Lifetime data may exhibit a decreasing, increasing, or a bathtub failure rate function when modeling and analyzing random phenomena. This arises in several areas of studies, such as biomedical studies, reliability, actuarial science, computer science, demography, and engineering. There are several lifetime models that have been used successfully for to analyze lifetime data in practical applications such as exponential, Weibull, Gompertz, generalized exponential, and half-logistic. For example, exponential distribution can accommodate lifetime data with a decreasing density despite having only a constant failure rate function. Gompertz distribution has a decreasing and unimodal density but has an increasing failure rate function. However, these distributions are unable to accommodate lifetime data with a non-monotone failure rate such as the bathtub or upside-down bathtub. Many researchers have attempted to provide several methods to generate new lifetime models with the ability to fit data with monotone or non-monotone failure rate to overcome these problems. These methods include the generalization of a distribution by exponentiation procedure.

Method of exponentiation is an of the important and commonly used techniques to add a parameter to a lifetime model, the new model becomes more flexible and can accommodate both monotones as well as non-monotone failure rate functions. For example, Kuş (2007) proposed en exponential Poisson (EP) distribution that possesses a decreasing failure rate function; however, Barreto-Souza and Cribari-Neto (2009) proposed generalized exponential Poisson (GEP) by exponentiation of the EP. The GEP distribution can accommodate decreasing, increasing and upside down bathtub failure rates. Similarly, Adamidis and Loukas (1998) introduced exponential geometric distribution, while Silva et al. (2010) come up with the generalized exponential geometric distribution. Tahmasbi and Rezaei (2008) introduced exponential-logarithmic distribution similarly, Pappas et al. (2015) proposed a generalized exponential-logarithmic. Silva and Cordeiro (2015) introduced a new class of distribution called the BurrXII Poisson distribution and recently Muhammad (2016a) proposed generalized BurrXII Poisson distribution. To read more about exponentiated distributions see Ali et al. (2007) and Raja and Mir (2011). A new two-parameter distribution known as the half-logistic Poisson (HLP) distribution was also introduced by Muhammad and Yahaya (2017) using the procedure followed by Kuş (2007) and Chung and Kang (2014).

The cumulative distribution function (cdf) of the HLP distribution is given by

with x > 0, α > 0, and λ > 0. The HLP distribution can accommodate data with increasing or decreasing hazard functions.

The first motivation of this study is to propose a new three-parameter probability distribution with increasing, decreasing and upside down bathtub-shaped failure functions. The second motivation of this study is to find the association of the density functions and hazard rate functions in practical applications. Last, we are motivated because the exponentiation method provides additional flexibility to a model especially in both the density and failure rate functions. The HLP distribution cannot fit data set with non-monotone failure rate functions; therefore, we hope that the new distribution that is the generalized half-logistic Poisson (GHLP) will provide solutions to many problems in various fields in practical applications.

The rest of the paper is arranged as follows, in Section 2 we provide the density of the GHLP and consider some important mathematical and statistical properties. Section 3 discusses the maximum likelihood estimate (MLE) and a simulation study. Section 4 provides applications of the new model to a real data set Section 5 provides the conclusions.

The cumulative distribution function of the GHLP distribution with parameters α, β, λ > 0 is given by

where the corresponding probability density and hazard rate functions are given by

respectively. The limiting distribution given by (2.1) when λ → 0⁺ is lim_λ→0⁺F(x) = ((1 − e^−αx)/(1 + e^−αx))^β for β > 0, which is the cdf of the exponentiated half-logistic distribution.

Theorem 1

The probability density function (pdf) given by (2.2) is decreasing function for β ≤ 1.

2.1. Moments

Here, we provide the following lemma which is very useful in computations of several important properties of the GHLP. first we recall that, B_kd(a, b) = (∂^k+dB(a, b))/(∂a^k∂b^d) for k+a > 0, b+d > 0, where B(a,b)=∫01ua-1(1-u)b-1du is a beta function.

Entropy is defined as a measure of uncertainty of a random variable. Here, we consider the two most important entropies known as the Shannon and Rényi entropies. The Shannon entropy is defined by E[− log f (x)]. For a random variable X with GHLP, the Shannon entropy can be computed by considering the Proposition 1 as follows.

Proposition 1

Let X be a random variable with pdf given by (2.2), then,

where J(·, ·, ·, ·, ·, ·) is given by (2.7).

Hence,

The Rényi entropy of a random variable X is defined by IR(ρ)=1/(1-ρ)log[∫0∞f(x)ρdx], where ρ > 0 and ρ ≠ 1. The Rényi entropy of X that has GHLP is obtain as follows.

by applying Lemma 1 the integral become

Hence, the Rényi entropy of X is

In this section, we discuss MLEs, one of the most popular and common methods used in inference. In this method the approximate MLEs are obtained either analytically or numerically using some mathematical packages. Let X_i (i = 1, 2, …, n) be a random sample of size n from the GHLP distribution with observed values x₁, x₂, x₃, …, x_n. The log likelihood function, i.e. log ℓ(θ) for complete data set of the GHLP distribution is given by

Hence, the MLEs of θ = (α, β, λ)^T, say θ̂ = (α̂, β̂, λ̂)^T is the solution of the nonlinear equations (3.2)–(3.4).

For the interval estimate and hypothesis tests of the parameters we required J(θ) the 3 × 3 Fisher information matrix defined by J(θ) = −{∂²(log ℓ(θ))/∂θ∂θ^T }. The approximate of the MLEs of θ, the θ̂, can be approximated as N₃(0, J(θ̂)⁻¹) under usual conditions for parameters in the interior of the parameter space, but not on the boundary. The asymptotic distribution of n(θ^-θ) is N₃(0, J(θ̂)⁻¹), where J(θ̂) is the unit information matrix evaluated at θ̂, which can be used to construct the approximate confidence interval for each parameter. A 100(1 − ε)% asymptotic confidence interval for each parameter θ_r is given by ACIr=(θ^r-Zɛ/2I^rr,θ^r+Zɛ/2I^rr), where I^rr is the (r, r) diagonal element of I_n(θ)⁻¹ for r = 1, 2, 3 and Z_ε/2 is the quantile (1 − ε/2) of the standard normal distribution. The elements of J(θ) can be obtained from the author under request. The existence and uniqueness of MLEs of a probability models based on some certain sufficient conditions have been considered in various literature by many researchers, the existence and uniqueness of maximum likelihood estimators of the EP was analyzed by Kuş (2007), for the exponential geometric by Adamidis and Loukas (1998), Generalized exponential-power series by Mahmoudi and Jafari (2012), extended exponentialgeometric by Adamidis at el. (2005), recently generalized BurrXII Poisson by Muhammad (2016a) and the complementary exponentiated BurrXII-Poisson by Muhammad (2017) among others. The following theorems provide the existence and uniqueness of the MLEs of the GHLP under some possible conditions with the proofs provided in Appendix B.

Theorem 2

Let g₁(α; β, λ, x_i) denote the function on the right hand side of the equation (3.2) where β and λ are the true values of the parameters, then, the equation g₁(α; β, λ, x_i) = 0 has at least one root for β ≠ 1 and for β = 1 the root lies in the interval(n/{(2+λ/2)∑i=1nxi},n/∑i=1nxi).

Theorem 3

Let g₂(β; α, λ, x_i) denote the function on the right hand side of the equation (3.3) where α and λ are the true values of the parameters, then, the equation g₂(β; α, λ, x_i) = 0 has at most one root forlog(1-e-λ)>n-1∑i=1nlog (1-e-λ{(1-e-αxi)/(1+e-αxi))}and is unique.

Theorem 4

Let g₃(λ; α, β, x_i) denote the function on the right hand side of the equation (3.4) where α and β are the true values of the parameters, then, the equation g₃(β; α, λ, x_i) = 0 has at least one root forβ>1/2+(1/n)∑i=1n{(1-e-αxi)/(1+e-αxi)}.

3.1. Simulation study

Here, we evaluate the performance of the MLEs given by equations (3.2)–(3.4) depending on sample size n for the GHLP distribution. In this process we generated 10,000 samples of size n = 20, 30, 40, 50, 100, and 150 from the GHLP distribution for some various values of α, β, and λ. MLEs are obtained by solving the nonlinear equations (3.2)–(3.4) using mlninb in R. The MLEs of α̂, β̂, and λ̂ and their standard deviations sd( α̂), sd(β̂), and sd(λ̂) of the parameters are given in Table 2 below. The results show that each MLE converges to its true value in all cases when the sample size increases and the standard deviations of the MLEs decrease as the sample size increases.

In this section, we provide an application of the GHLP distribution to a real data set. We used the Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), and Kolmogorov-Smirnov (K-S) test to compare the GHLP and some other existing distributions. The model with the smallest values of these measures fit the data better than the other distributions. The competing distributions are:

• The HLP distribution with cdf given by (1.1).

• Olapade-generalized half-logistic by Olapade (2014) with F(x) = 1 − (2^β(1 + e^x/α)^−β).

• Power-half-logistic by Krishnarani (2016) with F(x) = 1 − 2(1 + e^αxβ )⁻¹.

• The generalized half-logistic with F(x) = (1 − e^−αx)^β(1 + e^−αx)^−β. The distribution appears in the study of its estimation procedures by Arora et al. (2010), Kang and Seo (2011), Seo et al. (2012, 2013), and Kantam et al. (2013), where some of its important properties such as the r^th-moments, probability weighted moments, and Shannon and Rényi entropies can be obtained from Cordeiro et al. (2014).

• Poisson-half-logistic by Abdel-Hamid (2016) with F(x) = (e^{λ{(1−e−αx)/(1+e−αx)}} − 1)(e^λ − 1)⁻¹.

• Generalized exponential by Mudholkar and Srivastava (1993) with F(x) = (1 − exp(−αx))^β.

• The EP by Kuş (2007) with F(x) = (e^λe−αx − e^λ)(1 − e^−λ)⁻¹.

• Poisson-odd exponential uniform by Muhammad (2016b) with F(x) = [1−e^{−λ(1−e−α(x/(β−x)))}]/(1−e^−λ).

• Nadarajah and Haghighi (2011) exponential type (NH) with F(x) = 1 − e^{1−(1+λx)α}.

• The half-logistic distribution with cdf as F(x) = (1 − e^−αx)(1 + e^−αx)⁻¹.

The data set is the remission times (in months) of a random sample of 128 bladder cancer patients provided by Lee and Wang (2003).

In Figure 4 the TTT (total time on test)-plot show that the data has upside-down bathtub failure rate function and GHLP distribution has the ability to accommodate upside-down bathtub failure rate curve.

We estimate the unknown parameters of each model by the method of maximum likelihood. The numerical values of the log-likelihood (ℓ (θ)), AIC, CAIC, K-S, and its p-value obtained are presented in Table 3 below. We also used the muhaz package in R software to obtain the empirical hazard function of the given data set and then fitted with the estimated hazard function of the GHLP obtained using the MLEs in Table 3.

Table 3 indicates that our proposed model the GHLP distribution has the smallest values of the AIC, CAIC, and K-S; thus, GHLP provide better fit than the other competing models. Figure 5 provides the plots of the (i) histogram and estimated density (ii) empirical cdf and estimated cdf of the

GHLP distribution. Figure 6 shows (i) empirical and estimated hazard functions (ii) quantile-quantile of the GHLP distribution for the given data set.

We introduced a new three-parameter lifetime distribution with increasing, decreasing and upside-down bathtub-shaped hazard rate functions. We also provide explicit expressions for the r^th ordinary moment, moment generating function, Shannon and Rényi entropies. The estimation of the model parameters was conducted by the maximum likelihood method. The practical significance and applicability of the new distribution are demonstrated in an application to real data, which shows that the GHLP performs better than other existing distributions in terms of fit.

Proof of Lemma 1

For (2.6), by applying generalized binomial expanding on

then expanding

we get

Let u = 1 − e^−αx. By the generalized binomial expansion in the denominator we obtain

thus, we have (2.7).

Proof of Theorem 2

Let g₁(α; β, λ, x_i) be the right hand side of (3.2).

For β = 1, let

then,

On the other side

hence,

thus, the root of g₁(α; β, λ, x_i) = 0 lies in the interval (n/{(2+λ/2)∑i=1nxi},n/∑i=1nxi).

For β ≠ 1, lim_α→0g₁ = ∞ and limα→∞ g1=-∑i=1nxi<0, hence, g₁(α; β, λ, x_i) is a monotone decreasing function from positive to negative; thus, g₁(α; β, λ, x_i) = 0 has at least one root.

Proof of Theorem 3

Consider that lim_β→0g₂ = ∞, then we show that lim_β→∞g² < 0. But limβ→∞ g2=-n log(1-e-λ)+∑i=0nlog (1-e-λ((1-e-αxi)/(1+e-αxi))); thus, lim_β→∞g² < 0 only if log(1-e-λ)>n-1∑i=0nlog (1-e-λ((1-e-αxi)/(1+e-αxi))). To prove the uniqueness we show that g² is a decreasing function, that is g2′<0 and g2′=-n/β2<0.

Proof of Theorem 4

We start with limλ→∞ g3=-∑i=1n((1-e-αxi)/(1+e-αxi))<0; therefore, we show that lim_λ→0g³ > 0. And limλ→0 g3=nβ-n/2-∑i=1n((1-e-αxi)/(1+e-αxi)), thus, lim_λ→0g³ > 0 only if β>1/2+(1/n)∑i=1n((1-e-αxi)/(1+e-αxi)).