The odd Lindley Nadarajah-Haghighi distribution

In this paper, a new three-parameter model which can be used in lifetime data analysis is introduced. Its failure rate function can be decreasing, increasing, constant and bathtub-shaped depending on its parameters. We derive explicit expressions for some of its statistical and mathematical quantities including the ordinary moments, generating function, incomplete moments, order statistics, moment of residual life and reversed residual life. Some useful characterizations are presented. Maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimators are discussed in case of uncensored data. The censored maximum likelihood estimation is presented in the general case of the multi-censored data. We demonstrate empirically the importance and flexibility of the new model in modeling real data set. https://doi.org/10.28919/jmcs/3417


Introduction
Statistical distributions are very useful in describing and predicting real world phenomena.
Numerous classical models have been extensively used over the past decades for modeling real data sets in several areas.Recent developments focus on definining the new families of THE ODD LINDLEY NADARAJAH-HAGHIGHI DISTRIBUTION distributions that extend well-known distributions and at the same time provide great flexibility in modeling real data.
Recently, a new generalization of the exponential distribution as an alternative to the gamma, Weibull and exponentiated-exponential distributions was proposed by Nadarajah and Haghighi (2011).The cumulative distribution function (cdf) is given by (, , ) = 1 − exp[1 − (1 +  )  ],  > 0, (1) and the corresponding probability density function (pdf) is where the parameter  > 0 control the shape of the distribution and  > 0 is the scale parameter.
Nadarajah and Haghighi (2011) pointed out that the density function (2) has the attractive feature of always having the zero mode.They also showed that larger values of  in (2) will lead to faster decay of the upper tail.
We shall refer to the new distribution using ( 1) and ( 2 ], respectively, where (; ) and (; )are the baseline cdf and survival function which depends on a parameter vector  .where  is a positive shape parameter.To this end, we use (1), ( 2) and (3) to obtain the three-parameter OLNH pdf (for  > 0) (; , , ) = The corresponding cdf is given by Note that the Type I odd Lindley exponential model (TIOLE) arises when  = 1 and the TIIOLE model arises when  =  = 1.
The OLNH density function can be expressed as an infinite mixture of exponentiated-G (exp-G) density functions where model can be given by integrating (7) as where is the cdf of the exp-NH model with power parameter .Equation ( 8) reveals that the OLNH cdf is a linear combination of exp-G cdf's.So, some mathematical properties of this family can be determined from those of the exp-G distribution.Equations ( 7) and ( 8) are the main results of this section.

Moments and generating function
The th moment of , say   ′ , follows from (7) as where  , The skewness and kurtosis measures can be calculated from the ordinary moments using wellknown relationships.The mgf   () = (e   ) of  can be derived using (7) as For more details about the properties Exp NH see Lemonte (2013, p.155).

Incomplete moments
The main applications of the first incomplete moment refer to the mean deviations and the Bonferroni and Lorenz curves.These curves are very useful in economics, reliability, demography, insurance and medicine.The  th incomplete moment, say   () , of  can be expressed using (7) as

moment of residual life and reversed residual life
The  th moment of the residual life, say   () = [( − )  | > ],  = 1,2, … , uniquely determines ().The th moment of the residual life of  is given by   () = where

Order statistics and quantile spread order
Suppose  1 , … ,   is a random sample from an OLNH model.Let  : denote the  th order statistic.The pdf of  : can be expressed as we can write the density function of  : in (11) as Where Equation ( 12) is the main result of this section, it reveals that the pdf of the OLNH order statistics is a linear combination of exp-G density functions.So, several mathematical quantities of the OLNH order statistic such as ordinary, incomplete and factorial moments, mean deviations and several others can be determined from those quantities of the exp-G distribution.The th moment of  : is given by

Characterizations
This section deals with various characterizations of OLNH distribution.These characterizations are presented in two directions: () based on a truncated moment and () in terms of the hazard function.It should be noted that characterization () can be employed also when the  does not have a closed form.We present our characterizations () and () in two subsections.

Characterizations based on a truncated moment
Our first characterization employs a version of a theorem due to Glanzel (1987), see Theorem 1 of Appendix A .The result, however, holds also when the interval  is not closed since the condition of Theorem 1 is on the interior of  .We like to mention that this kind of characterization based on a truncated moment is stable in the sense of weak convergence (see, Glanzel 1990).
Proof.Let  be a random variable with  (5), then and hence and finally Conversely, if  is given as above, then and hence Now, according to Theorem 1,  has density (5).
Corollary 3.1.Let : Ω → (0, ∞) be a continuous random variable.Then,  has  (5) if and only if there exist functions  and  defined in Theorem 1 satisfying the differential equation The general solution of the differential equation in Corollary 3.1 is where  is a constant.Note that a set of functions satisfying the above differential equation is given in Proposition 3.1 with  = 0.

Characterization in terms of the hazard function
with the initial condition ℎ  (0) =  2  1+ .
Proof.If  has  (5), then clearly the above differential equation holds.Now, if the differential equation holds, then which is the hazard function of (5).
for characterizations of other well-known continuous distributions based on the hazard function, the reader should refer to Hamedani (2004) and Hamedani and Ahsanullah (2005).

maximum likelihood estimation
We consider the estimation of the unknown parameters of the OLNH model from complete samples only by maximum likelihood method.The MLEs of the parameters of the OLNH The multivariate normal  3 (0, ( ̂)−1 ) distribution, under standard regularity conditions, can be used to provide approximate confidence intervals for the unknown parameters, where ( ̂) is the total observed information matrix evaluated at  ̂.Then, approximate 100(1 − )% confidence intervals for ,  and  can be determined by:  ̂±  /2 √ ̂ ,  ̂±  /2 √ ̂ and  ̂±  /2 √ ̂ , where  /2 is the upper  ℎ percentile of the standard normal distribution.

Simulation studies
This Subsection assesses the performance of the maximum likelihood estimators (MLEs) in terms of biases, mean squared errors, coverage probabilities and coverage lengths by means of a simulation study ".But We didn't give coverage probability and coverage lengths.We only gave empirical means, sd, biases, and mean squared errors.
In this subsection, we perform two simulation studies using the OLNH distribution to see the performance of MLE's of this distributions.All results were obtained using optim routine in the R programme.We generate 1,000 samples of sizes 20, 50, 100, 200, 300 and 500 from OLNH distribution with  =1.5,  =0.5 and a=2.Secondly, we also generate 1000 samples of size n=20,21,…,100 from OLNH with  =0.25,  =5 and a=10.The random number procedure is THE ODD LINDLEY NADARAJAH-HAGHIGHI DISTRIBUTION obtained by using the inversion method of its cdf.We also compute the biases and mean squared errors (MSE) of the MLEs with and respectively, h = ,  and a.The results of the simulation are reported in Table 1 and Figure 2.
From this Table and Figure 2, we observe that when the sample size increases, the empirical means come close to true values whereas sd, biases and MSEs decrease in all cases, as expected.

Multi-censored maximum likelihood estimation
Often with lifetime data, we encounter censored observations.There are different forms of .
The normal equations are given in Appendix C.

Data analysis
In this section, we present an application based on the real data set to show the flexibility of the  ∑  =1  : and  = 1, . . .,  and  : are the order statistics of the sampleIt is convex shape for decreasing hazards and concave shape for increasing hazards.The TTT plot for the exceedances of flood peaks data in Figure 2 agree with a bathtub-shaped failure rate function.We also, analyzed the ordinary NH distribution with this data.For ordinary NH distribution, we obtained K-S statistics and its p-values as 0.1244 and 0.2153 respectively.The other results of this application are listed in Table 2.These results show that the OLNH distribution has the lowest AIC, CAIC, BIC, HQIC and K-S values and has the biggest estimated log-likelihood and p-value of the K-S statistics among all the fitted models.So it could be chosen as the best model under these criteria.andp-value of the K-S statistics among all the fitted models.So it could be chosen as the best model under these criteria.

Figure 1 .
Figure 1.(a) shows that the OLNH distribution has various pdf shapes.Further, Fig. 1.(b) shows that the OLNH model produces flexible hazard rate shapes such as constant, increasing, decreasing and bathtub.These plots indicate that the OLNH model is very useful in fitting different data sets with various shapes.

1 𝐹
) −ℎ .Another interesting function is the mean residual life (MRL) function or the life expectation at age  defined by  1 () = [( − )| > ], which represents the expected additional life length for a unit which is alive at age .The MRL of  can be obtained by setting  = 1 in the last equation.The th moment of the reversed residual life, say   () = [( − )  | ≤ ], for  > 0 and  = 1,2, …, uniquely determines () .We have   () = − )  () .Then, the  th moment of the reversed residual life of  becomes

.
The mean inactivity time (MIT), also called the mean reversed residual life function, is given by  1 () = [( − )| ≤ ], and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, ).The MIT of the ONH-G family can be obtained easily by setting  = 1 in the above equation.For more details see Navarro et al. (1998).THE ODD LINDLEY NADARAJAH-HAGHIGHI DISTRIBUTION

Subection 4 .
1 gives procedures for maximum likelihood estimation of the OLNH distribution.Subection 4.2 assesses the performance of the maximum likelihood estimators (MLEs) in terms of biases, mean squared errors, coverage probabilities and coverage lengths by means of a simulation study.Subection 4.3 gives procedures for maximum likelihood estimation in the presence of censored data.

Figure 2 :
Figure 2: Plots of the empirical mean, sd, biases and MSE of alpha, lambda and a versus n.

Fig. 3 :Fig. 4 : 1 + ( 1 +
Fig. 3: TTT plot of the exceedances of flood peaks data THE ODD LINDLEY NADARAJAH-HAGHIGHI DISTRIBUTION It is known that the hazard function, ℎ  , of a twice differentiable distribution function,  , satisfies the first order differential equation For many univariate continuous distributions, this is the only characterization available in terms of the hazard function.The following characterization establishes a non-trivial characterization of OLNH in terms of the hazard function which is not of the above trivial form.Let : Ω → (0, ∞) be a continuous random variable.Then,  has  (5) if and only if its hazard function ℎ  () satisfies the differential equation

Table 1 .
Empirical means, sd, bias and mean squared errors

Table 2 .
The application results of the exceedances of flood peaks data (and the corresponding standard deviations (sd) in parentheses) , we plot the estimated pdf, cdf and hrf of the OLNH for the exceedances of flood peaks data with Figure4.Clearly, the OLNH distribution provides a closer fit to the empirical pdf and cdf.Also, from this figures, we have a bathtub-shaped failure rate function for the exceedances of flood peaks data, which are in accordance with TTT plot. Finally