AbdulHadi N. Ahmed ^{1}, Zohdy M. Nofal ^{2}, Hager N. Ahmed ^{2}
^{1}Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt
^{2}Department of Statistics, Mathematics and Insurance Benha University, Benha, Egypt
Correspondence to: Hager N. Ahmed , Department of Statistics, Mathematics and Insurance Benha University, Benha, Egypt.
Email:  
Copyright © 2021 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract
In this thesis, We propose and study new compounding families, the truncated natural discrete Lindley with three distributions severally (Exponential distribution, Weibull distribution and Lomax distribution) this approach generates new distributions that extend wellknown families of distributions. At the same time, they offer more flexibility for modeling lifetime data. The structural properties of the new distributions are investigated. These include the compounding representation of the distribution, reliability analysis and statistical measures. Expressions for Lorenz and Bonferroni curves and Renyi entropy as a measure for uncertainty reduction are derived.. The maximum likelihood method is used to estimate the model parameters. Two reallife data applications are proposed to illustrate the importance of the new families.
Keywords:
Exponential distribution, Weibull distribution, Lomax distribution, Lindley distribution, Reliability analysis, Moment generating function, Order statistics and Quantile, Maximum likelihood estimation
Cite this paper: AbdulHadi N. Ahmed , Zohdy M. Nofal , Hager N. Ahmed , New Compounded Family of Distributions, with Applications, International Journal of Probability and Statistics , Vol. 10 No. 3, 2021, pp. 7493. doi: 10.5923/j.ijps.20211003.02.
1. Introduction
In statistical literature, we always assume that every real phenomenon can be modeled by some lifetime distributions. Modeling and analyzing lifetime data are important aspects of statistical research in many applied sciences such as engineering, medicine, economics and so on. Various recent probability distributions discussed modeling of such data by compounding the wellknown lifetime distributions such as exponential, Weibull, generalized exponential, exponentiated Weibull and etc. with some discrete distributions such as zerotruncated binomial, geometric, Poisson, logarithmic and the power series distributions in general. The compounding approach gives new distributions that extend wellknown families of distributions. At the same time, they offer more flexibility for modeling lifetime data. The flexibility of such compound distributions comes in terms of one or more failure rate shapes that may be decreasing, increasing, bathtub shaped or unimodal shaped. The relevance of mixtures in the analysis and improvements of coherent systems has been pinpointed in various investigations (see, for instance, Navarro et al. (2008) On the application and extension of system signatures in engineering reliability, where comparisons of coherent systems and mixed coherent systems are considered, Samaniego et al. (2007), Dynamic signatures and their use in comparing the reliability of new and used systems, where similar analysis are performed in a dynamic reliability setting, or Dugas et al. (2007), On optimal system designs in reliabilityeconomics frameworks, who showed that solutions to certain optimization problems in reliability turn out to be mixed rather than coherent systems). We propose and study new compounding families will be named the complementary exponential truncated natural discrete Lindley distribution (CETNDL), the complementary Weibull truncated natural discrete Lindley distribution (CWTNDL) and the complementary Lomax truncated natural discrete Lindley distribution (CLTNDL), this approach generates new distributions that extend wellknown families of distributions. At the same time, they offer more flexibility for modeling lifetime data. Adamidis and Loukas (1998) introduced the basic concept of these models. The basic idea of introducing compound models or families is that a lifetime of a system with N (discrete random variable) components and the positive continuous random variable, say Xi (the lifetime of the ith component), can be denoted by the nonnegative random variable (the minimum of an unknown number of any continuous random variables) or (the maximum of an unknown number of any continuous random variables), based on whether the components are in series or in parallel structure.If we define with cdf and N is a truncated discrete random variable with probability mass function the following steps explain how can we get the compounding of probability distribution.Step (1) obtain the conditional cdf of aswhere is the survival function.Step (2) obtain the joint cumulative distribution function as Step (3) The marginal cumulative distribution of for the compounding distribution is given by If we define = , the following steps explain how can we get the Compounding of probability distribution Step (1) Obtain the conditional cdf of asStep (2) obtain the joint cumulative distribution function as The marginal cumulative distribution of for the compounding distribution is given by The rest of paper is organized as follows: Section 2 introduces the definition of the probability density function (pdf) of the (CETNDL), (CWTNDL), (CLTNDL) distributions including its cumulative distribution function (cdf) and the submodels of the new suggested model. The reliability analysis including the survival function, the hazard (or failure) rate function, the reversed hazard rate function, the cumulative hazard rate function, and the mean residual lifetime is explored in Section 3. The statistical properties of the new distribution such as the moments, the moment generating function, and the distribution of order statistics are investigated in Section 4, with a proposed algorithm for generating random data from the new distribution in this section. Section 5 introduces Lorenz and Bonferroni curves and Renyi entropy as measures of inequality and uncertainty, respectively. Section 6 discusses the estimation of parameters by using maximum likelihood estimation. Finally, Section 7 provides an application for modeling real data sets to illustrate the performance of the new distribution.
2. Generalization and Related SubModels
In this section, we introduce the pdf and the cdf of the three new general compounding families and then the special cases of all distributions.
2.1. Generalization
the series systemLet be iid from and and N are independent.1. The conditional cdf of XN is given byNow the joint cdf is This would allow us to reduce the is given by If N follows a truncated natural discrete Lindley, then The Cumulative distribution function (cdf) of X is given byWe break the above sum into several components, as followsTo simplify, letandSumming up, one gets Or it follows that  (1) 
The associated pdf can be obtained as  (2) 
Next, specific well known distributions further explored.
2.1.1. N~ Complementary Exponintial Truncated Natural Discrete Lindley Distribution
Where Then the pdf of for the compounding (CETNDL) distribution can be written as  (3) 
The compounding (CETNDL) class of distributions is defined by the marginal cumulative distribution function of is given by  Figure 1. Increasing, Decreasing, Constant, Bathtub and UpsideDown Shapes for cdf, pdf of CETNDL distribution, and for different values 
 (4) 
2.1.2. N~ Complementary Weibull Truncated Natural Discrete Lindley Distribution
Every in exp. distribution can be replaced by  Figure 2. Increasing, Decreasing, Constant, Bathtub and UpsideDown Shapes for cdf, pdf of CWTNDL distribution,and for different values 
Then the pdf of for the compounding (CWTNDL) distribution can be written as  (5) 
the compounding (CWTNDL) class of distributions is defined by the marginal cumulative distribution function of is given by  (6) 
2.1.3. N~ Complementary Lomax Truncated Natural Discrete Lindley Distribution
where and Then the pdf of for the compounding (CLTNDL) distribution can be written as  (7) 
the compounding (CLTNDL) class of distributions is defined by the marginal cumulative distribution function of is given by  (8) 
 Figure 3. Increasing, Decreasing, Constant, Bathtub and UpsideDown Shapes for cdf, pdf of CLTNDL distribution,and for different values 
3. Reliability Analysis
The time for an event to take place in an individual is called a survival time. Examples include the time that an individual survives after being diagnosed with a terminal illness or the time that an electronic component functions before failing. Two important functions for describing survival data are the survival function, the hazard function, The Reversed Hazard Rate Function, The Cumulative Hazard Rate Function, The Mean Residual Lifetime.
3.1. Survival Functions
The survival function is the probability that an observation survives longer than that isBecause is a probability, it is positive and ranges from 0 to 1. It is defined as and as approaches to approaches to 0. (See Lee (1992), Therefore,
3.1.1. The Survival Probability for the CETNDL Distribution is Given by
 (9) 
3.1.2. The Survival Probability for the CWTNDL Distribution is Given by
 (10) 
3.1.3. The Survival Probability for the CLTNDL Distribution is Given by
 (11) 
3.2. Hazard Rate Function
Nelson, (1982) defined the hazard rate is the rate of death/failure at an instant given that the individual survives up to time t. It measures how likely an observation is to fail as a function of the age of the observation. This function is also called the instantaneous failure rate or the force of mortality. It is defined aswhere is the probability density function of therefor
3.2.1. The Hazard Rate Function for the CETNDL Distribution is Given by
 (12) 
 Figure 4. Increasing, Decreasing, Constant, Bathtub and UpsideDown Shapes for the Hazad Rate Function of the CETNDL 
3.2.2. The Hazard Rate Function for the CWTNDL Distribution is Given by
 (13) 
 Figure 5. Increasing, Decreasing, Constnt, Bathtub and UpsideDown Shapes for The Hazad Rate Function of the CWTNDL 
3.2.3. The Hazard Rate Function for the CLTNDL Distribution is Given by
 (14) 
 Figure 6. Increasing, Decreasing, Constant, Bathtub and UpsideDown Shapes for The Hazad Rate Function of the CLTNDL 
3.3. The Reversed Hazard Rate Function
The reversed (reverse) hazard rate, also named retro hazard, was first mentioned by the name ‘dual of the hazard rate’ in BARLOW et al. (1963). The name ‘reversed hazard rate’ was first used by LAGAKOS et al. (1988). 6 It extends the concept of hazard rate to a reverse time direction and is defined as:
3.3.1. The Reversed Hazard Rate Function for the CETNDL Distribution is Given by
 (15) 
3.3.2. The Reversed Hazard Rate Function for the CWTNDL Distribution is Given by
 (16) 
3.3.3. The Reversed Hazard Rate Function for the CLTNDL Distribution is Given by
 (17) 
3.4. The Cumulative Hazard Rate or Integrated Hazard Rate CHR is Defined as
3.4.1. The Cumulative Hazard Rate Function of the CETNDL Distribution is Given by
3.4.2. The Cumulative Hazard Rate Function of the CWTNDL Distribution is Given by
3.4.3. The Cumulative Hazard Rate Function of the CLTNDL Distribution is Given by
where is the total number of failure or deaths over an interval of time, and is a nondecreasing function of satisfying.
4. Statistical Properties
This section investigates the statistical properties of the (CETNDL), (CWTNDL) and (CLTNDL) distributions
4.1. Distribution of Order Statistics
4.1.1. Distribution of Order Statistics for (CETNDL)
Let be a simple random sample of size from CETNDL with cumulative distribution function and probability density function given by (1.1) and (1.2) respectively. Let denote the order statistics obtained from this sample. The probability density function and the cumulative distribution function of the order statistic, say are given by:thusand
4.1.2. Distribution of Order Statistics for (CWTNDL)
Let be a simple random sample of size from CWTNDL with cumulative distribution function and probability density function given by (1.1) and (1.2) respectively. Let denote the order statistics obtained from this sample. The probability density function and the cumulative distribution function of the order statistic, say are given by:thusand
4.1.3. Distribution of Order Statistics for (CLTNDL)
Let be a simple random sample of size from CLTNDL with cumulative distribution function and probability density function given by (1.1) and (1.2) respectively. Let denote the order statistics obtained from this sample. The probability density function and the cumulative distribution function of the order statistic, say are given by:thusand
4.2. The Quantile Function
4.2.1. The Quantile Function of (CETNDL) Distribution
The quantile function is given by:Substitute from (1.2), the quantile function isIn particular when the median can be defined as:
4.2.2. The Quantile Function of (CWTNDL) Distribution
The quantile function is given by:Substitute from (1.2), the quantile function isIn particular when the median can be defined as:
4.2.3. The Quantile Function of (CLTNDL) Distribution
The quantile function is given by:the quantile function isIn particular when the median can be defined as:
4.3. The NonCentral Moment
4.3.1. The NonCentral Moment for (CETNDL) Distribution
The noncentral moment is given by:
4.3.2. The NonCentral Moment for (CWTNDL) Distribution
The noncentral moment is given by:
4.3.3. The NonCentral Moment for (CLTNDL) Distribution
The noncentral moment is given by:
4.4. The Moment Generating Function
4.4.1. The Following Theorem Gives the Moment Generating Function (mgf) of (CETNDL) Distribution
5. Measures of Inequality and Uncertainty
In this section Lorenz and Bonferroni curves are introduced as measures of inequality. Also, Renyi entropy will be mentioned as an important measure of uncertainty.
5.1. Lorenz and Bonferroni Curves
5.1.1. Lorenz and Bonferroni Curves for (CETNDL) can be Obtained as Follows
A. Lorenz curve can be obtained byB. Bonferroni curve can be obtained by
5.1.2. Lorenz and Bonferroni Curves for (CWTNDL) can be Obtained as Follows
A. Lorenz curve can be obtained byB. Bonferroni curve can be obtained by
5.1.3. Lorenz and Bonferroni Curves for (CLTNDL) can be Obtained as Follows
Bonferroni curve can be obtained by
5.2. Renyi Entropy
According to Meeker and Escobar (1998) the entropy of random variable with density function is a measure of uncertainty or randomness of a system. One of the popular entropy measures is Rényi entropy which is defined by
5.2.1. Rényi Entropy for (CETNDL) is Obtained as Follows
5.2.2. Rényi Entropy for (CWTNDL) is Obtained as Follows
5.2.3. Rényi Entropy for (CLTNDL) is Obtained as Follows
6. Maximum Likelihood Estimation
In this section, we will discuss the estimation of the unknown parameters of the (CETNDL), (CWTNDL), (CLTNDL) using the maximum likelihood estimation (MLE) method
6.1. Maximum Likelihood Estimation for (CETNDL) as Follows
The estimators of unknown parameters of the CETNDL model are obtained based on maximum likelihood (ML) method. Let be a random sample of size from CETNDL, with set a parameters the likelihood function of the density is given by,Then, the log likelihood function is given byThusTo maximize taken first order partial derivatives of with respect to and equating the results equations to zeros and solve them.Taking derivatives to with respect to respectively, one get:By equating and to zero and solving for and the MLE of theses parameters are numerically solutions of the following two equations:
6.2. Maximum Likelihood Estimation for (CWTNDL) as Follows
The estimators of unknown parameters of the CWTNDL model are obtained based on maximum likelihood (ML) method. Let be a random sample of size from CWTNDL, with set a parameters the likelihood function of the density is given by,Then, the log likelihood function is given byThusTo maximize taken first order partial derivatives of with respect to and equating the results equations to zeros and solve them.Taking derivatives to with respect to respectively, one get:By equating and to zero and solving for the MLE of theses parameters are numerically solutions of the following three equations:
6.3. Maximum Likelihood Estimation for (CLTNDL) as Follows
The estimators of unknown parameters of the CLTNDL model are obtained based on maximum likelihood (ML) method. Let be a random sample of size from CETNDL, with set a parameters the likelihood function of the density is given by,Then, the log likelihood function is given byThusTo maximize taken first order partial derivatives of with respect to and equating the results equations to zeros and solve them.Taking derivatives to with respect to respectively, one get:By equating and to zero and solving for the MLE of theses parameters are numerically solutions of the following three equations:Table (1). Maximum likelihood, standard errors and associated asymptotic confidence interval estimates for CETNDL, CLTNDL and CWTNDL distributions based in series and systems given data set I 
 

Table (2). Maximum likelihood, standard errors and associated asymptotic confidence interval estimates for CETNDL, CLTNDL and CWTNDL distributions based in series and systems given data set II 
 

7. Application
In this section we present an application of the Complementary Exponential Truncated Natural Discrete Lindley Distribution, Complementary Weibull Truncated Natural Discrete Lindley Distribution and Complementary Lomax Truncated Natural Discrete Lindley Distribution to two real data sets and fit the Exponential Distribution, Weibull Distribution and Lomax Distribution. The first data set corresponding to remission times (in months) of a random sample of 128 bladder cancer patients given in Lee and Wang (2003). The data are given as follows: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.Table 4.3 The MLEs of the parameters, the values of KS statistic, BIC, AIC, HQIC, NLC and PValue are summarized in Table (1, 2 and 3). From this tables, we note that the TNLExponential distribution is better than the Exponential distribution in terms of fitting to this data, that the TNLWeibull distribution is better than the Weibull distribution in terms of fitting to this data and the TNLLomax distribution is better than the Lomax distribution in terms of fitting to this data.The AIC (Akaike Information Criterion) is given bywhere is the number of estimated parameters.where is the number of the sample size in the model. The Bayesian information criterion (BIC) is given bywhere is the sample size of the training set. The lower BIC score signals a better mode.The second data set is obtained from: Leukemia 33, the data are:c(194, 413, 90, 74, 55, 23, 97, 50, 359, 50, 130, 487, 102,15, 14, 10, 57, 320, 261, 51, 44, 9, 254, 493, 18, 209, 41,58, 60, 48, 56, 87, 11, 102, 12, 5, 100, 14, 29, 37, 186,29, 104, 7, 4, 72, 270, 283, 7, 57, 33, 100, 61, 502, 220,120, 141, 22, 603, 35, 98, 54, 181, 65, 49, 12, 239, 14, 18,39, 3, 12, 5, 32, 9, 14, 70, 47, 62, 142, 3, 104, 85, 67,169, 24, 21, 246, 47, 68, 15, 2, 91, 59, 447, 56, 29, 176,225, 77, 197, 438, 43, 134, 184, 20, 386, 182, 71, 80, 188,230, 152, 36, 79, 59, 33, 246, 1, 79, 3, 27, 201, 84, 27,21, 16, 88, 130, 14, 118, 44, 15, 42, 106, 46, 230, 59, 153,104, 20, 206, 5, 66, 34, 29, 26, 35, 5, 82, 5, 61, 31, 118,326, 12, 54, 36, 34, 18, 25, 120, 31, 22, 18, 156, 11, 216,139, 67, 310, 3, 46, 210, 57, 76, 14, 111, 97, 62, 26, 71,39, 30, 7, 44, 11, 63, 23, 22, 23, 14, 18, 13, 34, 62, 11,191, 14, 16, 18, 130, 90, 163, 208, 1, 24, 70, 16, 101, 52,208, 95).The MLEs of the parameters the values of KS statistic, BIC, AIC, HQIC, NLC and PValue are summarized in Table (1, 2 and 3). From this tables, we note that the TNLExponential distribution is better than the Exponential distribution in terms of fitting to this data, that the TNLWeibull distribution is better than the Weibull distribution in terms of fitting to this data and the TNLLomax distribution is better than the Lomax distribution in terms of fitting to this data.Table (1). Comparison between the Exponential Distribution and Complementary Exponential Truncated Natural Discrete Lindley Distribution 
 

The MLEs of the parameters, the values of KS statistic, BIC, AIC, HQIC, NLC and PValue are summarized in Table (1). From this table, we note that the TNLExponential distribution is better than the Exponential distribution in terms of fitting to this data.Table (2). Comparison between the Weibull Distribution and Complementary Weibull Truncated Natural Discrete Lindley Distribution 
 

The MLEs of the parameters, the values of KS statistic, BIC, AIC, HQIC, NLC and PValue are summarized in Table (2). From this table, we note that the TNLWeibull distribution is better than the Weibull distribution in terms of fitting to this data.Table (3). Comparison between the Lomax Distribution and Complementary Lomax Truncated Natural Discrete Lindley Distribution 
 

The MLEs of the parameters, the values of KS statistic, BIC, AIC, HQIC, NLC and PValue are summarized in Table (3). From this table, we note that the TNLLomax distribution is better than the Lomax distribution in terms of fitting to this data.
8. Conclusions
In this chapter a three new families of lifetime distributions called the complementary exponential truncated natural discrete Lindley distribution (CETNDL), the complementary Weibull truncated natural discrete Lindley distribution (CWTNDL) and the complementary Lomax truncated natural discrete Lindley distribution (CLTNDL) is introduced, this family is obtained by compounding the exponential distribution with truncated natural discrete Lindley distribution, compounding the Weibull distribution with truncated natural discrete Lindley distribution and compounding the Lomax distribution with truncated natural discrete Lindley distribution. It is observed the (CETNDL), (CWTNDL) and (CLTNDL) can have increasing, decreasing, constant, bathtub and upsidedown hazard rate functions which are eligible for data analysis purposes. Several properties of the (CETNDL), (CWTNDL) and (CLTNDL) distributions such as quantiles, median, mean deviations, moments, Lorenz and Bonferroni curves,. This family contains several new distributions such as complementary exponentiated truncated natural discrete Lindley distribution, complementary Weibull logarithmic truncated natural discrete Lindley distribution, complementary weibull geometric truncated natural discrete Lindley distribution and, complementary Weibull Poisson truncated natural discrete Lindley distribution, cdf, pdf, hazard function of these special submodels are presented. The method of maximum likelihood is used for estimating the model parameters. Finally the complementary exponential truncated natural discrete Lindley (CETNDL), the complementary Weibull truncated natural discrete Lindley (CWTNDL) and the complementary Lomax truncated natural discrete Lindley (CLTNDL) series models are fitted to real data sets to show the flexibility and potentiality of the proposed of distributions.
ACKNOWLEDGMENTS
The authorswould like to thank the Editor and the anonymous referees for very careful reading and valuable comments which greatly improved the paper.
Disclosure Statement
No potential conflict of interest was reported by the authors.
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