , distributions and obtain some important properties. In section + −x In section 2, we presented the definition of the proposed class of (1 e ) 3, we illustrate the usefulness of the distribution by utilizing a where xR∈ =( −∞,. +∞) The cumulative distribution function real life data set. (CDF) of the LD is given by Mixture of type I and type II logistic distributions 1 Fx( ) = , (2) 1 1+ e−x distribution and discuss some of its important properties. For xR∈ . Balakrishnan and Leung [13] proposed two First we present the definition of the proposed mixture generalized logistic distributions of type and I( LDI ) II( LDII ) Definition: A continuous random variable X is said to follow respectively through the following PDFs f2 (.) and f3 (.,) in which logistic mixture distribution (LMD) if its PDF is of the following xR∈>,0α and β > 0. form for xR∈ , 0≤≤p 1, and β > 0. e−x αα= −−xxβ fx2 ( , ) α +1 (3) ee 1+ e−x (7) ( ) fx( ) = pαβαβ++11+−(1 p) 11++ee−−xx −β ( ) ( ) βe x fx3 ( , β ) = β + (4) −x 1 (1+ e ) A distribution with PDF (7) we denoted by LMD( p,,αβ) . clearly when p =1 the LMD reduces to LD and when p = 0 the The corresponding CDFs of LDI and LDII are respectively I 1 LMD reduces to LDII . When either p =1 and α =1 or when Fx2 ( ) = α −x (5) (1+ e ) p = 0 and β =1, the LMD reduces to the LD. The probability and plots of the LMD( p,,αβ) .for particular choices of P,α and β are given in Figure 1.

Biostat Biometrics Open Acc J 2(5): BBOAJ.MS.ID.555598 (2017) 00122 Biostatistics and Biometrics Open Access Journal

LMD( p,,αβ) is the following, for any tR∈ , and i = −1. the definition, the characteristic function of the ∞ itx Φ=∫X (t) e f( x) dx −∞

∞∞ee(it−−1) x (itβ ) x =αβ ∫ +− ∫ p αβ++11dx(1 p) dx (10) −∞ (11++ee−−xx) −∞ ( )

1 if we put u = in (10) we get, 1+ e−x 11 α +−it 1 11−−it it β −−it 1 ΦX (t) = pαβ ∫ u(1 − u) du +−( 11 p) ∫ u( − u) du (11) 00 which implies (9), in the light of the beta function. In a similar way we can obtain the moment generating function of the LMD( p,,αβ) as given in the following result. Figure 1: Plots of LMD( p,,αβ) for particular values of p,,α and β Result 3: The moment generating function of LMD( p,,αβ) is

Mtx ( ) = pαα ×Β( + t,1 − t) +( 1 − p) ββ ×Β( − t,1 + t) , (12) Result 1 for max{− 1,αβ} <mean and variance of the LMD( p,,αβ) as given in the following −β x 1 e (8) result. Fx( ) =× p αβ +−(11p) ×− (11++ee−−xx) ( )  Result: The mean and variance of LMD( p,,αβ) with PDF (7) are The result directly follows from (5) and (6). Mean= pαψ( α) +−11( p) βψ( β) −( p α + p β − β) ψ (1) (13) Result 2 The characteristic function Φ X (t) of LMD( p,,αβ) 2 with PDF (7) is the following, in which B(.,.) is the complete beta var iance= pαψ′′( α) + ψ(1)  +−p α( 1 p α)  ψα( ) − ψ(1) + function. 2 (1−p) βψ ′′( β) + ψ(1) +−( 1pp) β  11 −−( ) β  ψ( 1) − ψ( β) − ΦX (t) = pαα × B( + it,1 − it) +−( 1 p) ββ × B( − it,1 + it) , (9) 21pp( −)αβ ψ( α) −− ψ(1)  ψ( 1) ψ( β ) , (14) for Re( 1−>it) 0, Re(β −>it) 0, and tR∈ , i = −1 in which ψ (.) is the psi or digamma function. Proof: Let X follows LMD( p,,αβ) with PDF (7). Then by Table 1: Estimated values of the parameters with the corresponding information criterion values.

Distribution

µσ LD( , ) LDI (µσα,,) , LDII (µσ,, β) LMD(µσ,,,, p α β)

µˆ 112.858 110.657 110.874 109.461

σˆ 5.26 5.691 4.804 5.781

αˆ _ 1.28 _ 1.713

βˆ _ _ 0.789 1.928

pˆ _ _ _ 0.879

AIC 657.274 652.906 653.124 646.303

BIC 657.153 655.352 655.57 650.381

CAIC 659.153 658.352 658.57 655.381

HQIC 654.424 649.603 649.821 640.798

How to cite this article: Satheesh K C, Manju L. A Note on Logistic Mixture Distributions. Biostat Biometrics Open Acc J. 2017; 2(5): 555598. DOI: 0123 10.19080/BBOAJ.2017.02.555598. Biostatistics and Biometrics Open Access Journal

On differentiating the PDF fx( ) of the LMD( p,,αβ) with References respect to x and equating to zero, we obtain the following result, 1. Oliver FR (1982) Notes on the logistic curve for human population. useful for the computation of the mode of the distribution. Journal of the Royal Statistical Society Series A 145: 359-363. 2. Peal R, Reed L J (1924) Studies in Human Biology. Williams and The mode of LMD( p,,αβ) Result: Wilkins, Baltimore. following equation αβ− −−xx with PDF (7) satisfies the 3. Pearl R, (1940) Medical Biometry and Statistics. Saunders, Philadelphia. pα (ee−+β )(1 ) = (15) (1− p) β 2 (αe−x −1) 4. Emmens CW (1941) The dose-response relation for certain principles Note that by using the mathematical softwares such as of the pituitary gland, and of the serum and urine of pregnancy. Journal of Endocrinology 2: 194-225. MATHCAD, MATHEMATICA, R we can easily evaluate the mean and variance. 5. Berkson J (1994) Application of the logistic function to bioassay. Journal of the American Statistical Association 37: 357-365. Application 6. Berkson J (1951) Why I prefer logits to probits. Biometrics 7: 327-339. For illustrating the usefulness of the LMD(µσ,,,, p α β) 7. Finney DJ (1947) The principles of biological assay. Journal of the model, here we considered the IQ data set of 87 white males Royal Statistical Society, Series B, 9: 46-91. hired by an insurance company in 1971 taken from Roberts 8. Finney DJ (1952) Statisitcal methods in Biological Assay. Hafner, New [14]. We obtain the maximum likelihood estimates (MLEs) of York. the parameters of the LMD(µσ,,,, p α β) by using the MaxLik 9. Plackett RL (1959) The analysis of life test data. Technometrics 1: 9-19. package in R software. The values of the Akaike information 10. Dyke GV, Patterson HD (1952) Analysis of factorial arrangements when criterion (AIC), Bayesian information criterion (BIC), Consistent the data are proportions. Biometrics 8: 1-12. Akaike information criterion (CAIC) and Hannan Quinn 11. Fisk PR (1961) The graduation of income distributions. Econometrica information criterion (HQIC) are computed for comparing 29: 171-185. the model LMD( p,,αβ) with the existing models - LD(µσ,,) 12. Balakrishnan N (1992) Handbook of Logistic Distribution, Dekker, µσα The results obtained are given in NewYork. LDI ( ,,) ,LMDLDII (µσ,, β) . Table 1. From Table 1 it is seen that the AIC, BIC, CAIC and HQIC 13. values are minimum for LD(µσ,,,, p α β) compared to other Press, Redwoodcity, California, USA. II Roberts HV (1988) Data Analysis for Managers with Minitab. Scientific models. Based on the computed values of the AIC, BIC, CAIC and 14. Balakrishnan N, Leung MY (1988) Order Statistics from the Type HQIC one can observe that the LMD model gives I Generalized logistic distribution. Communications in Statistics - LDII (µσ,,,, p α β) and Simulation and Computation 17(1): 25-50. LD, LDI LDII . better fit to the data set compared to

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How to cite this article: Satheesh K C, Manju L. A Note on Logistic Mixture Distributions. Biostat Biometrics Open Acc J. 2017; 2(5): 555598. DOI: 0124 10.19080/BBOAJ.2017.02.555598.