Classical Inference of the Cubic Transmuted Lindley Distribution under Type-II Censored Sample

Statistician always tries to find an easy method that gives a suitable fit. However, proposing a new distribution always solves many statistical problems. In this paper, we introduce a new extension of the Lindley distribution. We made a statistical inference under complete and Type-II censored sample on a new extension of the Lindley distribution. We have deduced its PDF and CDF using a cubic transmuted family that is because the new equations are very easy in computation. We called the new form the cubic transmuted Lindley distribution. The probability distribution function and the cumulative distribution function were also written as a closed form along with some mathematical properties. The classical method, which is the maximum likelihood estimation technique and maximum product of spacing technique, was used to find the estimators of the unknown model parameters. At last, to prove the superiority and applicability of the model, three real data sets are implemented and compared using the proposed method. We made a comparison with some of its baseline distributions and some other extensions, and our model outperforms the published ones.


Introduction
Probability distributions are used to explain real-world phenomena. ere are several practical life problems that do not t any of the basic probability models. In order to generate more exible statistical models, the generalization of well-known distributions has been widely used. ese real-world data sets are properly tted using generalized distributions. Amoroso [1] developed the generalized gamma distribution to explain the income rate distribution.Since then, a number of researchers have discussed the generalizations of distributions. ere are many researches that studied generalized distributions such as the inverse Gaussian distribution proposed by Good [2]; Lindley distribution developed by Lindley [3]; generalization of Pareto distribution by Ljubo [4]; Pickands III [5]; Hosking and Wallis [6] and McDonald and Xu [7] created the rst and second types of generalised beta to examine income distribution. A generalized form of the exponential distribution was proposed by Gupta et al. [8]; Gupta and Kundu [9]; and Gupta and Kundu [10]. e Quadratic Transmutation Map (QTM) was introduced by Shaw and Buckley [11] and used to generate non-Gaussian distributions. Rahman et al. [12,13] have introduced another two cubic transmuted families of distributions as the extension of the transmuted family of distributions. e Lindley distribution (LD) can be used in biology, engineering, and medicine sectors [14].
ere are several generalizations that have been made over Lindley distribution, such as beta Lindley distribution proposed by Merovci and Sharma [15]; cubic ranked transmuted Lindley proposed by [23]. Our model is a a special case of [23], by making some reparameterization; However, we did not propose a new distribution; we only made inferences on the cubic transmuted Lindley, which was generated from the cubic transmuted family proposed by Rahman et al. [16]; although our model is a special case from the cubic ranked transmuted Lindley proposed by [23], our distribution performs better in fitting real data; however, the proposed model has less number of parameters. e Weibull Lindley distribution by Asgharzadeh et al. [17], wrapped modified Lindley distribution, is the latest modification of Lindley distribution by Chesneau et al. [18]. e transmuted generalization of Lomax distribution has been introduced by Abu El Azm et al. [19]. ey introduced a new one-parameter circular distribution based on the wrapping method.
eir main objective was to extend the scope in astronomy, geology, and meteorology, that is, over circling objects. Hence, the main goal of this paper is to expand the Lindley distribution's goodness of fit test on further aspects over lifetime, time series, environmental, production sectors, and so on, so we applied the Type-II censored sample to the CT- Lindley. ere are various instances in life testing and dependability studies when observations are overlooked or trials are abandoned prior to failure. e tester may be unable to get comprehensive data on failure times for all experimental observations. e data from all of these trials are referred to as censored data. ere are several sorts of censored tests; the most common and widely utilised are Type-I censored and Type-II censored; see, for example, [20]. While on product spacing (PS), many authors used PS based on censored samples; for example, El-Sherpieny et al. [21] introduced progressive Type-II hybrid censored schemes based on PS. Alshenawy et al. [22] introduced progressive Type-II censoring schemes for the PS method by binomial random removal. Almetwally et al. [23] introduced PS estimation of Weibull distribution under adaptive Type-II progressive censoring schemes. Alshenawy et al. [24] obtained the PS estimation for a stressstrength model under progressive hybrid censored. Almongy et al. [25] used the PS method under progressive Type-II censoring schemes. Almetwally et al. [26] estimated parameters by using the PS method of flexible extension Weibull distribution based on Type-I and Type-II censoring. In the presence of fuzzy system, the PS has been used by Sabry et al. [27].
is paper firstly aims to insert and treatise a new form distribution defined by Lindley distribution and cubic transmuted family. Considerable statistical properties of CT-Lindley distribution are shown. Secondly, the estimate of parameters for the CT-Lindley distribution is discussed using the MLE and MPS approaches. irdly, we examined the estimation of parameters for the CT-Lindley distribution using Type-II censored data. Monte Carlo simulation is used to determine the estimators' efficiency. Fourthly, real-world data studies are conducted to verify the model and scheme's validity.
A second-order Lindley distribution is developed to accomplish so, which can capture the complex behavior in real-life experiences covering the lifetime, environmental, biological, engineering, and production data sets. e layout plan of the paper is as follows. e equations of the CT-Lindley distribution using the aid of the cubic transmuted family that was presented by Rahman et al. [16] are introduced in Section 2. Some of the distributional properties are illustrated in Section 3. e statistical inferences using both classical methods are described in Section 5. e simulation study associated with its results is recorded in Section 6. Four real-life applications of the distribution are shown in Section 7. In the end, some concluding remarks are listed in Section 8.

Cubic Transmuted Lindley Distribution
In this section, the Lindley distribution is introduced, and some properties are derived. e Lindley distribution was first introduced in the literature by Lindley [3]. e pdf and corresponding cdf are given as follows: and where θ ∈ R + [32] and it presents the shape of the distribution. e quadratic transmuted family of distribution has the following cdf form: where θ ∈ R + and λ ∈ [− 1, 1]. Rahman et al. [16] proposed a cubic transmuted family, of which the distribution function is written as where G(x) is the distribution function of any baseline probability distribution. e cubic transmuted Lindley is a special case from the ranked transmuted Lindley introduced by [23], by making some reparameterization; however, we used the cubic transmuted family proposed by Rahman et al. [38] to get the equations of the cubic transmuted Livley, which is a special case of the distribution proposed in [23]. Distribution is a special case from equation (4) that caused other new distributions to be introduced by Ansari et al. [39]; Ogunde et al. [40]; Wang and Bao [41]; Rahman et al. [38]; Akter et al. [42]; and so on, in order to introduce the proposed distribution, by substituting equation (2) into (4), which has the following distribution function: where θ ∈ R + and λ ∈ [− 1, 1]. e corresponding pdf of the cubic transmuted Lindley distribution is obtained by differentiating (6) with respect to x, which is defined as follows.

Definition 2.1. A continuous random variable X has a cubic transmuted Lindley distribution if its probability density function can be written as
where θ ∈ R + and λ ∈ [− 1, 1]. Figure 1 shows some possible shapes of the CT-Lindley distribution that indicates the capability of capturing the complex behavior of the real-life data sets. It can be seen that the shapes of the distribution approach zero as θ ⟶ ∞ and also become narrow.

Distributional Properties
Some important distributional properties of the CT-Lindley distribution that was not introduced by Rahman et al. [38], given in (7), are discussed in the following subsections.

Moments.
A moment is used to measure the shape characteristics of a distribution. A set of statistical parameters used to measure distribution is known as moments. Two most important approaches are measured, central tendency and dispersion. In probability, distribution mean and variance are the two most widely applied measurements. Skewness and kurtosis are also two additional measurements of shape characteristics. e r th raw moments of the CT-Lindley distribution take the following form: e variance of CT-Lindley distribution is obtained as Mathematical Problems in Engineering It is possible to obtain all other higher-order moments using r > 2 in equation (7).

Moment Generating Function.
A moment is calculated by either a sum or an integral function of a distribution. e computation of moments tends to be time-consuming. On the other hand, it is simply generated by the derivatives of a single expected value function.
is function is called a moment generating function, which is stated by the following theorem.
e moment generating function M x (t) is represented by the symbol M x (t) and has the following definition:  Mathematical Problems in Engineering Proof 1. e moment generating function is defined as where f(x) is given in (7). By using the series expansion of e tX given by Zwillinger and Jeffrey [43], we have By substituting (7) into (13), we have the moment generating function M X (t).
e Fourier transform of the distribution's density function is the characteristic function, which is analogous to the "logarithm table trick" for convolution. Characteristic functions are useful in probability theory because they may be used to derive the properties of distributions.
e characteristic function of the CT-Lindley distribution is stated by the following theorem.

Theorem 3.2. Let a continuous random variable X follow the cubic transmuted Lindley distribution, and then the characteristic function ϕ
where is the imaginary unit andt ∈ R.

Quantile Function and Median.
e quantile function for the CT-Lindley distribution is obtained by solving F(x) � q; see, for example, Rahman et al. [16] and further proceed as follows: which can be written as where Mathematical Problems in Engineering Using (16), it is easy to obtain the first quartile (Q 1 ), second quartile (Q 2 ), or median and third quartile (Q 3 ) by setting q � 0.25, 0.50, and 0.75, respectively.

Reliability Analysis.
e complement of a distribution function is known as the reliability function, and for CT-Lindley distribution, it is defined as follows: e hazard function is defined as the ratio of the probability distribution function as in equation (6) to the reliability function as in (18) and expressed as Using a sequence of values for the random variable, several plots of the reliability and hazard rate functions for the proposed distribution are shown in Figure 2. is distribution has the capacity to accommodate various increasing and decreasing hazard rate functions, as shown in the previous figure. It can be seen that this data range is suitable for the combination of parameter values used to draw these graphs.

Generating Random Sample.
e quantile function for the CT-Lindley distribution is obtained by solving F(x) � u; see, for example, Rahman et al. [13] and further proceed as follows: where u ∼ U(0, 1) and the above can be further obtained as with

Order Statistics
e probability density function of the r th order statistic for the proposed cubic transmuted Lindley distribution is given as follows: 6 Mathematical Problems in Engineering where r � 1, 2, . . . , n and r � 1 gives the lowest order statistic X 1: n density function, which is Mathematical Problems in Engineering Also, for using r � n in (23), the density function of the highest order statistic X n: n is obtained by For λ � 0, it has the density function of the r th order statistic for CT-Lindley distribution as follows; for more details, see Ghitany et al. [32].
e k th order moment of X r: n for the proposed cubic transmuted Lindley distribution is obtained by using the following equation:

Estimation Based on Complete Sample and
Censored Sample e estimation of model parameters for the CT-Lindley distribution has been done by the MLE method. For doing this, consider a random sample x 1 , x 2 , . . . , x n of size n from the proposed cubic transmuted Lindley distribution, which has the likelihood function as and the log-likelihood function is

Mathematical Problems in Engineering
So, by differentiating and maximizing the log-likelihood function in (29), with respect to unknown parameters, θ and λ are obtained as follows: and where e maximum value is obtained by setting zl/zθ � 0 and zl/zλ � 0, and solving the associated nonlinear system of equations gives the maximum likelihood estimate Θ � (θ, λ) ′ of Θ � (θ, λ) ′ . e theoretical solutions are often extremely complicated, and in order to get the numerical solution, we applied R-package "bbmle" [44]. Also as n ⟶ ∞, the asymptotic distribution of the MLE (θ, λ) is given by [12,38] (θ/λ) ∼ N (θ/λ), e asymptotic variance-covariance matrix V of the estimates θ, λ is obtained by inverting the Hessian matrix. Approximate 100(1 − α) two-sided confidence intervals for θ and λ are given by where Z α is the α percentile of the standard normal distribution.

Estimation Methods Based on Type-II Censored Sample.
In this section, the parameter estimation for the CT-Lindley distribution based on Type-II censored using MLE and MPS estimation methods will be discussed in detail. Let x 1 , x 2 , . . . , x n be a random sample of size n from the PDF of the CT-Lindley model based on Type-II censored sample; then, the likelihood function takes the form e data is made up on the observations: x 1: n < x 2: n < . . . < x r: n a, and the information that (n− r) objects persist further than the expiry of the period x r: n , where r represents the total number of uncensored items. Assume we have nCT-Lindley distribution observation, which is located in life testing. e number of errors r is random. In Type-II censorship, a life test is terminated after a predetermined number of errors here, n and r are fixed and predetermined, but x r: n is random. e following formula (19) represents the likelihood function of the CT-Lindley distribution, based on Type-II censoring.
x i:n + 1 1 + λ e − 2θx i: n (θ + 1) 2 6(θ + 1)e θx i: n θ + θx i: n + 1 − 6 θ + θx i:n + 1 2 − 1 R x r:n ; θ, λ e log-likelihood function for may be maximised directly (36). Using the software to solve the nonlinear likelihood equations produced, which is called R-package to solve the nonlinear likelihood equations obtained by differentiating (36) with respect to θ, λ and being equal to zero.  We employ a numerical methodology like the Newton-Raphson method to evaluate the MPS of the natural logarithm of the product spacing function under Type-II censored sample θ, λ.

Simulation
In this section, Monte Carlo simulation is done to estimate the parameters of CT-Lindley distribution based on Type-II censoring by using MLE and MPS methods. In the simulation algorithm, Monte Carlo experiments were carried out under the following data generated from CT-Lindley distribution by using the quantile (13), where x has CT-Lindley distributed for different parameters θ, λ as follows.
Different samples sizes have been obtained; n = 50, 100, and 200; also we used different censored samples schemes, where under Type-II, when p = 0.7 and 0.9 are the ratios of sample size. We can find the parameter estimation by using log-equations (35) and (36), R-package using 10000 iterations of the Newton-Raphson algorithm. e optimum technique is the one that minimizes bias and mean squared error.
From Tables 1-2, we can conclude the following: (i) As n increases, so does bias and MSE.
(ii) In Type-II censored samples, as the percentage of failure (p) grows, increases, then the values of the Bias and MSE for the CT-Lindley parameters go down.   (iii) e MPS estimates have more relative efficiency than MLE.

Simulation Algorithm.
In this part of the paper, we will discuss the simulation steps: (1) e first step is to set the parameters' starting values (2) Second, generate a random sample of size n for complete sample or size m for censored sample (3) Specify the number of failures in the censored sample (4) Specify the censoring scheme that will be used

Applications
In this section, we applied different kinds of data such as engineering data, scientific data, wind speed data, and agriculture data; all these kinds of data represent different shapes of skewness. ree real-life applications of the CT-Lindley distribution are conducted to check its applicability by the following four subsections. To evaluate the applicability of the CT-Lindley distribution, the two most related distributions, transmuted Lindley and Lindley and cubic transmuted Lindley [23], are selected. To know the   characteristics, summary statistics of the data sets used in this study are described briefly in Table 3.
To assess the practicality of the proposed model, certain model selection criteria such as − 2 log-likelihood, Akaike's information criterion (AIC), corrected Akaike information criterion (AICc), and Bayesian information criterion (BIC) are used.

e Wheaton River Data.
e data from Bourguignon et al. [45] refer to the Wheaton River at Carcross, Yukon Territory, Canada, exceeding flood peaks (in m 3 /s). e data are comprised of 72 exceedances from 1958 to 1984, rounded to the nearest decimal place. is data set is highly skewed and heavy-tailed platykurtic time series data. For comparison with other models, estimates of the model parameters with corresponding standard error and the loglikelihood values of the proposed model are shown in Table 4. e estimated plots for the selected models, as well as the proposed CT-Lindley distribution, are placed over the empirical cdf plot in Figure3(b), as well as the estimated density function in Figure 4(b). When compared to different models, the data set fits well with the proposed distribution, as seen in the figures.
e obtained values of the model selection criterion, shown in Table 5, imply that the proposed CT-Lindley distribution is more suitable to fit this highly skewed and heavy-tailed platykurtic time series data set than any other selected models used in this study.

Extreme Wind Speed Data.
One of EWS (Extreme Wind Speed) Chiodo and Noia [46] data of 52 weekly maximum wind speeds in (m/s) has been used in this study. is moderately skewed and light-tailed platykurtic data set is an example of environmental data. Estimated model parameters with the appropriate SE and log-likelihood values are displayed in Table 6. Figure 4 Table 7, which prove that the proposed CT-Lindley model provides better performance than other models in case of environmental data.

Fatigue Life.
e data are given by Birnbaum

Conclusions and Recommendations
e statistical inference has been made using the cubic transmuted Lindley distribution. We used two famous classical methods such as the MLE and MPS. We made a simulation study using both methods. To know how this distribution is performing, four different real-life data sets from the lifetime, time series, environmental, and production sectors have been considered. Reviewing the characteristics of the data set, it can be seen that they are not normal but highly skewed, negatively skewed, moderately skewed, heavy-tailed, or light-tailed platykurtic. Using estimated graphs as well as well-known model selection criterion values, it was shown both graphically and numerically that the proposed model gives better performance and captures more sectors of real-life problems than other models selected for comparison in this study. So, in the end, based on the analysis shown throughout this paper, this distribution is flexible enough to capture more complex reallife data sets that arise in various areas of life [47,48].

Future Work
In the next paper, we will apply an accelerated life test on the CT-Lindley; under Type-II censored sample, we will apply different kinds of classical and Bayesian estimators. We will use real data from Nelson's book for accelerated life tests. We will apply different kinds of acceleration models such as constant and partially accelerated experiments, and we will extend our work to get the optimal censoring scheme for the experiment and the optimal sample size. We can look for applying competing risk data to the proposed model and see its flexibility in fitting the data.
One last work will be done. We will work on the related parts between our paper and neutrosophic statistics.

Data Availability
e document contains all of the relevant data as well as references to that data.