The Alpha-Beta Skew Logistic Distribution: Properties and Applications

A new family of skew distributions is introduced by extending the alpha skew logistic distribution proposed by Hazarika-Chakraborty [9]. This family of distributions is called the alpha-beta skew logistic (ABSLG) distribution. Density function, moments, skewness and kurtosis coefficients are derived. The parameters of the new family are estimated by maximum likelihood and moments methods. The performance of the obtained estimators examined via a Monte carlo simulation. Flexibility, usefulness and suitability of ABSLG is illustrated by analyzing two real data sets.

The above integral can be obtained by the moments of the standard logistic distribution. Figure 1 presents the ABSLG pdf for different choices of the parameters α and β. It can be seen from Figure 1 that the proposed model has at most four modes, also the effects on the skewness can be seen. Some properties of the ABSLG(α, β) distribution are as follow: • For α = β = 0, we get the standard logistic distribution which is given by and denoted by Z ∼ LG(0, 1). • For β = 0 the pdf (4) is simplified as the pdf of the ASLG distribution. • For α = 0, the pdf (4) is simplified as the following pdf, The above equation is referred to as the beta skew logistic(BSLG) distribution. • If α −→ ±∞, then we get a bimodal logistic (BLG) distribution given by, Here, we prove that the ABSLG has at most four modes. The first derivative of the pdf f (z; α, β) of ABSLG(α, β) with respect to z is given by, This expression has at most seven zeros. Thus, the function f (z; α, β) can have at most four modes.

Moments
The mean and moments of distributions are usually used as measures of central indices in a population. In the following theorem, the moments of proposed distribution is investigated.
Theorem 2.1. The k th order moment of ABSLG (α, β) distribution is given by, Such that,
and the variance of ABSLG (α, β) is obtained by,

Skewness and Kurtosis
A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis. The skewness and kurtosis of proposed distribution is investigated in the following theorem.

Proof
By the moments and Theorem 2.1, and, 310 THE ALPHA-BETA SKEW LOGISTIC DISTRIBUTION And, such that, By using a numerical method the desired results follow.

Truncated ABSLG(α, β)
In this section alpha-beta skew logistic distribution truncated below 0 is presented as a potential life time distribution. The pdf of alpha-beta skew logistic distribution which is truncated below 0 is denoted as TABSLG(α, β) and is given by, The survival function S T (t; α, β) and the hazard function h T (t; α, β) of T ABSLG (α, β) can be easily expressed in terms of the pdf f T (t; α, β) and cdf F T (t; α, β) of T ABSLG (α, β) as, and, Remark 3.1. Clearly, for α, β = 0, T ABSLG (α, β) is simplified to the standard half logistic distribution, [3].

Method of Moments
Let m 1 , m 2 , m 3 are respectively the first, three sample raw moments for a given random sample z 1 , z 2, ..., z n of size n drawn from Z ∼ ABSLG (α, β, µ, σ) distribution. Then the moment estimates of the four parameters α, β, µ, σ are obtained by simulataneously solving the following equations derived by equating first three population moments with corresponding sample moments.
while the exact solution of the equation (8) is not tractable, it can be numerically solved to estimate α, β.

Maximum Likelihood Estimation
Let Y = (Y 1 , Y 2, ..., Y n ) denote a random sample of size n drawn from the ABSLG (α, β, µ, σ) distribution. From the equation (6) the likelihood function is given by, The MLE of α, β, µ and σ are obtained by numerically maximizing log L with respect to α, β, µ and σ. The variance-covariance matrix of the MLEs can be derived by using the asymptotic distribution of MLEs as,  When the exact expressions for various expectations above are cumbersome, in practice they are estimated as

Simulation
In this section, we conduct the Monte Carlo simulation studies to assess on the finite sample behavior of the MLEs of α, β, µ and σ. All results are obtained from 5000 Monte Carlo replications and the simulations were carried out using the package 'maxLik' in statistical software R. In each replication, a random sample of size n is drawn from the ABSLG(α, β, µ, σ) distribution. The true parameter values used in the data generating processes are µ = 0, and σ = 1 and different values for the parameters α = (−0.5, 0, 0.5, 1) and β = (−0.5, 0.0, 0.5, 1). Tables 1 represent the empirical means and the average mean squared errors (AMSE) of the corresponding estimators for sample sizes n = 25, 50, 100, 200 and 400. It is noticeable that the estimation of the parameters α and β are the most hard situations, whereas the estimation of µ and σ are the most tractable.
We note that in all cases the biases and AMSEs of the MLEs of α, β, µ and σ decay toward zero when the sample size increases, as expected. There is a small sample bias in the estimates of the model parameters. Figures 4 and 5 represents the AMSE of the parameters α and β for fixed values of another parameters in the model. As can be seen in the Figures, by increasing the sample size, the AMSE of estimated parameters decreases.
Aplication 6.1. For illustration purposes, we consider the uncensored part of a data set analysed by Leiva et al. (2007) corresponding to the survival times (T, in months) of 48 patients who were treated with alkylating agents for multiple myeloma. These data (which we will henceforth call myeloma) are :1, 1, 2, 2, 2, 3,5,5,6,6,6,6,7,7,7,9,11,11,11,11,11,13,14,15,16,16,17,17,18,19,19,24,25,26,32,35,37,41,42,51,52,54,58,66,67,88,89,92. By comparing the AICs and BICs, one can see the ABSLG distribution is the best among the fitted models. LR test result: the value of LR test statistics is 21.92 which exceed the 95% critical value. Thus there is evidence in favors of the alternative hypothesis that the sampled data comes from ABSLG (α, β, µ, σ) not from ASLG (α, µ, σ) .The values of these statistics for the models are listed in Table 2. Overall,by comparing the measures of these formal goodness-of-tests in Table 2, we conclude that the ABSLG distribution yields a better than the LG and ASLG distributions and therefore it can be an interesting alternative to these distributions for modeling lifetime data. These results illustrate the importance of the additional shape parameters of the new distribution to analyze real data.
Aplication 6.2. The second data set(n= 202) in Table 3

Conclusion
In this paper, we proposed a new family of distributions with two extra generator parameters, which includes as special cases of logistic, ASLG, BSLG and BLG distributions and some of its basic properties are investigated which include moments, Skewness and Kurtosis functions, mean deviations. Also the below zero truncated version of the proposed distribution was presented as a potential life time distribution. The application of the new family is straightforward. The model parameters are estimated by maximum likelihood and two real examples are used for illustration, where the new family does fit well both data sets.