Intervened Exponential Distribution: Properties and Applications

This manuscript aims to study the intervention-based probability model. Statistical and reliability properties such as the expressions for, cumulative density function (cdf), mean deviations about mean and median, r order central and non-central moments, generation functions for moments have been derived. Moreover, the expression for reliability function, hazard rate function, reverse hazard rate function, aging intensity, mean residual life function, stress-strength reliability, and entropy metrics due to Rényi and Shannon are also derived. Monte Carlo simulation study performance of maximum likelihood estimates (MLEs) has been carried out, followed by calculations of average bias (Abias), and Mean Square Error (MSE). The applicability of the model in real-life situations has been discussed by analyzing the two real-life data sets.


Introduction
In reliability and survival analysis, the lifetime of the system or an individual is viewed as, an essential characteristic. This important feature in reliability theory is investigated with traditional lifetime distributions available in the literature, such as Gamma, Exponential, Weibull, Rayleigh, Normal, Log-normal, etc. For a deep and more detailed summary about the lifetime models, one could refer to Barlow and Proschan (1975), Zacks (1992), Marshall and Olkin (2007), etc. In distribution theory, numerous continuous and discrete lifetime distributions along with modified versions are available in the literature. However, Shanmugam (1985) is the pioneer to develop intervention-based models in distribution theory, due to its impressive applications in several areas of statistics and other applied sciences, very useful publications on intervention-based models of Huang and Fung (1989), Scollnik (1995), Dhanavanthan (1998Dhanavanthan ( , 2000 , Shanmugam et al. (2002), Scollnik (2006), etc. are available in the statistical literature. Besides these few attempts observed in this direction, reliability theory is yet to explore with the idea. The most popular and frequently used distribution among all continuous distributions is considered Exponential distribution having a cluster of applications in reliability theory, survival analysis, engineering sciences, economics, physics, business statistics etc. In history, researchers have developed, different types of modified and generalized models of the Exponential distribution. The explanatory summary on Exponential distribution is given by Balakrishnan (2019). However, the recent developments in distribution theory, in the form of discrete intervened probability models motivated us to explore the continuous lifetime intervened model. In this connection, we have attempted to study the new extension of Exponential distribution namely intervened Exponential distribution developed by Shanmugam et al. (2002), and have also given a nice explanation of the intervention parameter (ρ) used in the model. We hope this move Intervened Exponential Distribution: Properties and Applications would enlighten distribution theory with new advantageous lifetime probability models such as intervened mixture probability models, generalized intervened probability models, and the intervention-based probability models developed by using different transformations or by adding new parameters are some possible directions. The distribution function (cdf ) of intervened Exponential distribution (I v ED) along with probability density function (pdf ) are given as and, where z > α > 0, and the parametric space ξ = {(ρ, α, β) : ρ > 0, α > 0, β > 0}, where ρ is the intervention parameter, α is the truncation and β is the rate parameter of the distribution. Also, some derived results mentioned in the article of Shanmugam et al. (2002) are noticed as, the mean (µ z ), µ z = α+(ρ+1)β, variance (σ 2 z ), σ 2 z = (ρ 2 +1)β 2 and the mode (M z ), M z = α + (ρβ/(ρ − 1)) ln |ρ| respectively. The graphical illustration of pdf given in equation (2) is shown in Fig.1, where as for cdf given in equation (1)  shown in Fig.2. The set of parametric values for both the plots are taken as, data1 = (ρ = 2.018, α = 0.010, β = 4.500), data2 = (ρ = 1.009, α = 0.050, β = 5.055), data3 = (ρ = 2.001, α = 0.020, β = 6.050), and data4 = (ρ = 2.018, α = 0.010, β = 8.500). It is quite visible from the graphical plots the pdf is exhibiting different shapes while changing the parametric values.

Inferential Study of I v ED
In this section, some interesting characteristics about I v ED are discussed in detail, these include mean deviations, central and non-central r th order moments, different generating functions for moments, etc. are some of the useful measures having considerable importance in distribution theory. Firstly we begin the section by the median (M d ) derivation. Let z be a continuous and absolutely non-negative random variable (r.v.) possessing I v ED(ρ, α, β), then we proceed the steps for derivation as, Intervened Exponential Distribution: Properties and Applications After solving the integral and simplifications, the final expression obtained is as, This derived expression for M d does not reveal the explicit solution, but we can find the approximate value for M d in R-software by using the uniroot function.

Mean Deviations
In statistics, the two prominent deviations are the mean deviation (D µz ) and median deviation (D M d ), which is the average of all the deviations taken from the mean and the respective median. These deviations have the advantage to measure the amount of scatteredness present in the data. Mathematically, the expressions are given by Theorem 2.1. If a continuous and absolutely non-negative r.v. z ∼ I v ED(ρ, α, β), then the mean deviations about mean (µ z ) and median (M d ) are given by Proof. (i) We have a non-negative r.v., z ∼ I v ED(ρ, α, β). Then the expression for D µz is given by, Hence, completes proof for part first.
(ii) Again, for a continuous, and non-negative r.v., z ∼ I v ED(ρ, α, β), we can write the mathematical expression for median deviation as, This completes proof for part (ii).

Moments and Generating Functions
In this subsection, we derive the expressions for generating functions, r th order central and non-central moments for I v ED(ρ, α, β), in the following subsequent theorems.
where, r = 1, 2, ..., n and Γ(s, x) = ∞ x t s−1 e −t dt is the upper incomplete gamma function. Proof. (i). The expression for moments about mean for a r.v., z ∼ I v ED(ρ, α, β) is given by Hence, completes proof for part (i).
Proof. (ii) Since, we know the mathematical expression for non-central moments, for a r.v. z ∼ I v ED(ρ, α, β), is defined by Hence, proved the result.
, then, we defined the expressions for, moment generating function M z (t), characteristic function φ z (t), and cumulant generating function K z (t) as , then mathematical expression for moment generating function is given by

Intervened Exponential Distribution: Properties and Applications
Hence, proved the result (i).
(ii) To prove part (ii) of the above theorem, similar steps have to be repeated while deriving the expression, but the only difference is that instead of t, we have to proceed with ιt to obtain the resulting equation for characteristic (iii) Since, we know that, if z ∼ I v ED(ρ, α, β), then the cumulant generating function is mathematically defined by: Hence, proved.

Reliability Characterization
Reliability or Survival function is defined as the failure-free operation of a system, during a particular interval of time.
For, many decades ago, this function is meant intrinsic features in performance activity measurements of the system, by considering failure data. Mathematically, the reliability R IvED (z; ξ), for a non-negative, and absolutely continuous By invariance property of M LE s , we can write the reliability estimate as given by The opposite of the reliability function is hazard rate function, often frequently named failure rate function, maybe constant or the function of the time. In literature, this function is widely acknowledged with alternative sub-titles in demographic and actuarial sciences. Among them, intensity rate, force of mortality, instantaneous force of mortality, and the mortality rate are sharply noticed. For a non-negative continuous The graphical plots of the reliability and hazard rate functions are shown in For both the plots the values of parameters are taken as, data1 = (ρ = 2.018, α = 0.010, β = 4.500), data2 = (ρ = 1.009, Intervened Exponential Distribution: Properties and Applications α = 0.050, β = 5.055), data3 = (ρ = 2.001, α = 0.020, β = 6.050), and data4 = (ρ = 2.018, α = 0.010, β = 8.500). The graphical illustration of hazard rate function as shown in Fig.(4) is clearly showing the increasing trend, which has useful application while analyzing the failure data in reliability.
In reliability theory, the well-known extended concept reverse hazard rate function is described as a hazard rate, but in a reverse direction of time. The ratio of pdf (f IvED (z; ξ)) upon cdf (F IvED (z; ξ)) is defined as a reverse hazard rate function. For I v ED it is obtained as Statistical literature revealed, the approximate constant, increasing or decreasing trends of unimodal hazard rate function. Therefore, for the system, aging representation is tedious. Thus, a new reliability function, named aging intensity (A.I) profounded by Jiang et al. (2003) would have the quantitative measurement for aging. Thus A.I for a r.v., z ∼ I v ED(ρ, α, β) is denoted by L IvED (z; ξ) and is defined as Thus, for I v ED our derived result for A.I are as where, f IvED (.) and R IvED (.) are respective pdf and reliability function of model, and the average failure rate function X Iv ED (z; ξ) = (1/z) z 0 h(z; ξ)dz.

Shape of Hazard Rate Function
The actual shape behavior of the hazard rate function given in equation (8) is not clear from the graphical plot as shown in Fig.(4). To provide the solid mathematical interpretation regarding the shape of the hazard rate function, the following lemma due to Glaser (1980) is defined as: Proof. For complete proof refer to Glaser (1980). Proof. The pdf f (z; ξ) of I v ED(α, β, ρ), when ρ = 1, is defined in equation (2). Therefor according to above Lemma 3.1., the function η(z; ξ) of a continuous non-negative r.v. z is defined as After differentiating η(z; ξ) with respect to z and on simplifying we get This is true for (α, β, ρ) > 0, when ρ = 1. Thus, we have η (z; ξ) > 0, for all z > 0, for given pdf f (z; ξ). Therefore according to Lemma 3.1., the statement of Theorem 3.2. is proved, the hazard rate function of I v ED(α, β, ρ), when ρ = 1, is increasing for all z > 0.
Intervened Exponential Distribution: Properties and Applications

Mean Residual Life Function
Specifically, to determine the distribution, an alternative approach Mean Residual Life (M RL) function does exist. Suppose a system working up to time t ≥ 0, then the remaining working hours until it fails i.e beyound time t is termed as residual life, defined by conditional r.v. Z − t| Z>t . Precisely, more attractive and interesting story about M RL function being that, this is advantageous in many branches of science, engineering science, economics, survival analysis, and reliability theory for characterizing the lifetime. Although, it is quite difficult to explore failure rate function without undertaking other measures. It is theoretically obvious, the most prominent among them being M RL function as they are complementary functions to each other Finkelstein (2008).

Mean Inactivity Time
Mean Inactivity Time (M IT ) have strong applications in different branches of applied sciences, such as, reliability analysis, survival analysis and economics are some areas. If z ∼ I v ED(ρ, α, β), then the mean inactivity time is given by

Entropy Measures
In this section, we discuss two popular entropy metrics namely Rényi entropy (see Rényi 1961) and Shannon entropy ( see Shannon 1948), which are used to measure the amount of information or uncertanity. Here, we derive Rényi and Shannon entropy mathematically.

Rényi Entropy
Let a r.v. z ∼ I v ED(ρ, α, β), then the Rényi entropy of order κ is given by which is required expression of Rényi entropy of order κ. Note: B(a, b) = 1 0 z a−1 (1 − z) b−1 dz is the beta function.

Shannon Entropy
Shannon entropy is the extended result of Rényi entropy, thus for a continuous r.v. z ∼ I v ED(ρ, α, β) this measure is obtained as After substituting the equation (2) of pdf , the simplified result we obtained for Shannon measure of entropy are given by

Order Statistics of I v ED
Order statistics are counted among important branches of statistics, having numerous applications in reliability analysis and life testing modeling used to study the system reliability characteristics. Let z = (z 1 , z 2 , z 3 , ..., z n ), be a random sample from I v ED(ρ, α, β) with cdf and its pdf defined in equations (1) and (2) respectively. Let us consider the ordered random sample as z (1:n) ≤ z (2:n) ≤ z (3:n) ≤ ... ≤ z (n:n) , so that z (i:n) ; [i = 1, 2, ..., n] having the life time of [n − i + 1] out-of-n system consisting n i.i.d components. The pdf of i th order statistic z (i:n) ; 1 ≤ i ≤ n, is given by: The joint (i, j)th order density function of (z (i:n) , z (j:n) ) for 1 ≤ i ≤ j ≤ n is given below: where, Intervened Exponential Distribution: Properties and Applications Thus, for an ordered random sample of n observations, 1 st order statistic is the smallest observation given by z (1) = min.{z (1) , z (2) , z (3) , ..., z (n) }, while the n th order statistic is the largest sample observation given by z (n) = max. { z (1) ,z (2) ,z (3) , ... ,z (n) }, and the median order statistics is the middle observation given by z m+1 .

Joint Order Density
The joint (i, j) th order statistic pdf of I v ED is given by:

Stress-Strength Reliability
Stress-strength modeling, is another famous system reliability measurement technique, having a wide variety of applications in different areas of statistical sciences and engineering. Suppose a system, having specific strength Z 1 subjected to common stress Z 2 . Whenever stress exceeds the strength, failure will occur and functions smoothly when Z 1 > Z 2 . Since, R = P r.(Z 1 > Z 2 ) is the desired model for system reliability under stress strength modeling. Now, for independent r.v. s Z 1 ∼ I v ED(α, ρ 1 , β 1 ) and Z 2 ∼ I v ED(α, ρ 2 , β 2 ), having the same parameter (α), with pdf of Z 1 and cdf of Z 2 , when ρ i = 1, (i = 1, 2), the derivation for R is given below: and, Intervened Exponential Distribution: Properties and Applications We have, Now, let us consider ρ 1 = 1 and ρ 2 = 1. Then, in this situation, we have derived the expression for stress-strength reliability, which is given by The next possibility is if ρ 1 = 1 but ρ 2 = 1. Then, in this case, the obtained stress-strength reliability is given by The last possibility is when both ρ 1 and ρ 2 are equal to 1. Then, in this case, we have derived the stress-strength reliability given by

Stochastic Ordering
To judge the performances of lifetime distributions stochastic ordering is considered to be the critical and essential technique briefly discussed by Shaked and Shanthikumar (2007). Let Z 1 and Z 2 , be two r.v. s each following I v ED having cdf s F 1 (z) and F 2 (z), with respective pdf s f 1 (z) and f 2 (z) respectively. Then, we say Z 1 is smaller than Z 2 , according to the below-mentioned orderings: [a 4 ] Likelihood ratio order (Z 1 ≤ LR Z 2 ), if f1(z) f2(z) decreasing in z. Hence, the above stochastic ordering reveal the following implications.
The following, theorem defines likelihood ratio ordering of I v ED with respect to their strongest likelihood.
Intervened Exponential Distribution: Properties and Applications

Estimation Procedure for Parameters
Let z = (z 1 , z 2 , z 3 , ..., z n ) be a sample of random observations drawn form I v ED with some desired parameters ρ, α and β. Let us consider ω = (ρ, α, β) T be a k × 1 vector of parameters. Then the complete sample log likelihood function when ρ = 1 is defined below: Let us consider, G 1 = e −(z−α)/ρβ and G 2 = e −(z−α)/β . Therefore the above equation can be written as, Now differentiate above equation partially with respect to given parameters ρ, α and β we get. For all the three parameters of I v ED(ρ, α, β), the second order partial derivatives of log-likelihood function exists. Hence the obtained Inversion dispersion matrix is given below: Therefore, After finding the Inverse dispersion matrix, we can obtain easily asymptotic variances and co-variances for M LE s (ρ, α, β). Now, to determine the approximate confidence interval of 100(1 − δ)% by using (25) as given below: where y δ is defined as upper 100δth quantile of standard normal distribution.
Intervened Exponential Distribution: Properties and Applications

Simulation Study Performance
For I v ED(ρ, α, β), a Monte Carlo simulation study by repeating the process 1000 times has been carried out for different sample sizes, to check the performances of M LE s . Since the quantile function of the proposed model is not in closed form. Thus inverse transformation method is not applicable to generate the data from IvED. However, in such a situation other data generation techniques namely, the acceptance-rejection algorithm is used to simulate the data. The results are shown in table 1.  The data set second contains the survival lifetimes of forty-four patients, suffering from the Head and Neck cancer disease given by Efron (1988  From the results shown in Table 2 and Table 3 it is clear that I v ED performs better than ED and GED respectively. Hence, this confirms the model applicability for future data analysis purposes.

Conclusion
In this article, we introduced intervened Exponential distribution (I v ED) as a lifetime model in reliability and survival analysis. The derived statistical and reliability properties has been presented in this study. We have plotted graphs of different functions, for pdf it exhibits different shapes, whereas the hazard rate function plot shows this could be useful to model the increasing failure rate data sets. Simulation study of I v ED based on M LE s were also discussed to judge the performance of the parameters. Finally, we have done the comparison study with the pre-exists models by analyzing two real data sets, and this study revealed that the I v ED outperformed among the two well-known models, namely ED and GED available in statistical literature.

Acknowledgment
The first author is very grateful to Pondicherry University for providing him with a research fellowship to carry out this work.
Intervened Exponential Distribution: Properties and Applications