Inference for a Kavya–Manoharan Inverse Length Biased Exponential Distribution under Progressive-Stress Model Based on Progressive Type-II Censoring

In this article, a new one parameter survival model is proposed using the Kavya–Manoharan (KM) transformation family and the inverse length biased exponential (ILBE) distribution. Statistical properties are obtained: quantiles, moments, incomplete moments and moment generating function. Different types of entropies such as Rényi entropy, Tsallis entropy, Havrda and Charvat entropy and Arimoto entropy are computed. Different measures of extropy such as extropy, cumulative residual extropy and the negative cumulative residual extropy are computed. When the lifetime of the item under use is assumed to follow the Kavya–Manoharan inverse length biased exponential (KMILBE) distribution, the progressive-stress accelerated life tests are considered. Some estimating approaches, such as the maximum likelihood, maximum product of spacing, least squares, and weighted least square estimations, are taken into account while using progressive type-II censoring. Furthermore, interval estimation is accomplished by determining the parameters’ approximate confidence intervals. The performance of the estimation approaches is investigated using Monte Carlo simulation. The relevance and flexibility of the model are demonstrated using two real datasets. The distribution is very flexible, and it outperforms many known distributions such as the inverse length biased, the inverse Lindley model, the Lindley, the inverse exponential, the sine inverse exponential and the sine inverse Rayleigh model.


Introduction
Accelerated life tests (ALTs) are applied to gain rapid information on the lifetime distribution of materials or products. In ALTs, the units' test is performed at higher-than-normal levels of stress (voltage, vibration, pressure, temperature, etc.) to induce early failures. Data obtained at the accelerated conditions are analyzed in terms of an appropriate statistical model and then extrapolated to the specified normal stress to estimate the lifetime distribution in normal use conditions. There are different methods to apply the stress. Commonly used methods are constant-stress, step-stress and progressive-stress; see, for example, Nelson [1], AL-Hussaini and Abdel-Hamid [2,3], Abdel-Hamid and AL-Hussaini [4] and Abdel-Hamid In recent years, many various statisticians have been drawn to create families of distributions such as Marshall-Olkin-G [18], Kumaraswamy-G (Kum-G) in [19], odd Lomax-G [20], sine-G in [21], odd Dagum-G [22], Type II half logistic-G in [23], transmuted geometric-G [24], odd Perks-G in [25], odd Lindley-G in [26], truncated Cauchy power Weibull-G [27], generalized transmuted-G [28], truncated Cauchy power-G in [29], Burr X-G (BX-G) class [30], transmuted odd Fréchet-G in [31], Type II exponentiated half logistic-G in [32], Topp Leone-G in [33], exponentiated M-G by [34], odd Nadarajah-Haghighi-G in [35], exponentiated truncated inverse Weibull-G in [36] and T-X generator proposed in [37], among others.
Additional parameters give greater flexibility, but they also increase the complexity of estimation. To counter this, Ref. [38] proposed the Dinesh-Umesh-Sanjay (DUS) transformation to obtain new parsimonious classes of distributions. This is as follows. If G(x) is the baseline cumulative distribution function (CDF), the DUS transformation generates a new CDF F(x) expressed as: The merit of using this transformation is that the resulting distribution is parameterparsimonious because no extra parameters are added. In this way, Ref. [39] proposed a new class of distributions that includes many flexible hazard rates. They explored using the DUS transformation using the exponentiated cdf, introducing the generalized DUS The fundamental goal of the article under consideration is to introduce the KMILBE model, as a new one-parameter lifetime model based on the KM transformation family, ILBE distribution, and also to investigate its statistical characteristics. The following points provide sufficient incentive to study the KMILBE distribution. We specify it as follows: (i) It is remarkable to observe the flexibility of the proposed model with the diverse graphical shapes of pdf and hrf. Thus, the the pdf of the KMILBE distribution can be unimodal and right-skewed, with very heavy tails, but the hrf of the KMILBE distribution can be increasing, J-shaped form; (ii) The KMILBE distribution have a closed form of the quantile function; (iii) The KMILBE is a good alternative to several lifetime distributions for modeling skewed data in applications; (iv) Different types of entropy and extropy are computed; (v) Based on progressive type-II censoring, we have discussed some estimation methods on a progressive-stress model when the lifetime of a product follows the KMILBE distribution. The methods that have been discussed are maximum likelihood (ML), least squares (LS), weighted least squares (WLS) and maximum product of spacing (MPS) estimation.
This paper is organized as follows: In Section 2, a new lifetime model using inverse length biased distribution as the baseline distribution in the KM transformation family is presented. In Section 3, we demonstrate the statistical features of the KMILBE model. Different measures of entropy are discussed in Section 4. In addition, some measures of extropy are proposed in Section 5. Model description and progressive type-II censoring by using ML, LS, WLS, and MPS are studied in Section 6. The simulation study and the numerical results are discussed in Section 7. Application to two real datasets is discussed in Section 8. Finally, concluding remarks are proposed in Section 9.

Construction of the Kavya-Manoharan Inverse Length Biased Exponential Distribution
In this section, we construct a new flexible distribution called the Kavya-Manoharan transformation inverse length biased exponential (KMILBE) distribution by inserting Equation (6) into Equation (1), to obtain and the corresponding PDF is The survival function (SF), HRF, reversed HRF and cumulative HRF for the KMILBE distribution are . Figures 2 and 3 show graphical representations of the PDF and the HRF of the KMILBE distribution with various values for the parameter θ. Forms of the PDF include right skewness and unimodal as shown in Figure 2. In addition, the forms of the HRF include increasing and J-shaped form, as shown in Figure 3. The KMILBE distribution is a very flexible model that provides different distributions when its parameters are changed.

Statistical Features of the New Suggested Model
This section provides the structural properties of the KMILBE, defined in Equation (9), including explicit expressions for quantile function (QF), linear representation of the density, rth ordinary and sth incomplete moments, and moment generating function.

Quantile Function
The QF, say Q(u) = F −1 (u), u ∈ (0, 1), is obtained by inverting Equation (8) as follows: By multiplying the both sides by e −1 , then we have the Lambert equation Hence, we have the negative Lambert W function of the real argument where u ∈ (0, 1) and W −1 (.) is the negative Lambert W function. By replacing u = 0.5 in Equation (10), the median (Q2) of the KMILBE is readily available.

Useful Expansion
Here, we showed the useful expansion of the pdf, cdf and survival for the KMILBE distribution which can be used to drive several important properties of the KMILBE. According to the next exponential expansion By inserting the previous Equation (11) in Equation (9), we obtain by applying the binomial expansion (1 + z) b = ∞ ∑ j=0 b j z j , in the last equation, we can rewrite it as follows: In addition, we can obtain the expansion of f δ KMILBE (x; θ) by using the last two expansions as follows: where, A gain using the previous expansions, then we can write the expansion of R 2 KMILBE (x; θ) as follows: where

rth Moment
The rth ordinary or raw moments is an important measure to find measures of dispersion of the distribution. The following relationship is used to obtain the central or actual moments; the first moment about mean is always equal to zero, and the second moment about mean is equal to variance as , respectively. Suppose that X ∼ KMILBE (θ) for x ∈ (0, ∞) and θ > 0; then, its rth ordinary moment is given by Let y = (i+1)θ x ; then, For r = 1, the mean of KMILBE is yielded as

sth Incomplete Moment
The sth incomplete moment is an important measure and has wide applications in order to compute mean deviation from mean and median, mean waiting time, conditional moments and income inequality measures.

Moment Generating Function
By definition, the moment generating function, M(t) = E e tX = e tx f (x)dx, can be yielded as Assume that X ∼ KMILBE (θ) for x ∈ (0, ∞) and θ > 0; then, its moments generating function can be obtained by using (12) and replacing e tx = ∑ ∞ r=0 t r r! x r is given by where j + 2 < r.

Entropy Measures
Entropy is a measure of a system's variation, instability or unpredictability.

The Rényi Entropy
The Rényi entropy [44] is important in ecology and statistics as an index of diversity. For δ > 0 and δ = 1, it is defined by the following expression: By using Equation (13), we obtain

The Tsallis Entropy
The Tsallis entropy measure (see [45]) is defined by: By using Equation (13), we obtain

The Havrda and Charvat Entropy
The Havrda and Charvat entropy measure (see [46]) is defined by: By using Equation (13), we obtain

The Arimoto Entropy
The Arimoto entropy measure (see [47]) is defined by: By using Equation (13), we obtain

Extropy
Recently, an alternative measure of uncertainty, named by extropy was proposed by [48]. For an absolutely continuous non-negative random variable X with PDF f and CDF F, the extropy is defined as By using Equation (13), and putting δ = 2, we obtain

The Cumulative Residual Extropy
The cumulative residual extropy (CREX) was proposed by [49] analogous with (23) as a measure of uncertainty of random variables. The CREX is defined as It is always non-positive. By using Equation (14), we obtain

The Negative Cumulative Residual Extropy
Refs. [49,50] studied and investigated the negative CREX (NCREX) can be presented as By using Equation (14), we obtain

Cumulative Exposure Model
The cumulative exposure model (CEM) enables us to relate the distribution under progressive stress to the distribution under constant stress.
If the stress υ is a function of time y, υ = υ(y), and influences the scale parameter θ of the considered failure distribution, then θ becomes a function of y, θ(y) = θ(s(y)). Hence, the CEM takes the form; see Nelson [1], .
The CDF under progressive stress becomes where F(.) is the assumed CDF with scale parameter equal to 1.

1.
First assumption: The relationship between the stress s and the scale parameter β satisfies the inverse power law i.e., where υ is the applied stress and (η, µ) are two positive parameters to be estimated.

2.
Second assumption: The stress υ(y) is a linearly increasing function in time y, i.e.,

3.
Third assumption: During the test process, the M units to be tested are divided into (>1) groups; each group includes m k units and is run under progressive stress. Thus,
Fifth assumption: The failure mechanisms of the failures are the same under any stress level.
From the first and second assumptions, the CEM (26) takes the form From (8), CDF (27) under progressive stress takes the form The corresponding PDF is given by

Progressive Type-II Censoring
The progressive type-II censoring under progressive stress model can be applied as follows: Under Assumption 3, for k = 1, . . . , , suppose that r k (<m k ) and R k1 , R k2 , . . . , R kr k are fixed before the experiment. R k1 surviving units are randomly removed from the test, when the first failure time in group k occurs and R k2 surviving units are randomly removed from the test when the second failure time in group k occurs. The test continues in the same manner until the r k -th failure at which all the remaining surviving units The data from progressively type-II censored samples are as follows: (y k1:r k :m k ;R k1 ), . . . , (y kr k :r k :m k ;R kr k ) where y k1:r k :m k < . . . < y kr k :r k :m k denote the r k ordered observed failure times, and R k1 , . . . , R kr k denote the number of units removed from the experiment at failure times y k1:r k :m k , . . . , y kr k :r k :m k .
Based on progressively type-II censored samples, under progressive stress ALT, the likelihood function is given by where y = (y 1 , y 2 , . . . , y ), y k = (y k1 , . . . , y kr k ), and y kj ≡ y kj:r k :m k , k = 1, . . . , , j = 1, . . . , r k , Using Equations (29) and (30), the log-likelihood function takes the form where D = ∑ k=1 r k and Then, the likelihood equations take the forms where The MLEsμ andη of µ and η could be obtained by solving the likelihood equations, ∂ log[L(µ, η; y)] ∂µ = 0 and ∂ log[L(µ, η; y)] ∂η = 0, with respect to µ and η and solving these equations simultaneously to obtain the MLEs. These equations can be numerically solved using iterative techniques using statistical software, since it is not possible for analytical solutions to obtain the roots. Based on the common asymptotic normality theory of MLEs, we can consider that can be approximated by a standard normal distribution, i.e., where Var(μ) and Var(η) are the variance ofμ andη, which can be obtained from the inverse of the local Fisher information matrix (FIM), where where the caretˆdenotes that the derivative is evaluated at (μ,η). The second partial derivatives of the natural logarithm of the likelihood function with respect to µ and η can be obtained without difficulty.
Sometimes, the lower bound of NACI may have a negative value for the positive parameter. Thus, Meeker and Escobar [12] suggested using a log transformation confidence interval (LTCI) for this parameter. The normal approximation of log-transformed MLE, lnζ i − ln ζ i Var(lnζ i ) , i = 1, 2, can be approximated to a standard normal distribution i.e., Therefore, a 100(1 − ε)% LTCI for ζ i can be defined as

Least Squares and Weighted Least Squares Estimations
The LS and WLS methods were introduced by Swain et al. [51] to estimate the Beta distribution parameters. Based on progressive type-II censoring, Abdel-Hamid and Hashem [52], and Hashem and Alyami [17], used these two methods to estimate the parameters included in the doubly Poisson-exponential and exponential-doubly Poisson distributions. They can be performed as follows: Let (Y k1 , . . . , Y kr k ), k = 1, . . . , , be the ordered progressively type-II censored sample of size r k from the KMILBE distribution, under progressive stress ALT. The LS estimates (LSEs) of the unknown parameters can be obtained by minimizing the following quantity with respect to the unknown parameters: , where E[ G k (y kj ) is the expectation of the empirical CDF, see Aggarwala and Balakrishnan [15], which is given by Therefore, the LSEsμ andη of µ and η can be obtained by minimizing the following quantity with respect to µ and η These estimates can also be obtained by solving the nonlinear equations simultaneously to obtain the LSEs. These equations can be numerically solved using iterative techniques using statistical software since it is not possible for analytical solutions to obtain the roots: where and ϕ kj , A kj and B kj are given by (33), (36) and (37), respectively. The WLS estimates (WLSEs) of the unknown parameters can be obtained by minimizing the following quantity with respect to the unknown parameters: where V[ G k (y kj ) is the variance of the empirical CDF, see Aggarwala and Balakrishnan [15], which is given by , s = 1, . . . , r k , k = 1, . . . , , The WLSEsμ andη of µ and η can be obtained by minimizing the following quantity with respect to µ and η These estimates can also be obtained by solving the nonlinear equations simultaneously to obtain the WLSEs. These equations can be numerically solved using iterative techniques using statistical software since it is not possible for analytical solutions to obtain the roots: where Υ kj and Ω kj are given by (42) and (43), respectively.

Maximum Product of Spacing Estimation
Cheng and Amin [53] introduced an alternative method to the ML method for estimating the unknown parameters in univariate continuous distributions. Based on progressive type-II censoring, Ng et al. [54] used this method to estimate the parameters included in the Weibull distribution. The Maximum product of spacing estimates (MPSEs) of the unknown parameters can be obtained by maximizing the following product of spacing with respect to the unknown parameters: where G k (y k0 ) = 0 and G k (y kr k +1 ) = 1. Using (29), the MPSEsμ andη of µ and η can be obtained by maximizing the following product of spacing with respect to the µ and η These estimates can also be obtained by solving the nonlinear equations simultaneously to obtain the MPSEs. These equations can be numerically solved using iterative techniques using statistical software since it is not possible for analytical solutions to obtain the roots: where Υ kj and Ω kj are given by (42) and (43), respectively.

Simulation Study
As it is theoretically difficult to assess the efficiency of estimation methods, a Monte Carlo simulation is used to overcome this difficulty. In the current section, through Monte Carlo simulation, we conduct a numerical study to assess the efficiency and performance of the estimation methods according to the following steps: 1.

2.
For k = 1, . . . , , generate a progressively type-II censored sample of size r k from the KMILBE distribution with CDF (29), according to the algorithm given in Balakrishnan and Sandhu [14].
Ifβ is an estimate of β, then the average of estimates, mean squared error (MSE) and relative absolute bias (RAB) ofβ overh samples are given, respectively, by

7.
Calculate the average of estimates of the parameters µ and η and their MSEs and RABs as shown in Step 5. Calculate also the mean of the MSEs (MMSE) and mean of the RABs (MRAB) according to the following two relations:

8.
Calculate the average interval lengths (AILs) and coverage probability (COVP) of the parameters µ and η.
• CS3: For k = 1, . . . , The computational results are presented in Tables 1-3 taking into account the population parameter values: µ = 0.2 and η = 1.5. For the sake of comparison among the MLEs, MPSEs, LSEs, WLSEs, NACIs and LTCIs of the parameters µ and η, the total number of observations M is divided into two groups, = 2, and another time into three groups, = 3.

Numerical Results
From Tables 1-3, we observe the following: 1.
The MLEs are better than the LSEs and WLSEs through the AMSEs and ARABs; 2.
The MLEs are better than the MSPEs through the AMSEs and ARABs for the parameter µ; 3.
The WLSEs are better than the LSEs through the AMSEs and ARABs; 4.
The MPSEs are better than the LSEs and WLSEs through the AMSEs; 5.
The NACLs are better than the LTCIs via the AILs and COVP; 6. For = 2, 3, and fixed values of the total number of items to be tested, M, and hence fixed sample sizes, m k , by increasing the failure times, r k , the MSEs, AMSEs, RABs, ARABs and AILs of the considered parameters decrease. 7. For = 2, 3, and fixed values of the failure times, r k (=50%, 75% and 100% of the sample size m k ), by increasing the total number of items to be tested, M, the MSEs, AMSEs, RABs, ARABs and AILs of the considered parameters decrease. 8.
For fixing the total number of items to be tested, by increasing , the MSEs, AMSEs, RABs and ARABs decrease. 9.
By increasing the sample and failure time sizes (r k , m k ), the COVP are close to 95%. 10. For fixed values of the sample and failure time sizes (r k , m k ), the third CS gives more accurate results through the MSEs, AMSEs, RABs, ARABs and AILs than the other two CSs.
The above results are satisfied except for some rare cases; this may be due to fluctuation in the data.

Real Data Analysis
In this section, we illustrate the importance of the newly KMILBE distribution by utilizing two real-life datasets. We shall compare the fits of the KMILBE distribution with the following competing continuous distributions, which are reported in Table 4.
Inverse exponential IE The first data set we consider in this paper is taken from [55] 38. These data are carbon retained by leaves measured in kilogram/hectare for thirty-eight different plots of mountainous regions of Navarra (Spain), depending on the forest classification: areas with ninety percent or more beech trees (Fagus Sylvatica) are labeled monospecific, while areas with many species of trees are labeled multi specific.
The second data set: we consider data of times to infection of kidney dialysis patients in months, as described by [56]. The "times of infection" data set is: 2.5, 2.5, 3.5, 3.5, 3.5, 4.5, 5.5, 6.5, 6.5, 7.5, 7.5, 7.5,7.5, 8.5, 9.5, 10.5, 11.5, 12.5, 12.5, 13.5, 14.5, 14.5, 21.5, 21.5, 22.5, 22.5, 25.5, 27.5. Now, we make a normalization operation by divided these data by 30 Tables 5 and 6 for the both datasets, respectively. For visual comparisons, the fitted CDF of the competitive distributions are depicted in Figures 4 and 5, the fitted PDF of the competitive distributions are depicted in Figures 6 and 7, the fitted sf of the competitive distributions are depicted in Figures 8 and 9 respectively. Furthermore, P-P (probability-probability) plots of fitted distributions are displayed in Figures 10 and 11 for the analyzed datasets, respectively. The findings in Tables 5 and 6 illustrate that the KMILBE model provides a superior fit over other competing continuous models, since it has the lowest values for all measures and lowest value of the Kolmogorov-Smirnov distance (KS).      Figure 11. The P-P plots of the competing continuous models for data set 2.

Conclusions
In this study, we explore a new one parameter model, which is called a Kavya-Manoharan inverse length biased exponential model. Its statistical and mathematical features (quantile, moments, inverse moments, incomplete moments and moment generating function) are derived. Different types of entropies such as Rényi entropy, Tsallis entropy, Havrda and Charvat entropy and Arimoto entropy are computed. Different measures of extropy such as extropy, cumulative residual extropy and the negative cumulative residual extropy are computed. Based on progressive type-II censoring, we have discussed some estimation methods on the progressive-stress model when the lifetime of a product follows the Kavya-Manoharan inverse length biased exponential distribution. The methods that have been discussed are ML, MPS, LS and WLS estimations. The approximate CIs for the unknown parameters have been established. The performance of these methods has been investigated through a simulation study, based on three different progressive CSs. The relevance and flexibility of the KMILBE model are demonstrated using two real datasets.