1. Introduction
Generally, there is a rising interest in the introduction of new generators for univariate continuous families of distributions by adding one or more additional shape parameter(s) to the baseline distribution. This introduction of parameter(s) was proven useful in the exploration of tail properties and the improvement of the goodness-of-fit of the family under investigation. There are well-known generators like the following: the beta-G distribution by Eugene et al. [1], the gamma-G type 1 distribution by Zografos and Balakrishanan [2] and Amini et al. [3], the Kumaraswamy-G (Kw-G) distribution by Cordeiro and de Castro [4], the McDonald-G (Mc-G) distribution by Alexander et al. [5], the gamma-G type 2 distribution by Ristic and Balakrishanan [6] and Amini et al. [3], the odd exponentiated generalized (odd exp-G) distribution by Cordeiro et al. [7], the transformed transformer (T-X) (Weibull-X and gamma-X) distribution by Alzaatreh et al. [8], the odd Weibull-G distribution by Bourguignon et al. [9], the Lomax-G distribution by Cordeiro et al. [10], the Kumaraswamy odd log-logistic-G distribution by Alizadeh et al. [11] and the logistic-X distribution by Tahir et al. [12], a new family by using survival function called weighted exponential-G family introduced by Abd El-Bar et al. [13].
The class of weighted exponential (WE) distributions was obtained by Gupta and Kundu [14] by implementing Azzalini’s method to the exponential distribution. Shakhatreh [15] generalized the WE distribution to the two-parameter weighted exponential distributions (TWE). Finally, Kharazmi et al. [16] proposed a generalized weighted exponential distribution (GWE). It is observed that the WE, TWE and GWE distribution can provide a better fit for survival time data relative to other common distributions such as gamma, Weibull, or generalized exponential distribution.
The aim of this paper is to introduce a new family of distributions generated by generalized weighted exponential (GWE) distribution. We expect that the proposed family may be better (at least in terms of model fitting) than other classes of distributions in certain practical situations. The proposed GWE-G distribution provides WE-G and TWE-G distributions as its sub-models. This extension offers more flexible distributions with applications in reliability engineering and lifetime modeling. In addition, several properties of the GWE-G distribution have been established.
In the following sections, we study the properties of a special case of this family, when
is the CDF of the Gompertz distribution. In this case, the random variable X is said to have the generalized weighted exponential-Gompertz (GWE-G) distribution.
The reminder of this paper is organized as follows. We discuss the generalized weighted exponential-G family in Section 2. In Section 3, the generalized weighted exponential-Gompertz distribution is studied in detail. In Section 4, we provide expansions for weighted exponential-Gompertz cumulative and density functions. In Section 5, we present various properties of the new model such as moment generating function, moments and conditional moments. Rényi Entropy introduced in Section 6. Also, some reliability properties of our model are discussed in Section 7. In Section 8, the maximum likelihood estimator of the parameters of our model is obtained. Section 9 gives an application to a real data set.
2. Generalized Weighted Exponential-G Family
Ristić and Balakrishnan [6] introduced a new family of distributions generated by gamma random variable with survival function (SF) given by:
(1)
where
is a cumulative distribution function (CDF) of baseline distribution which is used to generate a new family of continuous distributions. On the same line, we provide a new family of distributions generated by generalized exponential distribution.
If a random variable X has the generalized weighted exponential distribution GWE
with integer
, shape parameter
and scale parameter
, if its probability density function(PDF) is given by:
(2)
where,
, and the corresponding CDF is given by:
(3)
Now, by using an integral transform of the PDF of a random variable T which follows GWE distribution we generate a new family of distributions. The SF of generalized weighted exponential-G (GWE-G) family is defined as:
(4)
and the PDF of GWE-G family is obtained by:
(5)
where,
and
respectively, are the baseline CDF and PDF which depends on a
parameter vector
and
are two additional parameters.
3. Generalized Weighted Exponential-Gompertz Distribution
The PDF of GD with location parameter
and shape parameter
, is given by:
(6)
The CDF associated with Equation (6) is obtained as:
(7)
The GD plays an important role in modeling reliability, human mortality and actuarial data that have hazard rate with exponential increase. Some applications of this distribution can be found in Pollard and Valkovics [17]. An extension version of Gompertz is the weighted Gompertz distribution discussed by Bakouch and Abd El-Bar [18].
Now, we introduce the PDF and CDF of GWE-G distribution by using Equations (4)-(6) respectively, are:
(8)
and,
(9)
where,
.
Figure 1 shows the shapes of GWE-G distribution or some various parameters. It can be summarized some of the shape properties of our model as: (a) The PDF is left-skewed when
and
and 7. (b) The PDF is right-skewed when
and
and
and the PDF is reserved-J when
and
and
. (c) The PDF is monotonically decreasing when
and
and
. (d) The PDF is reserved-J when
and
and
. (e) The PDF is left-skewed when
and
and
. (f) The PDF is left-skewed when
and
and
.
The SF and HR of the GWE-G distribution respectively, are:
(10)
and,
Figure 1. The PDF curves of GWE-G with
.
(11)
Figure 2 gives some of the possible shapes of the hazard rate function of the GWE-G distribution for some various values of parameters n,
and
. It can be summarized some of the shape properties of the hazard rate function of
Figure 2. The hazard curves of GWE-G with
.
WE-G as: (a) The hazard rate function is increasing function when
and
and 10. (b) The hazard rate function is bathtub shaped when
and
and
. (c) The hazard rate function is increasing function when
and
and
.
4. Expansions for the Cumulative, Density and Survival Function
In this section, we discuss some useful expansions for the CDF and PDF of GWE-G distribution. The following mathematical relations will be used in these expansions.
(12)
for any positive real non-integer k, and
. We also use this relation for any positive integer m:
(13)
4.1. Expansion for the CDF
In this subsection, we introduce the expansion forms for the CDF for GWE-G (
) distribution. We can writ Equation (9) in another form:
(14)
where,
, and
denotes the exponentiated Gompertz (EG) distribution with parameters
and
.
We also obtain another expansion for the CDF of GWE-G as:
From Equation (9) and expanding the term
, the GWE-G can be rewritten as:
Using power series expansion for
, we have:
Now, using the binomial expansion for
, the CDF admits the following expansion:
(15)
Also, the GWE-G CDF can be rewritten as:
(16)
where,
4.2. Expansion for the SF
In this subsection, we introduce the expansion forms for the SF for GWE-G (
) distribution. From Equation (10) and using Equation (13), the SF of GWE-G can be rewritten as:
expanding the term
, we have:
Using power series expansion for
, we have:
Now, using the binomial expansion for
, the CDF admits the following expansion:
(17)
where,
4.3. Expansion for the PDF
Here, we provide simple expansion for the GWE-G density function. Firstly, applying Equation (13) into Equation (8) and expansion the term:
,yields:
Using again the series expansion (13), we can express the PDF of GWE-G distribution as:
(18)
where,
Also, we can write the PDF of GWE-G in the form:
Now, using the power series (12) in the last term of above equation:
(19)
where,
and
denotes the density function of Gompertz distribution with parameters
and
. Therefore, the density function of GWE-G can be expressed as an infinite linear combination of Gompertz densities.
5. Statistical Properties
In this part, Let us view different moments of GWE-G (
) distribution. By using the moment we can study some of the most important characteristics and features of a distribution, such as moment generating function, the moments, and interesting reliability properties such as mean residual lifetime.
5.1. Moment Generating Function
If X has the GWE-G (
) distribution, then the MGF is:
(20)
where,
defined by Equation (18).
5.2. Moments
The GWE-G random variable has the rth moments about the origin are:
(21)
For
Equation (21), yields the mean of GWE-G that is given by:
(22)
5.3. Central Moments and Cumulants
The central moments (
) and cumulants (
) of X can be calculated as
(23)
and
respectively, where
. Then,
,
,
, etc. Also, the skewness
and kurtosis
follow from the second, third and fourth moments.
Variance
For
Equation (23), the variance of GWE-G distribution is given by:
(24)
5.4. The Mean Deviation
Let X be a random variable that follows GWE-G distribution with median m and mean
. In this subsection, we inferred the mean deviation from the mean and the median.
5.4.1. The Mean Deviation from the Mean Can Be Found from the following Theorem
Theorem 1. The form of the mean deviation from the mean of the GWE-G distribution is
(25)
Proof: The mean deviation from the mean can be defined as
where,
. Using Equation (15) we obtain
Hence, the theorem is proved.
5.4.2 The Mean Deviation from the Median Can Be Found from the Following Theorem
Theorem 2. The mean deviation from the median (m) of the GWE-G distribution is in the form
(26)
Proof: The mean deviation from the median can be defined as
Using Equation (15) we obtain
Hence, the theorem is proved.
5.5. The Mode
The mode for the GWE-G distribution can be found by differentiating
with respect to x; thus, from Equation (8)
(27)
By equating Equation (27) with zero, we get
(28)
Then, the mode of GWE-G distribution can be found numerically by solving Equation (28).
6. Rényi Entropy
Entropy is a measurement representation of the degree of disorder or uncertainty in a system. The Rényi entropy is defined by
In this case
We expand the term
, the
can be rewritten as:
Using power series expansion for
, we have:
when we expand the term
Expand the term
and using power series expansion of
Hence, the Rényi entropy is given by
(29)
when
7. Reliability Measures of GWE-G
Here, we derive the expression for the mean and strong mean inactivity time functions, mean of residual lifetime of the GWE-G model.
7.1. Mean Residual Time
One of the well known properties of the lifetime distribution is mean residual lifetime (MRL). For the GWE-G distribution, it can be written as
by using Equation (17) the MRL can be written as:
(30)
7.2. Mean Inactivity and Strong Mean Inactivity Time
The mean inactivity time (MIT) and strong mean inactivity time (SMIT) functions are significant to describe the time which had elapsed since the failure in many applications. Many properties and applications of MIT and SMIT functions can be found in Kayid and Ahmad [19], and Kayid and Izadkhah [20]. Let X be a lifetime random variable with CDF,
. Then the (MIT) and (SMIT) are defined as, respectively:
and
by using the expansion form of CDF that given in Equation (16) and after some simple calculations, we obtain the MIT of GWE-G distribution as:
(31)
Similarly, we obtain the SMIT of the GWE-G distribution as:
(32)
We observed from Table 1 and Table 2 that the MIT and SMIT are decreasing for increasing values of
and
. Also from Table 3 the MIT and SMIT are increasing for increasing values of n.
8. Estimation and Inference
In this section, the ML method is considered to estimate the parameters of GWE-G (
) distribution. Let
be a random sample with size m from the GWE-G with PDF and CDF given, respectively, by Equation (8) and Equation (9). Also, we assume that
is the
unknown parameter vectors. Based on the Equation (8), the log-likelihood function is defined by:
where,
Table 1. MIT and SMIT of GWE-G distribution at t = 8 and n = 2.
Table 2. MIT and SMIT of GWE-G distribution at t = 8 and n = 3.
Table 3. MIT and SMIT of GWE-G distribution at t = 8.
is the likelihood function. We can derive the likelihood function of GWE-G distribution as:
thus, the log-likelihood function of GWE-G distribution is obtained as:
(33)
The first derivatives of Equation (33) with respect to
and
respectively is given by:
(34)
(35)
(36)
(37)
where,
is the polygamma function of order m and it is defined by
derivative of the logarithm of the gamma function, as
particular case,
is called digamma function.
Equate the Equations (34)-(37) to zero and solving them simultaneously yield the maximum likelihood estimators (MLEs) of GWE-G distribution parameters. Clearly, these equations are not in explicit form, the solutions can be found by using a numerical method such as the Newton-Raphson procedure to obtain the MLEs of the parameters
and
.
To obtain the asymptotic confidence intervals (CIs) for the parameters of the GWE-G distribution, the
is required. Under certain regularity conditions, the MLEs asymptotically have (for large samples) a multivariate normal distribution with mean vector
and variance-covariance matrix, which is given by the inverse of Fisher information matrix:
for more details about asymptotic confidence intervals. The
depends on
, the observed Fisher information matrix
may be used instead of the
in the estimation of the variance of MLEs. The
observed information matrix,
is:
where,
Regrettably, the accurate mathematical expressions for the above expectation are very hard to obtain. Therefore, the observed Fisher information matrix is given by
which is obtained by dropping the expectation on operation E.
The elements of the
are given by the following equations:
The approximate
CIs of the parameters of GWE-G (
) are respectively, given by:
,
and
,
where,
,
,
and
are the variances of
and
which are given by the diagonal elements of
and
is the upper (
) percentile of the standard normal distribution.
9. Real Data Application
In this section, we illustrate an application of the GWE-G distribution to the total milk production in the first birth of 107 cows from SINDI race. These cows are property of the original data is not in the interval (0, 1), and it was necessary to make Carnaúba farm which belongs to the Agropecuária Manoel Dantas Ltda (AMDA), located in Taperoá City, Paraíba (Brazil). a transformation given by
, for
. These data are presented in Table 4 and the values of
are given in Table 3.1 of Brito ( [21], p. 46). Also, Descriptive statistics of these data are tabulated in Table 5.
Table 4. Proportion of total milk production.
Table 5. Descriptive statistics for the real data.
We apply the values of negative log likelihood function (-LOG), Kolmogorov-Smirnov (K-S), P-value of (K-S), Androson-Darling (A*), Cramér-Von Mises (W*) and Watson statistics to verify which distribution better fits these data. The model selection was carried out using the AIC (Akaike information criterion), the BIC (Bayesian information criterion), the CAIC (consistent Akaike information criterion), Second Order of Akaike Information Criterion (AICc) and Hannan-Quinn Information Criterion (HQC):
and
where,
denotes the log likelihood function evaluated at the maximum likelihood estimates, k is the number of parameters and m is the sample size. The MLEs, AIC, BIC, AICc and HQIC for the parameters of
,
,
and
are given in Table 6. Table 7 provides the values of negative log likelihood function (-LOG), Kolmogorov-Smirnov (K-S), P-value of (K-S), Androson-Darling (A*), Cramér-Von Mises (W*) and Watson statistics.
It is evident from Table 6, Table 7 that, the GWE-G when (n = 7) distribution has the lowest statistics among all fitted model. Hence, this distribution can be chosen as the best model for fitting this data set.
Figure 3(a): the plots of estimated density function of the real data set. Figure 3(b): the plots of empirical distribution and estimated CDF for the real data set.
Table 6. MLEs, AIC, BIC, HQIC, and AICC for data set.
Table 7. The statistics -LOG, K-S, WT, A* and W* for data set.
Figure 3. (a) Estimated density for real data set; (b) Estimated CDF for real data set.
10. Conclusion
In this paper, we discuss a new extension version of the Gompertz distribution generated by integral transform of the PDF of the generalized weighted exponential distribution. Statistical properties of the GWE-G are viewed. Maximum likelihood estimators of the GWE-G parameters are obtained. Moreover, the new model with its sub-models is fitted to real data set and it is shown that this model has a better performance among the compared distributions.