Mohamed Kayidand Tareq Alsayed

Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh, Saudi Arabia

Abstract: An important property that any lifetime model should satisfy is scale invariance.In this paper, a new scale-invariant quasi-inverse Lindley(QIL)model is presented and studied.Its basic properties, including moments,quantiles, skewness, kurtosis, and Lorenz curve, have been investigated.In addition, the well-known dynamic reliability measures, such as failure rate(FR), reversed failure rate (RFR), mean residual life (MRL), mean inactivity time (MIT), quantile residual life (QRL), and quantile inactivity time (QIT)are discussed.The FR function considers the decreasing or upside-down bathtub-shaped, and the MRL and median residual lifetime may have a bathtub-shaped form.The parameters of the model are estimated by applying the maximum likelihood method and the expectation-maximization (EM)algorithm.The EM algorithm is an iterative method suitable for models with a latent variable, for example, when we have mixture or competing risk models.A simulation study is then conducted to examine the consistency and efficiency of the estimators and compare them.The simulation study shows that the EM approach provides a better estimation of the parameters.Finally, the proposed model is fitted to a reliability engineering data set along with some alternatives.The Akaike information criterion (AIC),Kolmogorov-Smirnov (K-S), Cramer-von Mises (CVM), and Anderson Darling (AD)statistics are used to compare the considered models.

Keywords: Inverse Lindley distribution; reliability measures; maximum likelihood estimation; EM algorithm

1 Introduction

Lindley [1] and inverse Lindley models have attracted much attention in the last decade.There is a long list of research on the Lindley model and its generalizations.Ghitany et al.[2] studied some features of the Lindley model.Sankaran [3] applied the Lindley model to define a compound Poisson-Lindley distribution.Ghitany et al.[4] considered the distribution introduced by Sankaran [3] to study a compound Poisson-Lindley model truncated to zero.Zamani et al.[5] proposed and studied a ccompound negative binomial Lindley model.Ghitany et al.[6] considered a power Lindley model with two parameters.Al-Mutairi et al.[7] estimated the probability of stress strength for two independent Lindley distributions.Al-babtain et al.[8] generalized the Lindley model to a distribution with fiveparameters.Shanker et al.[9,10] introduced an extended version of the Lindley model.Shankeret al.[11] investigated somemathematical properties of the extended Lindley model defined by Shankeret al.[10].Moreover, Shanker et al.[12] introduced and studied a new quasi-Lindley model.Merovci et al.[13] proposed a beta Lindley model and studied its properties.Zakerzade et al.[14], Ibrahim et al.[15], and Shanker et al.[16] defined generalizations of the Lindley distribution with three parameters.Moreover, Broderick et al.[17] proposed a generalization of the Lindley model with four parameters.

Sharma et al.[18] introduced the inverse Lindley distribution and considered it as a stress-strength model.The probability density function (PDF) of the inverse Lindley distribution is

and follows an upside-down bathtub (unimodal) hazard rate function, so it is useful when the data has a unimodal hazard rate.Sharma et al.[19] have presented some data examples that follow models with such a hazard rate function.Alkarni [20] and Sharma et al.[19] proposed an extension of the inverse Lindley distribution with three parameters and a new generalized inverse Lindley distribution, respectively.Also, Barco et al.[21] have obtained a new distribution from the power Lindley distribution and the inverse Lindley distribution.Recently, Eltehiwy [22] studied the logarithmic transformation of the inverse Lindley distribution.

An important property that any lifetime model should satisfy is scale invariance.A distribution family with scale parameter θ and PDFfθ(x) is scale invariant if the change fromxtokx,k＞0, does not change the family.More precisely,fθ(kx) =Jfθ′(x) whereJis the Jacobian of the transformation.Unfortunately, the inverse Lindley and the generalized versions mentioned above are not scale invariant.In this paper, we therefore present and study a scale-invariant extension of the inverse Lindley distribution.

The rest of the paper is organized as follows.In Section 2, we introduce the new quasi-inverse Lindley model and study some of its properties.In Section 3, we apply maximum likelihood and EM to estimate the parameters of the model.In Section 4, we investigate the behaviour of the estimators through a simulation study.In Section 5, we fit the proposed model to a reliability engineering data set to show its applicability.Finally, we conclude the paper in Section 6.

2 Quasi Inverse Lindley Distribution

The scale invariant quasi inverse Lindley distribution,QIL(α,θ), is defined by the cumulative distribution function (CDF)

The PDF of QIL is

It can be checked by differentiation that the sign of thef′(x) is equal with the sign of -+(α- 3)x+θ.Thusf′(0)＞0 andfincreases at an early interval and then decreases, i.e., the PDF is unimodal for all values of the parameters.The PDF of the inverse gamma (IG) distribution with parameters (ν,θ),IG(ν,θ), is of the form

so the PDF of QIL is a mixture of the PDF of inverse gamma distributionsIG(1,θ) andIG(2,θ) with weightsand.The QIL is an scale invariant extension of the the inverse Lindley distribution investigated by Sharma et al.[18], see scale invariance property discusssed in the Introduction section.If α be replaced by θ this new model will reduce to inverse Lindley.Moreover, if a random variableXfollows the QIL, then its reciprocal follows the model proposed by Shanker et al.[11].

Proposition 1.Fork＜1, thekthmoment of theQIL(α,θ) is finite and equals

while fork≥1 it is infinite.

Proof:Thekthmoment of theQIL(α,θ) is

The first integral of (6) simplifies to

fork＜1 equal to αθk-2Γ(1 -k).But, fork= 1,I1reduces to

in which arbitrarya＞0.It shows that fork≥1,E(Xk) is infinite.On the other hand, the second integral of (6) is

Thus by (6) the proof is completed.□

As a result of Proposition 1, and by the fact that

it follows that the moment generating function is infinite.

The quantile functionq(p) is defined by the inverse of the CDF.The quantile function has not a closed form for QIL and should be computed through the following

The quantile functionq(p) can be used to describe distribution characteristics, especially skewness kurtosis and to estimate parameters.The skewness of theQILcan be measured by

whereu∈(0,0.5), (see MacGillivray [23]).The special caseu= 0.25 is referred to Bowley’s measure of skewness (Bowley [24]).Also, the kurtosis of QIL is measured by (see Moors [25])

The well-known Lorenz curve is a graphical representation for the inequality of distribution of wealth or income.It measures the proportion of overall wealth or income of the bottomppercent of the people.The line of perfect equality is represented by the straight line between (0, 0) and (1, 1).The Lorenz curve is also a curve connecting these points and lies below the perfect equality line, see Bishop et al.[26].For a model with the CDFF, the Lorenz curve is defined by

For QIL, due to the fact that μ is infinite, but μpis finite for everypsuch thatq(p)＜∞, the Lorenz curve is zero and gives no information about income inequality.Fortunately, Prendergast et al.[27] defined three quantile versions of the Lorenz curve that can be applied toQIL.The main idea is to replace μ by median of the distribution and μpby its alternative quantile, i.e.,q.So, their alternatives for Lorenz curve are as follows.

and

Dynamic Measures

The FR, RFR, MRL, MIT,p-QRL andp p-QIT play key role in the reliability and survival analysis.Let the reliability function of QIL beR(x) = 1 -F(x), then the FR λ(x) and RFR η(x)functions of QIL are respectively,

and

The FR function λ(x) tends to zero whenxtends to zero or infinity.By differentiating from λ(x)with respect tox, we find that the FR function shows a unimodal form.The RFR function η(x) tends to infinity at zero and tends to zero at infinity and is a decreasing function, refer to Lai et al.[28].

The MRL functionm(x) represents the expectation of the conditional remaining life of an object given that it has been survived up to timexand equalsm(x) =E(X-x|X≥x), see Lai et al.[28].

Proposition 2.The MRL functionm(x) of the QIL is infinite forx≥0.

Proof:The MRL can be expressed by

With straightforward algebra we have

The first integral (8) is not finite since

which shows the proposition.□

The MIT ν(x) describes the conditional expectation of the elapsed time of an event given that it has been happened sometime beforex, more formally ν(x) =E(x-X|X＜x), see Lai et al.[28].The following proposition shows that the MIT of QIL is finite for allx＞0.

Proposition 3.The MIT is finite for allx＞0 and

Proof:The MIT can be written as

and for the QIL, we have

The first integral (10) can be simplified to

The second integral of (10) reduces to

which completes the proof.□

Thep-QRL function, denoted byqp(x), represents the conditionalpth quantile of the remaining life of an object, given that it has been survived up to timex,

where the quantile functionqis defined in (7) and= 1 -p.

Thep-QIT(x), represents the conditionalpth quantile of the elapsed time of an event, given that it has been happened sometime beforex, more formally

Similar to the quantile functionqdefined by (7), thep-QRL andp-QIT have not closed form for QIL.Thus, we should be computed them numerically.Whenp= 0.5,thep-QRLandp-QITareknown as the median residual life and the median inactivity time respectively.

Fig.1 draws the density andFRfunctions of QIL for some parameters.The density ismore skewed to right for larger θ.The FR shows upside down bathtub shape (unimodal form).

Figure 1: The PDF and FR function of QIL(α,θ) for some values of parameters

3 Estimation of the Parameters

In this section, we discuss the maximum likelihood estimation (MLE) method and the EM algorithm for estimating the parameters of the proposed model (1).

3.1 The MLE

Letx1,x2,...,xnbe an independent and identically distributed (iid) realization fromQIL(α,θ),the log-likelihood function of the parameters is

Then, the likelihood equations are and

The MLE can compute by maximizing the log-likelihood function directly, or solving the likelihood equations.The first approach has applied in the next sections.

Letl= lnf(X), then the following matrix shows the Fisher information.

3.2 The EM Algorithm

As explained earlier, theQIL(α,θ) is a mixture of two inverse gamma distributionsIG(1,θ) andIG(2,θ) and α determines the weights of the mixture.Letxibe an instance of the QIL model.Imagine one unobserved latent random variableZiwhich determines thatxicomes fromIG(1,θ) orIG(2,θ).LetXi,i= 1,2,...,nbe an iid random sample,Xi|Zi=j～IG(j,θ),j= 1,2,P(Zi= 1) =andP(Zi= 2) =.ThenXifollowsQIL(α,θ).For simplicity, let β= (α,θ).The likelihood function is

whereI(zi=j) equals 1 whenzi=jand equals 0 otherwise.Also,PZi(j) =P(Zi=j) =and

is the PDF of theIG(j,θ).Then the log-likelihood function can be simplified to the following form.

The following expectation (E) and maximization (M) steps:

The E Step

Given the estimate of the parameters at iterationt,βt, the conditional distribution ofZican compute by Bayes theorem as

which are known as membership probabilities at iterationt, and applied to obtain the expectation functionQ(β|βt) as follows.

Thus the expectation function can be arranged as a sum of two expressions which one expression just depends on α and the other term just depends on θ.

where

and

The M Step

To find the estimation of the parameters att+1 iteration, the objective expressionQ(β|βt) should be maximized in terms of β.So, it results that

which by (18) can reduce to the two following separate maximization problems

and

whereQ1(α|βt) andQ2(θ|βt) are determined by (19) and (20) respectively and by differentiating fromQ1(α|βt) andQ2(θ|βt) with respect to α and θ respectively, it follows that

and

The iterative process can conclude for some predefined small ?＞0,

4 Simulation

By a simulation study, the efficiency of the MLE and EM estimator have been investigated and compared.The fact that QIL is a mixture of two inverse gamma distribution to provide random samples.More specifically, the following steps should be performed:

?Simulate one sample of multinomial distribution with parametersn,p=and 1 -p.Let the generated instance ben1andn2, corresponding to probabilitiespand 1 -prespectively.

?Generate one sample with sizen1from gamma distributionG(1,θ) and another sample fromG(2,θ) with sizen2.Combine two generated samples to provide one sample ofQIL(α,θ) with sizen.

In each run, some suitable values for the parameters are selected.Then,r= 5000 repetitions of random samples of QIL are simulated.The size of samples aren= 50, 150 or 250.For each repetition,the parameters were estimated using the maximum likelihood method or the EM approach.

The function“nleqslv”of the library“l(fā)neqslv”in R was used to calculate the MLE.This function solves the likelihood Eqs.(12) and (13) to find the MLE.The initial values were randomly generated by a uniform distribution in both the MLE and EM approaches.In the EM algorithm, checking the termination condition in each EM iteration causes the runs very slow.Therefore, the EM algorithm was tested many times to find out how many iterations are sufficient.We found that 5 iterations is sufficient.Tab.1 shows the bias (B) and mean square error (MSE) of the computed estimators.In every cell of this table the first and second lines show B and MSE for α and θ respectively.Letrepresent the MLE of α, then its corresponding B and MSE are defined by the following relations:

and

Table 1: Simulation results for MLE and EMestimator of the parameters of QIL distribution.In every cell the first and second lines are related to α and θ respectively

Other measurements are defined similarly.Some of the simulation results are listed in the following:

?As sample size increases, the MSE decrease, in both MLE and EM approaches, i.e., the MLE and EM estimators are consistent.

?The MSE of MLE shows unexpectedly large values especially for α.Fortunately, the EM approach shows far smaller MSEs for both α and θ.Thus, the EM outperforms MLE in terms of the MSE.

5 Applications

In this section, we fit the proposed model to a data set to show its applicability.Tab.2 represents one data set consists of 46 observations reported on active repair times in terms of hours for an airborne communication transceiver discussed by Alven [29].TheQILis fitted to this data set and the parameters have been estimated by the maximum likelihood method and EM algorithm.The computed MLE and EM estimation are (?,) = (14.997, 1.2066) and (?,?) = (19.6777, 1.1911)respectively.In a comparative analysis, the IG, inverse Weibull (IW) and power inverse Lindley (PIL)distributions are fitted to this dataset.The PDF of the IG is defined by (4) and the PDF of the IW and PIL are respectively

and

Table 2: Active repair times (hours) for an airborne communication transceiver

The results of fitting models are abstracted in Tab.3.The estimates of the parameters, AIC, K-S,CVM and A-D statistics are computed.The QIL model outperforms the other candidates in terms of the AIC, K-S, CVM and A-D statistics.Also, the great p-values (near one) indicates good and competitive fits for allmodels.The empirical and fitted CDFs ofQIL along with the alternative models aredrawnin Fig.2.Also,the rightside of Fig.3shows upside down bathtub shape for the FRfunction of all of the estimated models.The total time on test plot of the data set presented by Fig.4 shows a plot which is above the identity line at the beginning and then falls below the identity line.Thus, Fig.4 confirms an upside down bathtub shape for the FR function, too.

Table 3: The results of fitting the QIL model and some alternative models to dataset of Tab.2

Figure 2: The 0.5-QRL function and the MIT function of QIL(α,θ) for some values of parameters

Figure 3: Left: The empirical distribution and fitted distributions to the data set of Tab.2.Right: The FR function of the estimated distributions

Figure 4: The total time on test plot for data set of Tab.2.This plot reveals an upside-down bathtub shape form for the FR function

6 Conclusions

A new scale-invariant quasi-Lindley distribution was introduced and studied.It is useful for the analysis of lifetime data with an upside-down bathtub shape FR function.The elementary properties of the proposed model were explored.Also, some dynamic reliability measures were investigated.The maximum likelihood and EM methods were discussed.A simulation study was conducted to investigate and compare the behavior of the two approaches.It found that the EMmethod can estimate the parameters more efficiently.

Acknowledgement:The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.

Funding Statement:This work is supported by Researchers Supporting Project Number (RSP-2021/392), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.