Chi-squared type test for distributional censored and uncensored validity with numerical assessments and real data applications

In this work, we introduce a new chi-squared type test the odd Lindley exponentiated gamma distribution. The new test is an extension of the Nikulin–Rao–Robson test. The new test is tailored to fit the right censored data. The performance of the new test, as well as the baseline Nikulin–Rao–Robson test, are evaluated via numerical simulation. The new test, as well as the baseline Nikulin–Rao–Robson test, are also evaluated using the data. Furthermore, we present some characterization results.


Introduction and motivation
A chi-squared test (χ 2 ), especially Pearson's chi-squared test and its modifications, are statistical hypothesis testing that may be used when the test statistic is chi-squared distributed under the null hypothesis.If there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table, it may be determined using Pearson's chi-squared test.The Pearson's chi-squared test is a statistical procedure used to assess the likelihood that any observed difference between two sets of categorical data resulted from chance.
It is the chi-squared test that is most regularly utilized (e.g., Yates, likelihood ratio, portmanteau test in time series, etc.).In the last five decades, many researchers have taken up the chi-squared test and made many extensions of it and new generalizations of it.Some of these extensions touch on statistical hypothesis tests in the case of censored data (see Bagdonavičius & Nikulin, 2011a, b;Nikulin, 1973a, b, c;Rao & Robson, 1974).
The so called Nikulin-Rao-Robson (N-RR) statistic (Y 2 α (r − 1)) is a well-known variant of the traditional chi-squared tests in the situation of full data.It is based on differences between two estimators of the probability to fall into grouping intervals.One estimator is based on the empirical distribution function, while the other uses MLEs of the tested model's unknown parameters and initial non-grouped data (for more information, see Nikulin, 1973a, b, c;Rao and Robson, 1974, as well as Goual et al., 2019;Goual and Yousof, 2020a).
However, techniques for evaluating the censored validity of parametric distributions are growing although they are not yet available due to censoring.Based on the wellknown Kaplan-Meier estimators, Habib and Thomas (1986) and Hollander and Pena (1992) suggested a modified chi-squared test for randomly censored data.For models of accelerated failure, Galanova et al. (2012) took into account various nonparametric adjustments to the Anderson-Darling, Kolmogorov-Smirnov, and Cramer-Von-Mises statistics.For the correct censored data, Bagdonavičius and Nikulin (2011a) introduced a new chi-squared goodness-of-fit test statistic (or see Bagdonavičius & Nikulin, 2011b).The chi-squared goodness-of-fit test statistic of Bagdonavičius-Nikulin can be applied for distributional validation under the right censoring case.The right censor scenario is used in this study to validate a modified chi-squared goodness-of-fit test statistic based on the N-RR test (M 2 α (r )) for the odd Lindley exponentiated gamma (OLEG) distribution.To evaluate the proper censored estimate approach, a simulation study using the Barzilai-Borwein (BB) algorithm is first conducted.The BB algorithm is an optimization method that is widely used in machine learning and numerical optimization.It is an iterative algorithm that is designed to efficiently minimize a given objective function, and is particularly useful in scenarios where the objective function is difficult to compute or has a large number of variables.One of the key advantages of the BB algorithm is its ability to converge quickly, even in situations where other optimization methods may struggle.This makes it well-suited to applications where speed and efficiency are important, such as in real-time data analysis or large-scale numerical simulations.Another important feature of the Barzilai-Borwein (BB) algorithm is its robustness (for more details see Ravi & Gilbert, 2009).Unlike some other optimization methods, it is less susceptible to problems such as getting stuck in local minima, and can often find the global minimum of a given objective function.This makes it a valuable tool for a wide range of optimization problems in fields such as engineering, finance, and data science.Overall, the BB algorithm is an important optimization method due to its speed, efficiency, and robustness, and it is likely to continue to play a key role in numerical optimization in the years to come.
In this regard, we will describe a few recent research that used the N-RR goodnessof-fit test or provided new modified related expansions.The N-RR goodness-of-fit test has specific criteria, it can be considered as a tight method and calls for censored data, thus it is important to note that the browser for statistical literature on this topic will not discover many new N-RR goodness-of-fit extensions and will find little study that has performed this test.As it is generally known, obtaining fresh censored data to apply to and highlight the significance of the new test is difficult.We will go through a couple recent studies that looked at using this test on real data that where censored appropriately in the next few paragraphs.In the next few lines, we will shed light on some studies and research closely related to the modified tests in this paper.For the purpose of distributional validation, Mansour et al. (2020b) used the Bagdonavicius-Nikulin goodness-of-fit test on a new log-logistic model.The updated test is applied to the "right censored" real dataset of survival times.All of the components of the new test are logically derived and presented.Three uncensored real data applications are offered to test the applicability and importance of the new model inside the unfiltered framework under the Y 2 α (r − 1) test.Moreover, three censored real datasets are evaluated for filtered validation under the M 2 α (r ) statistic.The modified chi-squared goodness-of-fit test, also known as the modified Bagdonavicius-Nikulin goodness-of-fit test, is investigated in this paper and employed for distributional validation in the proper censored situation.Introduced and utilized with the proper censored data sets is the revised goodness-of-fit test.The validity of the proposed test is assessed using the censored BB algorithm through a detailed simulation examination.The modified Bagdonavicius-Nikulin test is applied to four real and right censored data sets.A novel distribution is compared to a large number of existing competing distributions using the updated Bagdonavicius-Nikulin goodness-of-fit test statistic.In a new updated form, the Bagdonavicius and Nikulin goodness-of-fit test statistic validity for the right censor case under the double Burr type X distribution is demonstrated.In the case of censored data, the maximum likelihood estimate approach is utilized.The optimum censored estimate method is determined by simulations utilizing the BB algorithm.Another simulation study is offered to evaluate the null hypothesis using a modified version of the Bagdonavicius and Nikulin statistical goodness-of-fit test.Four right censored data sets are investigated using the new modified test statistic in order to assess the distributional validity (see Aidi et al., 2021).In this work, simulations using the BB algorithm are run to determine the best censored estimating technique.For the purpose of examining the distributional validity, four right censored data sets are examined using the new modified test statistic.For more details, information, applications, and new extensions of this test in the case of censored data from the right, see: Yousof et al. (2021a) (for a new parametric lifetime model along with modified chi-squared type test for right censored distributional validation, characterizations and many estimation methods), Ibrahim et al. (2021) (for a new exponential generalized log-logistic model with the Bagdonavičius and Nikulin testing for distribution validation and some non-Bayesian estimation methods), see also Ibrahim et al. (2019Ibrahim et al. ( , 2020) ) and Yadav et al. (2022) for some related details about the Nikulin-Rao-Robson goodness-of-fit test.
To show the adaptability and efficacy of the tests described in this study, we conduct a substantial examination utilizing numerical simulations.Then, to execute these tests, we use actual data from reliability and survival analyses.The first simulation results are for evaluating the ML technique under the BB algorithm, and the second simulation results are for evaluating the Y 2 α (r − 1) statistic.Next, we give two simulations using complete data.Second, we demonstrate two censored simulations, the first of which evaluates the censored ML technique under the BB algorithm and the second of which evaluates the M 2 α (r ) statistic.

The new model
The gamma distribution is an important probability distribution that arises in a wide variety of contexts.In this section we present are some reasons why we the gamma distribution is chosen and extended.The gamma distribution is often used to model the time between events in a Poisson process, such as the time between arrivals of customers at a service center or the time between failures of a machine.The gamma distribution is also used to model waiting times in queueing theory and other applications where time is a critical factor.Moreover, the gamma distribution can be used to model income and other variables that are positively skewed.This is because the gamma distribution has a long right tail, which allows for the modeling of extreme values.In particular, the two-parameter gamma distribution can be used to model income data, while the three-parameter gamma distribution can be used to model income data that is censored at zero.The gamma distribution is often used as a prior distribution in Bayesian inference.This is because the gamma distribution is a conjugate prior for the exponential distribution, which is often used to model waiting times and other phenomena.The gamma distribution is also a conjugate prior for the Poisson distribution, which is used to model counts of rare events.Additionally, the gamma distribution is used to model the failure times of systems and components.By fitting the distribution to failure data, engineers can estimate the probability of failure at a given time and make decisions about maintenance, replacement, and warranty claims.The gamma distribution is particularly useful for modeling systems with a bathtub-shaped failure rate, which is a common pattern in reliability analysis.The gamma distribution is a versatile probability distribution that is closely related to several other important distributions.Its relationships with other distributions make it a valuable tool for modeling and analysis in a variety of fields.The OLEG distribution is derived based on the odd Lindley G (OL-G) family of distributions (Gomes-Silva et al., 2017) and the exponentiated gamma (EG) model (Gupta et al., 1998).The cumulative distribution function (CDF) of the OLEG distribution can be expressed as (for z ≥ 0) where τ λ (z) = (1 + λz) exp (−λz) and = (a, θ, λ) .The corresponding probability density function (PDF) can be derived as (for z > 0) (2) In this work, our primary task is to employ the OLEG distribution in the statistical modeling processes and to judge the quality of fitting through the N-RR goodnessof-fit test and the modified N-RR goodness-of-fit test.Following Ravi and Gilbert (2009), and using the BB algorithm, we generated N = 10,000 with different sample sizes from the OLEG model using some carefully selected initial values.The mean square error (MSEs) are used for assessing the performance of the censored maximum likelihood.Then, the modified N-RR test is applied using three right censored real data sets for distributional validation.The first simulation results are for evaluating the ML technique under the BB algorithm, and the second simulation results are for evaluating the Y 2 α (r − 1) statistic.Next, we give two simulations using the complete data.Second, we demonstrate two censored simulations, the first of which evaluates the censored ML technique under the BB algorithm and the second of which evaluates the M 2 α (r ) statistic.In this study, we provide six applications to actual data sets, three for evaluating the Y 2 α (r − 1) statistic and three more for evaluating the M 2 α (r ) statistic.Then, we provide six applications to actual data sets, three for evaluating the Y 2 α (r −1) statistic and three more for evaluating the M 2 α (r ) statistic.Regarding the Y 2 α (r − 1) test statistic: some real data applications for assessing the Y 2 α (r − 1) statistic are given according the following scenarios:

Characterizations
Theoretically, we present some characterization results based on some characterization theories, but in reality we ignored many statistical properties to focus on numerical and applied results.Algebraic derivations and mathematical results are often presented without providing any practical applications.In this work, we will focus our attention on the choice of statistical hypotheses and related simulation studies and applications to environmental data.This section considers the characterizations of the OLEG distribution via: (i) two truncated moments; (ii) the hazard function and (iii) the conditional expectation of a function of the random variable.For the characterization (i) the CDF need not to have a closed form.The characterizations (i)-(iii) will be presented in the following subsections.

Characterizations based on two truncated moments
In particular, it is sometimes possible to characterize a distribution based on two truncated moments.Two truncated moments are moments that are calculated using only the data within two specific ranges.For example, the first truncated moment might be the mean of the data within the range [a, b], and the second truncated moment might be the variance of the data within the range [c, d].The specific form of the characterization will depend on the distribution being considered.The use of two truncated moments can be a useful method for characterizing distributions when full data are not available, or when it is desirable to focus on a specific range of the data.However, the specific form of the characterization will depend on the distribution being considered, and care must be taken to ensure that the assumptions underlying the method are appropriate for the data being analyzed.This subsection deals with the characterizations of OLEG distribution based on a relationship between two truncated moments.
The first characterization applies a theorem of Glänzel (1987), Theorem 2.1.1 given below.Clearly, the result holds as well when the ϒ [d,e] is not a closed interval.This characterization is stable in the sense of weak convergence, please see Glänzel (1990)."Theorem 2.1.1Let ( , F, P) be a given probability space and let ϒ [d,e] = [d, e] be an interval for some d < e (d = −∞, e = ∞ might as well be allowed) .Let Z : → ϒ [d,e] be a continuous random variable with the distribution function F and let g(z) and h(z) be two real functions defined on ϒ [d,e] such that is defined with some real function τ (z).Assume that g and F is twice continuously differentiable and strictly monotone function on the set ϒ [d,e] .Finally, assume that the equation τ (z) h (z) = g (z) has no real solution in the interior of ϒ [d,e] .Then F is uniquely determined by the functions g(z), h(z) and τ (z), particularly where the function s is a solution of the differential equation s

and assume the h
and and finally Conversely, if τ (z) has the above form, then = λ, if and only if there exist functions g(z) and τ (z) defined in Theorem 2.1.1 satisfying the following first order differential equation Corollary 2.1.2The general solution of the above differential equation is where D is a constant.A set of functions satisfying this differential equation is presented in Proposition 2.1.1 with D = 0. Clearly, there are other triplets (h(z), g(z), τ (z)) satisfying the conditions of Theorem 2.1.1.

Characterization based on hazard function
The use of the hazard function is a powerful tool for characterizing survival distributions and estimating parameters of interest.However, it is important to ensure that the assumptions underlying the analysis (such as the assumption of independent censoring) are appropriate for the data being analyzed.The hazard function, h F , of a twice differentiable distribution function, F with density f , satisfies the first following trivial first differential equation As we mentioned in our previous works, for many univariate continuous distributions, this is the only characterization based on hazard function.The proposition presented below, provides a non-trivial characterization of OLEG distribution.
Proof Is straightforward and hence omitted.

Characterizations based on conditional expectation
The use of conditional expectation is a powerful tool for characterizing statistical relationships and properties.However, the specific form of the characterization will depend on the specific problem being considered, and care must be taken to ensure that the assumptions underlying the analysis are appropriate for the data being analyzed.
The following proposition can be found in Hamedani (2013), so we will use it to characterize the OLEG distribution.
and δ = a a+1 , Proposition 2.3.1 presents a characterization of OLEG distribution.Clearly, there are other possible function.

Estimation
In this section, we consider the complete maximum likelihood estimation method and provide all related derivations.Then, we consider the right censored maximum likelihood estimation.We must use methodologies that give numerical solutions due to the theoretical complexity and the fact that the quantile function is not known in a specific closed form.We'll use tools like "R" and "MATHCAD" to make the numerical operations easier.Numerous factors have contributed to the recent rise in popularity of numerical approaches.The prevalence of several mathematically complicated distributions and models, as well as the availability of well-known statistical programs, is the most significant of them.The complexity of modeling techniques is no longer the main issue facing researchers in the fields of statistical analysis and mathematical modeling, as statistical programs and packages have made significant contributions to simplifying these complexities by offering numerical solutions.This is a fact that has become widely acknowledged and cannot be ignored.In this work, we employed numerical approaches to the estimate, statistical analysis, and assessment procedures (see Sect. 4), and we also applied numerical methods to the issue of distributional validation under the NRR and its new equivalent version (see Sect. 5).

Complete maximum likelihood estimation
Here, the parameters of the OLEG distribution are estimated using the method of maximum likelihood.Let z 1 , z 2 , . . .z n be random samples distributed according to the OLEG, the likelihood function is obtained by the relationship By taking the natural logarithm, the log-likelihood function is

123
where The MLEs a, λ and θ of the unknown parameters a, λ and θ are derived from the following nonlinear score equations , and .
The importance of uncensored maximum likelihood estimation lies in its wide range of applications in various fields, including engineering, economics, finance, biology, and more.Uncensored maximum likelihood estimation can be used to estimate the parameters of a distribution for a manufacturing process to ensure that products meet certain quality standards.For example, if the distribution of product weights follows a normal distribution, the mean and standard deviation can be estimated using maximum likelihood estimation.In engineering, maximum likelihood estimation can be used to estimate the parameters of a distribution for the failure time of a product or system, which can be used to assess reliability and inform maintenance decisions.In this work, two applications under uncensored maximum likelihood estimation are presented, the first one is for the lifetime data and the other one is for failure times data.

Right censored maximum likelihood estimation
Right censored maximum likelihood estimation is a method for estimating the parameters of a probability distribution based on a sample of data, where some of the data is censored on the right, meaning that the exact value of the observation is unknown but only known to be above a certain value.The maximum likelihood estimator is the set of parameter values that maximize the likelihood function, which is the probability of observing the data given the parameter values and taking into account the censoring information.The importance of right censored maximum likelihood estimation lies in its wide range of applications in various fields, including survival analysis, reliability engineering, and medical research.Let us consider z = (z 1 , z 2 , . . ., z n ) T a sample from the OLEG with the parameter vector = (a, λ, θ) T which can contain right censored data with fixed censoring time τ.Each z i can be written as where The censorship is assumed to non-informative, so the likelihood function can be given by where , and the log-likelihood function log where Then, we obtain The maximum likelihood estimator (MLEs) = ( â, λ, θ) T of the unknown parameters = (a, λ, θ) T are derived from the following nonlinear score equations and As in complete data case, to calculate the MLEs â, λ and θ , we use numerical methods such as Newton-Raphson method, Monte Carlo method or BB-solve package.

Goodness-of-fit testing
A statistical model's validity may be assessed using a variety of criteria.When data are not censored, tests based on empirical functions, such as the likelihood ratio test, Akaike information criteria, Bayesian Akaike information criterion, or chisquared tests, are the most widely used, these also include test statistics such as the Kolmogorov-Smirnov test, the Anderson-Darling test, among others.In this work, we are interested in the well-known N-RR statistic Y 2 α (r − 1) (see Nikulin, 1973a, b, c;Rao & Robson, 1974), which is based on MLEs on initial non-grouped data, among these goodness-of-fit assessments.The Y 2 α (r −1) statistic restores information lost during data grouping and has a chi-squared distribution.However, censoring renders all traditional goodness-of-fit assessments ineffective.As a result, researchers provided several adjustments to the available statistics.A modified N-RR statistic, M 2 α (r ), was recently proposed by Bagdonavičius and Nikulin (2011a) for continuous distributions with unknown parameters and right censoring.Since it recovers all information lost during data regrouping, the new version of the N-RR statistic may be used to fit data from the fields where data is frequently censored, such as survival analysis, reliability, insurance, and others.In this part, we create modified chi-squared goodness-of-fit test statistics for fitting complete and the appropriate censored data to the recommended model.

N-RR statistic under the uncensorship case
Consider testing the null hypothesis H 0 according to which a sample z 1 , z 2 , . . .z n belongs to a parametric family F (z), where = ( 1 , 2 , . . ., s ) T is the parameter vector Assume r equiprobable grouping intervals I 1 , I 2 , . . ., I r Let υ = (υ 1 , υ 2 , . . ., υ r ) T represents the number of observed z i grouping into these intervals I j , and the vector T n,Z is (3) The N-RR statistic Y 2 α (r − 1) proposed by Nikulin (1973a, b, c) and Rao and Robson (1974) is defined by where I and J are the estimated information matrices on non-grouped and grouped data respectively, and is the vector of the MLEs on initial data.The elements of the vector l = l k T 1×s are where s is the number of the model parameters.The distribution of Y 2 α r − 1, is a chi-squared with r −1 degrees of freedom.To construct the test statistic Y 2 α r − 1, corresponding to the OLEG with a parameter vector , first we calculate the MLEs = ( â, λ, θ) T and the limit intervals b j (r ) .Then, the derivatives ∂ ∂ k p j are deduced as follow Then we calculate the estimated information matrices I and J .They are not given in this paper but we make them available to the users upon request.Finally we obtain the statistic Y 2 α r − 1, which allows us to verify if data belong to the OLEG distribution.

N-RR statistic under right censorship case
To verify if a right censored sample z = (z 1 , z 2 , . . ., z n ) T with fixed censored time τ, follows a parametric model F 0, (z) Bagdonavičius and Nikulin (2011b) introduced a modification of the N-RR statistic described above.This one is based on the vector where U j,Z and e j,Z are the observed and expected numbers of failures to fall into the grouping intervals I j , the statistic M 2 α (r ) is defined by where − is a generalized inverse of the covariance matrix .For calculation purposes, the authors write this statistic as follow with the quadratic form Q obtained as and Under the null hypothesis H 0 , the limit distribution of the statistic M 2 α (r ) is a chi-squared with r = rank( ) degrees of freedom.For more details on modified chi-square tests, one can see the book of Voinov et al. (2013).For testing the null hypothesis that a right censored sample is described by the OLEG distribution, we develop M 2 α (r ) corresponding to this distribution.At that end, we have to compute the MLEs = ( â, λ, θ) T on initial data, the estimated information matrix i ll which can be deduced from the score functions and the estimated limit intervals b j,Z .To apply this test statistic, the expected failure times e j,Z to fall into the grouping intervals I j must be the same for any j, so the estimated interval limits b j,Z are equal to where and So the numbers e j,Z and U j,Z can be obtained.After that, we calculate the components of the estimated matrix Ĉ which are obtained as follows: and the estimated matrix Ŵ is derived from the matrix Ĉ. Therefore the test statistic can be obtained easily: where 6 Assessing the Y 2 ˛(r − 1) and M 2 ˛(r) statistics under BB algorithm We performed a significant investigation using numerical simulations to demonstrate the flexibility and effectiveness of the tests suggested in this work.We then used the actual data from reliability and survival analysis to run these tests.Firstly, we present two simulations for complete data, the first simulation results for assessing the ML method under BB algorithm, the second simulation for assessing the Y 2 α (r − 1) statistic.Secondly, we present two simulations for censored, the first simulation results for assessing the censored ML method under BB algorithm, the second simulation for assessing the M 2 α (r ) statistic.In this work, we present six applications to real data sets, three applications to real data sets for assessing the Y 2 α (r − 1) statistic, and another three applications to real data sets for assessing the M 2 α (r ) statistic.Here are the steps to use the BB algorithm for optimization: 1. Define the objective function: The first step in using the BB algorithm is to define the objective function that needs to be minimized.This function can be a complex mathematical function that depends on one or more variables.2. Initialize the variables: Once the objective function is defined, the next step is to initialize the variables that the function depends on.This can involve setting initial values for the variables, or using random values if the problem is too complex.3. Calculate the gradient: The BB algorithm requires the gradient of the objective function to be calculated at each iteration.The gradient provides information about the direction of steepest descent for the function.4. Compute the step size: Using the gradient, the BB algorithm computes the step size for the next iteration.The step size is determined using a simple formula that involves the current gradient and the previous gradient.5. Update the variables: Using the step size, the BB algorithm updates the variables for the next iteration.The new values of the variables are obtained by subtracting the step size times the current gradient from the current values of the variables.6. Check for convergence: After each iteration, the BB algorithm checks for convergence by comparing the value of the objective function at the current iteration with the value at the previous iteration.If the difference is below a certain threshold, the algorithm stops.Repeat until convergence: steps 3-6 are repeated until the objective function converges to a minimum value, or until a maximum number of iterations is reached.Overall, the BB algorithm is a relatively simple iterative method that requires only a few steps to be implemented.However, the effectiveness of the algorithm depends on the specific problem being solved and the quality of the initial values for the variables.

Parameter estimation
We consider the OLEG model.The data were simulated N = 10,000 times (with the sample sizes n = 25, 50,130,350,500,1000) and the values of the parameters a = 1.6, λ = 0.8, θ = 2. Using the BB algorithm and the R software, the means of the simulated values of the MLEs a, λ, θ of the parameters and their mean square errors are calculated and presented in Table 1.

The Y 2 ˛(r − 1) statistic
To test hypothesis H 0 according to which the variable follows a OLEG distribution, N = 10,000 times are generated, samples of respective sizes n = 25, 50, 130, 350, 500 and 1000, of data coming from this distribution.We calculate the M 2 α (r ) values of the proposed N-RR test criterion.Then, the different empirical levels of rejection of the null hypothesis H 0 , when Y 2 α (r − 1) > χ 2 α (k − 1) are compared to their levels of theoretical significance α (α = 0.01, 0.05, 0.10).The results are given in Table 2.
We observe, taking into account the simulation errors, that the levels simulated for the statistic Y 2 α (r − 1) coincide with those corresponding to the theoretical levels of the chi-squared distribution with (r − 1) degrees of freedom.Consequently, we can say that the test proposed in this work, can suitably adjust the data coming from a OLEG model

Parameter estimation
We perform N = 10,000 simulations of censored data samples of the proposed OLEG model, of respective sizes n = 25, 50, 130, 350, 500 and 1000.The mean values of the MLEs a = 2, λ = 1.5, θ = 3 and their mean square errors are brought together in Table 3.The results of the simulations confirm the fact that for regular models, the MLEs are convergent.

˛(r)
In order to study the maneuverability and the effectiveness of the modified chi-squared type adjustment test for the OLEG model in the case where the data is censored, proposed in this work, an important study was carried out by numerical simulations.Thus and to test the null hypothesis H 0 according to which a sample of data comes from a OLEG model, we generate 10,000 samples of censored data of OLEG distribution whose respective sizes of the samples are n = 25, 50, 130, 350, 500 and 1000.The values of the criterion of the statistic M 2 α (r ) are calculated as indicated above.Then, we calculate the number of cases of rejection of the null hypothesis H 0 , i.e., when M 2 α (r ) > χ 2 α (r ) (in our case χ 2 α (r ) is the quantile of the chi-squared distribution at r degrees of freedom), for the significance levels α (α = 0, 10, α = 0.05, α = 0, 01).The values of the empirical significance levels are compared with their corresponding theoretical values (see Table 4).
From the results obtained, we note that the empirical significance levels of the statistic M 2 α (r ) coincide with those corresponding to the theoretical levels of chisquared distributions at r degrees of freedom.Therefore, we can say that the proposed test can efficiently adjust censored data from the OLEG distribution.

Real applications for assessing the Y 2 ˛(r − 1) statistic
In this section, some real data applications for assessing the Y 2 α (r − 1) statistic are given according the following scenarios: In this uncensored application, we need to test: The null hypothesis H 0 : The uncensored lifetime data follows the OLEG model, versus the alternative hypothesis H 1 : The uncensored lifetime data do not follow the OLEG model.In this Subsection, the application of the OLEG distribution is demonstrated the lifetime data of 20 electronic components (see Murthy et al., 2004).The uncensored lifetime data is given as follows: (0.03, 0.22, 0.73, 1.25, 1.52, 1.8, 2.38 , 2.87, 3.14, 4.72, 0.12, 0.35, 0.79, 1.41, 1.79, 1.94, 2.4, 2.99, 3.17, 5.09).Using the BB algorithm, the MLEs of the vector of parameters is given by = a, θ, λ T = (0.456, 1.425, 2.478).Then, taking for example 5 intervals (r = 5), we calculate the Fisher information matrix I 3×3 on the initial data, the Fisher information matrix can be expressed as We also calculated the Y 2 α=0.05 (4) test statistics to adapt these data sets to the competing model, where Y 2 α=0.05 (4) = 6.4252.
After calculate,we give the N.R.R statistic test and the critical value as: Y 2 α=0.05 (4) = 6.4252 and the critical value χ 2 α=0.05 (4) = 9.3877, this shows the importance and the usefulness of this distribution in the modeling of different data.In other words, for the uncensored lifetime data: By accepting the null hypothesis, we can conclude that the uncensored lifetime data also follow the OLEG distribution and that the uncensored lifetime data can be modeled using the OLEG distribution.
Then, taking for example 5 intervals (r = 5), we calculate the Fisher information matrix I 3×3 on the initial data, the Fisher information matrix can be expressed as We finally obtain the statistical N-RR, where Since Y 2 0.05 (4) < χ 2 α=0.05 (4) = 9.4877, we can say that with a risk α = 0.05, the distribution of these data is a OLEG distribution.In other words, for the uncensored failure times data: By accepting the null hypothesis, we can conclude that the uncensored failure times data also follow the OLEG distribution and that the uncensored failure times data can be modeled using the OLEG distribution.
8 Real applications for assessing the M 2 ˛(r) statistic In the statistical literature, many authors paid a great attentions to the real applications for assessing the M 2 α (r ) statistic.For the right censored validation of the Burr X Weibull model, Mansour et al. (2020a) proposed and implemented a modified chi-squared goodness-of-fit test employing the Bagdonavicius-Nikulin method.The modified goodness-of-fit statistics test is run on the relevant censored real dataset.The grouped data follows the chi-square distribution, whereas the modified goodness-of-fit test recovers the information loss based on the censored MLEs on the initial data.The elements of the modified criteria tests are drawn from it.Validation is a real data application in the unfiltered method.A recently published study by Yousof et al. applied a modified chi-squared type test for distributional validity on right-censored reliability and medical data (2021a).In this section, some real data applications for assessing the M 2 α (r ) statistic are given according the following scenarios: 1. Scenario 1: Assessing the M 2 α (r ) test statistic under the censored cancer of the tongue data.2. Scenario 2: Assessing the M 2 α (r ) test statistic under the censored lymphoma data.3. Scenario 3: Assessing the M 2 α (r ) test statistic under the censored survival data.4.1938 4.1938 4.1938 4.1938 4.1938 8.1 Assessing the M 2 ˛(r) statistic under the censored cancer of the tongue data In medical research, right censored maximum likelihood estimation can be used to estimate the survival function of patients with a certain disease.For example, in a clinical trial, patients may be followed for a certain period of time, and those who have not experienced the event of interest (e.g., death) by the end of the study are rightcensored.Right censored maximum likelihood estimation can be used to estimate the survival function of the population and to compare the effectiveness of different treatments.In this censored application, we need to test: The null hypothesis H 0 : The censored cancer of the tongue data follows the OLEG model, versus the alternative hypothesis H 1 : The censored cancer of the tongue data do not follow the OLEG model.
A study was conducted on the effects of ploidy on the prognosis of patients with cancers of the mouth.Patients were selected who had a paraffin-embedded sample of the cancerous tissue taken at the time of surgery.Follow-up survival data was obtained on each patient.The tissue samples were examined using a flow cytometer to determine if the tumor had an aneuploidy (abnormal) or diploid (normal) DNA profile using a technique discussed in Sickle-Santanello et al. (1988).The following data below is on patients with cancer of the tongue.Times are in weeks.The data below relates to patients with tongue cancer.The deadlines are in weeks is given as follows: Death times: 1,3,3,4,10,13,13,16,16,24,26,27,28,30,30,32,41,51,65,67,70,72,73,77,91,93,96,100,104,157,167. Censored observations: 61,74,79,80,81,87,87,88,89,93,97,101,104,108,109,120,131,150,231,240,400.We use the statistic test provided above to verify if these data are modeled by OLEG distribution, and at that end, we calculate the MLEs of the unknown parameters, where = a, θ, λ T = (0.769, 2.746, 4.621) T .
The censored cancer of the tongue data are grouped into r = 5 intervals I j (Table 5).
For significance level α = 0.05, the critical value χ 2 α=0.05 (5) = 11.0705 is superior than the value of M 2 0.05 (5) = 8.347, so we can say that the proposed model OLEG fit By accepting the null hypothesis, we can conclude that the censored cancer of the tongue data also follow the OLEG distribution and that the censored cancer of the tongue data can be modeled using the OLEG distribution.

Assessing the M 2 ˛(r) statistic under the censored lymphoma data
In this censored application, we need to test: The null hypothesis H 0 : The censored lymphoma data follows the OLEG model, versus the alternative hypothesis H 1 : The censored lymphoma data do not follow the OLEG model.We have analyzed lymphoma dataset consisting of times (in months) from diagnosis to death for 31 individuals with advanced non Hodgkin's lymphoma clinical symptoms, by using our model.This data has been analyzed by Gijbels and Gurler (2003) by using exponential change point model.Among these 31 observations 9 of the times are censored, because the patients were alive at the last time of follow-up.The data is given as: Times (in months) from diagnosis to death: 2.5, 4. 1,4.6,6.4,6.7,7.4,7.6,7.7,7.8,8.8,13.3,13.4,18.3,19.7,21.9,24.7,27.5,29.7,32.9,33.5,42.6,45.4. Censored observations: 30.1,35.4,37.7,40.9,48.5,48.9,60.4,64.4,66.4.We use the statistic test provided above to verify if these data are modeled by the OLEG distribution, and at that end, we calculate the MLEs of the unknown parameters, where = a, θ, λ T = (0.8369, 2.4136, 3.7852) T .
The censored lymphoma data are grouped into r = 5 intervals I j (Table 6).
For significance level α = 0.05, the critical value χ 2 α=0.05 (5) = 11.0705 is superior than the value of M 2 0.05 (5) = 7.2346, so we can say that the proposed model OLEG By accepting the null hypothesis, we can conclude that the censored lymphoma data also follow the OLEG distribution and that the censored lymphoma data can be modeled using the OLEG distribution.

Assessing the M 2 ˛(r) statistic under the censored survival data
In reliability and survival analysis, right censored maximum likelihood estimation can be used to estimate the reliability function of a system, such as a machine or a bridge.For example, if the failure time of a machine follows a Weibull distribution, the parameters of the distribution can be estimated using right censored maximum likelihood estimation, and the reliability of the machine can be assessed based on the estimated distribution.In this censored application, we need to test: The null hypothesis H 0 : The censored survival data follows the OLEG model, versus the alternative hypothesis H 1 : The censored survival data do not follow the OLEG model.Woolson et al. (1981) has reported survival data on 26 psychiatric inpatients admitted to the university of Iowa hospitals during the years 1935-1948.This sample is part of a larger study of psychiatric inpatients.Data for each patient consists of age at first admission to the hospital, sex, number of years of follow-up (years from admission to death or censoring) and patient status at the followup time.The data is given as: 2,11,14,22,22,24,25,26,28,40. Censored observations: 30,30,31,31,32,33,33,34,35,35,35,36,37,39.We use the statistic test provided above to verify if these data are modeled by the OLEG distribution, and at that end, we calculate the MLEs of the unknown parameters, where = a, θ, λ T = (1.4253,5.32641, 4.6325) T .
The censored survival data are grouped into r = 4 intervals I j (Table 7).
For significance level α = 0.05, the critical value χ 2 α=0.05 (4) = 9.4877 is superior than the value of M 2 0.05 (4) = 6.1935, so we can say that the proposed model OLEG fit these data.
By accepting the null hypothesis, we can conclude that the censored survival data also follow the OLEG distribution and that the censored survival data can be modeled using the OLEG distribution.

Conclusions
This article introduces and studies an unique continuous probability distribution called the odd Lindley exponentiated gamma (OLEG) distribution.On the basis of several characterization theories, we theoretically offered some novel characterization results, but in fact we overlooked numerous statistical features in favor of our numerical and practical findings.For the purpose of testing statistical hypotheses in the case of censored data, we have introduced a new modification to the famous test known as Nikulin-Rao-Robson statistic (Y 2 α (r − 1)).The new test is a version of the Nikulin-Rao-Robson test and is called the modified Nikulin-Rao-Robson (M 2 α (r )).We presented four comprehensive simulation experiments with specific conditions mentioned in the paper.The first simulation results are for evaluating the technique under the BB algorithm, and the second simulation results are for evaluating the Y 2 α (r − 1) statistic.Then, we demonstrate two censored simulations, the first of which evaluates the censored maximum likelihood estimation technique under the BB algorithm and the second for assessing the M 2 α (r ) test statistic.We provide six applications to actual data sets, three for evaluating the Y 2 α (r − 1) statistic and three more for evaluating the M 2 α (r ) statistic.Then, we provide six applications to actual data sets, three for evaluating the Y 2 α (r − 1) statistic and three more for evaluating the M 2 α (r ) statistic.Regarding the Y 2 α (r − 1) test statistic: three uncensored real data applications for assessing the Y 2 α (r − 1) statistic are given, the following results can be highlighted: 1.Under the uncensored lifetime data: Y 2 0.05 (4) = 6.4252 < χ 2 α=0.05 (4) = 9.3877 ⇒ Accept H 0 .
By accepting the null hypothesis, we can conclude that the uncensored survival times data also follow the OLEG distribution and that the uncensored survival times data can be modeled using the OLEG distribution.2. Under the uncensored failure times data: Y 2 0.05 (4) = 5.2398 < χ 2 α=0.05 (4) = 9.4877 ⇒ Accept H 0 .
By accepting the null hypothesis, we can conclude that the uncensored failure times data also follow the OLEG distribution and that the uncensored failure times data can be modeled using the OLEG distribution.3.Under the uncensored survival times data: Y 2 0.05 (6) = 10.5236< χ 2 α=0.05 (6) = 12.59159 ⇒ Accept H 0 .
By accepting the null hypothesis, we can conclude that the uncensored survival times data also follow the OLEG distribution and that the uncensored survival times data can be modeled using the OLEG distribution.
By accepting the null hypothesis, we can conclude that the censored cancer of the tongue data also follow the OLEG distribution and that the censored cancer of the tongue data can be modeled using the OLEG distribution.2. Under the censored lymphoma data: By accepting the null hypothesis, we can conclude that the censored lymphoma data also follow the OLEG distribution and that the censored lymphoma data can be modeled using the OLEG distribution.3.Under the censored survival data: M 2 α=0.05 (4) = 6.1935 < χ 2 α=0.05 (4) = 9, 4877 ⇒ Accept H 0 .
By accepting the null hypothesis, we can conclude that the censored survival data also follow the OLEG distribution and that the censored survival data can be modeled using the OLEG distribution.
Funding Open access funding provided by The Science, Technology amp; Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
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Table 5
Values of b j,Z , e j,Z , U j,Z , Ĉ1 j,Z , Ĉ2 j,Z , Ĉ3 j,Z for the censored cancer of the tongue data

Table 6
Values of b j,Z , e j,Z , U j,Z , Ĉ1 j,Z , Ĉ2 j,Z , Ĉ3 j,Z for the censored lymphoma data

Table 7
Values of b j,Z , e j,Z , U j,Z , Ĉ1 j,Z , Ĉ2 j,Z , Ĉ3 j,Z for the censored survival data