Abstract

A new lifetime distribution, called exponential doubly Poisson distribution, is proposed with decreasing, increasing, and upside-down bathtub-shaped hazard rates. One of the reasons for introducing the new distribution is that it can describe the failure time of a system connected in the form of a parallel-series structure. Some properties of the proposed distribution are addressed. Four methods of estimation for the involved parameters are considered based on progressively type II censored data. These methods are maximum likelihood, moments, least squares, and weighted least squares estimations. Through an extensive numerical simulation, the performance of the estimation methods is compared based on the average of mean squared errors and the average of absolute relative biases of the estimates. Two real datasets are used to compare the proposed distribution with some other well-known distributions. The comparison indicates that the proposed distribution is better than the other distributions to match the data provided.

1. Introduction

Sometimes, there is a need to generate a new distribution from a given distribution if the new distribution gives more flexibility to the data analysis in the sense of having a better fit, more variations of the hazard rate function (HRF), etc. Several methods can be implemented to generate new distributions. One of them is the compounding of distribution functions.

Suppose that and are two probability functions and can take countable numbers with masses at , . Then, the technique of compounding distributions takes the formwhere and .

In lifetime testing experiments and reliability studies, exponential distribution (ED) is one of the most discussed distributions. However, for some practical applications where associated hazard rates are not constant, presenting monotone shapes, the exponential distribution does not provide an appropriate parametric fit. In the last decades, new classes of distributions have been proposed based on modifications of the ED to overcome such a problem.

Kuş [1] presented the exponential Poisson distribution (EPD), which has a decreasing hazard rate, as a lifetime distribution of a system, connected as a series structure, where the number of its components is a random variable (RV) subjecting to the truncated Poisson distribution (at zero) and the underlying lifetime distribution is exponential. Cancho et al. [2] obtained the Poisson exponential distribution (PED), which has an increasing hazard rate, as the lifetime of a system, connected as a parallel structure, in which the number of its components is a RV subjecting to the truncated Poisson distribution (at zero). A distribution named exponentiated exponential Poisson distribution (EEPD) for the series-parallel structure in which the number of components included in the series subsystem is random and that of the parallel subsystem is fixed was considered by Ristić and Nadarajah [3]. For more details about the compounding of distribution functions based on failures of a system, see, for example, Louzada et al. [4, 5], Mahmoudi and Sepahdar [6], Abdel-Hamid and Hashem [7, 8], and Nadarajah et al. [9].

In the implemented paper, we use the compounding of distributions concept to generate a new distribution based on the failures of a system connected as a parallel-series structure where the number of units in both the parallel and series subsystems is random. We name the proposed distribution exponential doubly Poisson distribution (EDPD) and then compare it with exponentiated Weibull Poisson distribution (EWPD), EEPD, EPD, PED, and ED. The EDPD accommodates decreasing, increasing, and upside-down bathtub-shaped hazard rates. These various shapes make it more versatile to suit multiple real data and are therefore regarded as one of the motivations of the EDPD. Some properties of the proposed distribution are studied. Four estimation methods for the included parameters, based on progressive type II censoring, are considered.

The EDPD’s further motivation is focused on the failure of a parallel-series system. Suppose the system of units in a factory is designed according to the parallel-series structure. As shown in Figure 1, let denote the number of parallel subsystems functioning independently at a certain time, where is a RV (with a realization ) subjecting to a discrete distribution (geometric, power series, zero-truncated Poisson, etc.). Assume that each system of the parallel systems consists of (where , , are RVs with realizations ) units, which are connected as a series structure. Their failure times, , , are independent and identically distributed (iid) RVs subjecting to a continuous lifetime distribution (Lomax, exponential, Weibull, Burr type XII, etc.). The success of the -th system depends on the success of all the system units. In other words, the system fails just a one unit fails, i.e., . Since the systems are connected as a parallel structure, then the factory’s system can succeed when at least one of its parallel subsystems succeeds. In other words, it fails if all of the included subsystems fail. Let represent the failure time of the factory’s system. Thenwhere are iid RVs.

Figure 1 

System with a parallel-series structure.

The rest of the paper is structured as follows: the EDPD and some of its properties are presented in Section 2. Section 3 explores several methods of estimation. In Section 4, a real example is given. Section 5 presents a simulation analysis accompanied by conclusions in Section 6.

2. Exponential Doubly Poisson Distribution

In the current section, we obtain the EDPD by compounding Poisson and EDs based on failures of the parallel-series system. The probability density function (PDF), cumulative distribution function (CDF), survival function (SF), HRF, -th moment, mean, variance, PDF of the -th order statistic, Bonferroni curve, Lorenz curve, Rényi entropy, and Shannon’s entropy of the EDPD are obtained and studied in detail.

The distribution of the RV , presented in (2), is achieved in the following theorem.

Theorem 1. For , suppose that , are iid RVs with the PDF and the CDF . Suppose also that and are two independent zero-truncated Poisson RVs with probability mass functions ,  and , , respectively. Then, the distribution of the RV , presented in (2), has an EDPD with PDF and CDF given, respectively, bywhere

Proof. Suppose that , .
The conditional density function of given is then given by Using equation (1), the marginal PDF of and its corresponding CDF are given, respectively, bySince , then the conditional density function of given is given by Therefore, the marginal PDF of is given bywhere is given by (5). Thus, the CDF of takes the form The SF and HRF of the EDPD with CDF (4) are given, respectively, bywhere is given by (5).

Remark 1. The EDPD, with CDF (4), tends to the(1)EPD, proposed by Kuş [1], if ,(2)PED, proposed by Cancho et al. [2], if ,(3)ED if and .

Theorem 2. A sufficient condition for PDF (3) to be decreasing (unimodal) is

Proof. The derivation of with respect to takes the formwhereThe first derivative of the function takes the formIt is clear that . Therefore, the function is increasing.
Now, if , then (since is increasing). Then , and hence, the is a decreasing function, which leads to (PDF (3)) is also decreasing.
But, if and , then has a unique root and and , and hence, the is a unimodal function, which leads to (PDF (3)) is also unimodal.

Theorem 3. HRF (12) of the EDPD is(1)a decreasing (an increasing) function if ,(2)an upside-down bathtub-shaped function if .

Proof. Suppose that , where . Hence,and its first derivative is whereIts first derivative takes the formIt is clear that . Therefore, the function is increasing.
Now, if , then . Thus, , and from Glaser’s theorem (see Glaser [10]), HRF (12) is decreasing.
If and , then . Thus, , and from Glaser’s theorem, HRF (12) is increasing.
If and , then has a unique root and , , and , . Thus, , , and , , and from Glaser’s theorem, HRF (12) is upside-down bathtub.
For different values of , and , PDF (3) and HRF (12) of the EDPD are plotted in Figure 2.

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Figure 2 

(a) The PDF and (b) HRF of the EDPD for different values of , and .

Theorem 4. Suppose that is a random sample from the EDPD with CDF (4). Then, the CDF and its corresponding PDF of the -th order statistic, say , are given, respectively, bywhere is given by (5).

Proof. As shown in Arnold et al. [11] and David and Nagaraja [12], the PDF of the -th order statistic can be written as follows:If the PDF and CDF are given by equations (3) and (4), respectively, thenwhere is given by (5).
The corresponding CDF is given bywhere is given by (5).

Theorem 5. Let and be the smallest and largest order statistics of a random sample from the EDPD with PDF (3) and CDF (4), respectively. Thenwhere ,

Proof. The proof can be easily shown by Theorem 8.3.1–Theorem 8.3.4 in Arnold et al. [11].

Theorem 6. Assume that are order statistics from the EDPD. Then, for .where , , and are given in Theorem 5.

Proof. Using equations (8.4.2) and (8.4.3) in Arnold et al. [11], the proof yields directly.

Theorem 7. If the RV has the EDPD with PDF (3), then the -th moment of is given bywhere, as in Canuto et al. [13], and represent the zeros and corresponding Christoffel numbers of the Legendre–Gauss quadrature formula on the interval (−1, 1).

Proof. The -th moment of the RV is given byWe can approximate the previous integral, by using Legendre–Gauss quadrature formula aswhereand is the Legendre polynomial of degree .
Figure 3 plots the relationship between the mean and the degree of the Legendre polynomial in which we can observe the volume of required to obtain a satisfactory approximation to the true mean.
The mean, variance, skewness, and kurtosis of the EDPD are plotted in Figure 4. We observe the following:(1)By increasing , the mean and variance decrease, but they increase by increasing .(2)By increasing , the skewness and kurtosis decrease, but they increase-decrease by increasing .The Bonferroni and Lorenz curves have applications in economics to study income and poverty, insurance, demography, reliability, and medicine. In the following theorem, we discuss the Bonferroni and Lorenz curves of the EDPD with PDF (3).

Figure 3 

The relationship between the mean and the degree of Legendre polynomial.

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Figure 4 

The mean, variance, skewness, and kurtosis of the EDPD.

Theorem 8. Suppose that the RV has the EDPD with PDF (3). Then, the Bonferroni curve and Lorenz curve are given, respectively, bywhere is given by (29) and

Proof. The Bonferroni curve of the EDPD is given byThe Lorenz curve of the EDPD is given byThe Bonferroni and Lorenz curves are plotted in Figure 5.
Entropy is applied to determine dynamical systems’ randomness or uncertainty and is commonly applied in science and engineering. In the following theorem, we discuss two popular entropy measures that are the Rényi and Shannon’s entropies (see Shannon [14] and Rényi [15]).