Review Article

A Note on Logistic Mixture Distributions

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: drcsatheeshkumar@gmail.com

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: 10.19080/BBOAJ.2017.02.555598.

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

Introduction

The logistic function has wide applications in several areas of research such as demographic studies to estimate the growth of human population [1], as a growth model in biology [2], bioassay problems [3-7], survival data [8], public health [9], income distributions [10] etc. For a detailed account of the properties and applications of the logistic model see Balakrishnan [11]. A continuous random variable X is said to follow the standard logistic distribution (LD) if its probability density function (PDF) is of the following form.

where x ∈ R = (-∞,+∞). The cumulative distribution function (CDF) of the LD is given by

For x∈R Balakrishnan and Leung (1988) proposed two generalized logistic distributions of type I ( LD_I) and II (LD_II) respectively through the following PDFs f₂ (.) and f ₃(.), in which x ϵ R,α> 0 and β> 0.

If Z follows LD_I , then Y = —Z follows LD_II. LD_I is negatively skewed for 0 < α <1 and positively skewed for α> 1. LD_II is positively skewed for β < 1 and negatively skewed for β > 1. Both these classes of distributions have applications in several areas of scientific research. Through this paper we introduce a new class of distributions which is a convex mixture of the LD_I and the LD_II and examine its important properties. In section 2, we presented the definition of the proposed class of distributions and obtain some important properties. In section 3, we illustrate the usefulness of the distribution by utilizing a real life data set.

Mixture of type I and type II logistic distributions

First we present the definition of the proposed mixture distribution and discuss some of its important properties.

Definition: A continuous random variable X is said to follow logistic mixture distribution (LMD) if its PDF is of the following form for x ∈ R, 0≤p ≤ 1, and β> 0.

A distribution with PDF (7) we denoted by LMD (p,α, β). clearly when p = 1 the LMD reduces to LD_I and when p = 0 the LMD reduces to LD_II. When either p = 1 and α = 1 or when p = 0 and β= 1, the LMD reduces to the LD. The probability plots of the LMD (p,α,β ). for particular choices of P,α and β are given in Figure 1.

Result 1

The CDF of LMD (p,α, β), with PDF (7) is the following, for x ∈ R.

The result directly follows from (5) and (6).

Result: The characteristic function Ф _X (t) of LMD (p,α, β) with PDF (7) is the following, in which B (.,.) is the complete beta function.

Proof: Let X follows LMD (p,α, β) with PDF (7). Then by the definition, the characteristic function of the LMD (p,α, β) is the following, for any t ∈ R, and i = .

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,α, β) as given in the following result.

Result The moment generating function of LMD(p,α, β) is

From Result 4.3, by differentiation techniques, we obtain the mean and variance of the LMD(p,α,β) as given in the following result.

Result: The mean and variance of LMD (p,α,β) with PDF (7) are

On differentiating the PDF f (x) of the LMD (p,α,β) with respect to x and equating to zero, we obtain the following result, useful for the computation of the mode of the distribution.

Result: The mode of LMD(p,α,β) with PDF (7) satisfies the following equation

Note that by using the mathematical softwares such as MATHCAD, MATHEMATICA, R we can easily evaluate the mean and variance.

Application

For illustrating the usefulness of the LMD(μ ,σ, p,α, β) model, here we considered the IQ data set of 87 white males hired by an insurance company in 1971 taken from Roberts [12]. We obtain the maximum likelihood estimates (MLEs) of the parameters of the LMD(μ ,σ, p,α, β) by using the MaxLik package in R software. The values of the Akaike information criterion (AIC), Bayesian information criterion (BIC), Consistent Akaike information criterion (CAIC) and Hannan Quinn information criterion (HQIC) are computed for comparing the model LMD(p,α,β) with the existing models - LD(μ,σ), LD_I (μ,σ, α), LD_II (μ ,σ, β).The results obtained are given in Table 1 [13,14]. From Table 1 it is seen that the AIC, BIC, CAIC and HQIC values are minimum for LD_II (μ ,σ, p,α, β) compared to other models. Based on the computed values of the AIC;BIC;CAIC and HQIC one can observe that the LMD, (μ ,σ, p,α, β) model gives better fir to the data set compared to LD,LD_I and LD_II.

References

1. Oliver FR (1982) Notes on the logistic curve for human population. Journal of the Royal Statistical Society Series A 145: 359-363.
2. Peal R, Reed L J (1924) Studies in Human Biology. Williams and Wilkins, Baltimore.
3. Pearl R, (1940) Medical Biometry and Statistics. Saunders, Philadelphia.
4. Emmens CW (1941) The dose-response relation for certain principles of the pituitary gland, and of the serum and urine of pregnancy. Journal of Endocrinology 2: 194-225.
5. Berkson J (1994) Application of the logistic function to bioassay. Journal of the American Statistical Association 37: 357-365. 6.
6. Berkson J (1951) Why I prefer logits to probits. Biometrics 7: 327-339.
7. Finney DJ (1947) The principles of biological assay. Journal of the Royal Statistical Society, Series B, 9: 46-91.
8. Finney DJ (1952) Statisitcal methods in Biological Assay. Hafner, New York.
9. Plackett RL (1959) The analysis of life test data. Technometrics 1: 9-19.
10. Dyke GV, Patterson HD (1952) Analysis of factorial arrangements when the data are proportions. Biometrics 8: 1-12.
11. Fisk PR (1961) The graduation of income distributions. Econometrica 29: 171-185.
12. Balakrishnan N (1992) Handbook of Logistic Distribution, Dekker, NewYork.
13. Roberts HV (1988) Data Analysis for Managers with Minitab. Scientific Press, Redwoodcity, California, USA.
14. Balakrishnan N, Leung MY (1988) Order Statistics from the Type I Generalized logistic distribution. Communications in Statistics - Simulation and Computation 17(1): 25-50.