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

Authors: Md. Awwal Islam Khan; Amirah Saeed Alharthi; Hassan M. Aljohani; Taghreed M. Jawa; Neveen Sayed-Ahmed; Md. Shohel Rana; Md. Mahabubur Rahman
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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.


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## Body

## 1. Introduction

Probability distributions are used to explain real-world phenomena. There are several practical life problems that do not fit any of the basic probability models. In order to generate more flexible statistical models, the generalization of well-known distributions has been widely used.These real-world data sets are properly fitted 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. There 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 first 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]. The 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. The Lindley distribution (LD) can be used in biology, engineering, and medicine sectors [14].There 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.The Weibull Lindley distribution by Asgharzadeh et al. [17], wrapped modified Lindley distribution, is the latest modification of Lindley distribution by Chesneau et al. [18]. The transmuted generalization of Lomax distribution has been introduced by Abu El Azm et al. [19]. They introduced a new one-parameter circular distribution based on the wrapping method.Their 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.There are various instances in life testing and dependability studies when observations are overlooked or trials are abandoned prior to failure. The tester may be unable to get comprehensive data on failure times for all experimental observations. The data from all of these trials are referred to as censored data. There 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 stress-strength 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].Based on the complete sample, many authors used the PS method, for example, Almetwally et al. [28]; Ahmad and Almetwally [29]; Almetwally [30]; and Almongy et al. [31]. This 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. Thirdly, 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.The layout plan of the paper is as follows. The 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. The statistical inferences using both classical methods are described in Section 5. The 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.


## 2. Cubic Transmuted Lindley Distribution

In this section, the Lindley distribution is introduced, and some properties are derived.The Lindley distribution was first introduced in the literature by Lindley [3]. The pdf and corresponding cdf are given as follows:(1)gx;θ=θ2θ+1x+1e−θx,x∈ℝ+,and(2)Gx;θ=1−θ+θx+1θ+1e−θx,x∈ℝ+,where θ∈ℝ+ [32] and it presents the shape of the distribution. The quadratic transmuted family of distribution has the following cdf form:(3)Fx=λ+1Gx−λG2x,x∈ℝ,where λ∈−1,1 and Gx is the base distribution function of any standard probability distribution. For, λ=0, (3) becomes the baseline probability distribution. Many new distributions have already been developed by using (3) in the literature of probability models, including Aryall G. R. [33]; Merovci and Elbatal [34]; Mahmoud and Mandouh [35]; Alizadeh et al. [36]; Ahmed and Qasim [37]. Merovci and Elbatal [34] developed transmuted Lindley distribution that has the following cdf:(4)Qx;θ,λ=1−θ+θx+1θ+1e−θx1+λθ+θx+1θ+1e−θx,x∈ℝ+,where θ∈ℝ+ and λ∈−1,1. Rahman et al. [16] proposed a cubic transmuted family, of which the distribution function is written as(5)Fx=1−λGx+3λG2x−2λG3x,x∈ℝ,where G(x) is the distribution function of any baseline probability distribution. The 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:(6)Fx;θ,λ=1θ+13e−3θxθ+1eθx−1−x+1θ×e2θxθ+12+eθxθ+1λθ+θx+1−2λθ+θx+12,x∈ℝ+,where θ∈ℝ+ and λ∈−1,1. The 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(7)fx;θ,λ=θ2θ+1x+1e−xθ1+λe−2θxθ+12×6θ+1eθxθ+θx+1−6θ+θx+12−1,x∈ℝ+,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.Figure 1
The density and cumulative distribution functions of CT-Lindley distribution with different values ofλ and θ.
(a)(b)(c)(d)


## 3. 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.


### 3.1. 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. Therth raw moments of the CT-Lindley distribution take the following form:(8)μr′=Γr+13θrθ+136−r+2−2r+327θ+13+r333θ+4r2+9θ3θ+7+38rλ−3r+3×θ+1θ+12r+2θ+r+1λ−1−3λ4θ+12+r2+4θ+5r.Letr = 1 in (8); we have(9)μ=1108θθ+13108θ+12θ+2−3θ6θθ+4+19+19λ.The variance of CT-Lindley distribution is obtained as(10)σ2=111664θ2θ+16−11664θ+14θθ+4+2+36θ+12377+θ15089θ22θθ+6+225λ+3θ6θθ+4+19+192λ2.It is possible to obtain all other higher-order moments usingr>2 in equation (7).


### 3.2. Moment Generating Function

A moment is calculated by either a sum or an integral function of a distribution. The 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. This function is called a moment generating function, which is stated by the following theorem.Theorem 3.1.
The moment generating functionMxtis represented by the symbolMxtand has the following definition:(11)MXt=∑r=0∞trr!6−r+2Γr+13θrθ+13−2r+327θ+13+r333θ+4r2+9θ3θ+7+38rλ−3r+3θ+1θ+12r+2θ+r+1λ−1−3λ4θ+12+r2+4θ+5r.Proof 1.
The moment generating function is defined as(12)MXt=EetX,=∫0∞etXfxdx,where fx is given in (7). By using the series expansion of etX given by Zwillinger and Jeffrey [43], we have(13)MXt=∫0∞∑r=0∞trr!xrfxdx,=∑r=0∞trr!EXr.
By substituting (7) into (13), we have the moment generating function MXt.


### 3.3. Characteristic Function

The 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.The 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 functionϕxtis(14)ϕXt=∑r=0∞itrr!6−r+2Γr+13θrθ+13−2r+327θ+13+r333θ+4r2+9θ3θ+7+38rλ−3r+3θ+1θ+12r+2θ+r+1λ−1−3λ4θ+12+r2+4θ+5r,where i=−1is the imaginary unit andt∈ℝ.Proof 2.
The proof is simple.
Instead of calculating the density functions directly, characteristic functions are an alternate way for simplifying the probability distributions function. The manipulation of probability distributions, the formation of moments, or the shape of a group of points for any random variable, as well as the fitting of probability distributions to large samples of data, are all examples of practical applications of characteristic functions.


### 3.4. Quantile Function and Median

The quantile function for the CT-Lindley distribution is obtained by solvingFx=q; see, for example, Rahman et al. [16] and further proceed as follows:(15)1θ+13e−3θxθ+1eθx−1−x+1θ×e2θxθ+12+eθxθ+1λθ+θx+1−2λθ+θx+12=q,which can be written as(16)xq=−1θθ+1+We−θ−1θ+1y−1,where(17)y=12−232/3λ−13Δ1+9Δ23−12λ+233Δ1+27Δ2−1/3,Δ1=−λ3λλλ+6−108q−1q−15+8,Δ2=λ22q−1,W=LambertWfunction.Using (16), it is easy to obtain the first quartile Q1, second quartile Q2, or median and third quartile Q3 by setting q = 0.25, 0.50, and 0.75, respectively.


### 3.5. Reliability Analysis

The complement of a distribution function is known as the reliability function, and for CT-Lindley distribution, it is defined as follows:(18)Rt;θ,λ=1−1θ+13e−3θtθ+1eθt−1−t+1θ×e2θtθ+12+eθtθ+1λθ+θt+1−2λθ+θt+12.The 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(19)ht=e2θtθ2t+11+−1+e−2tθ6eθtθ+1θ+tθ+1−6θ+θt+12/θ+12λθ+θt+1−e2θtλ−1+3λeθtθ+θt+1/θ+1−2λθ+θt+12/θ+12.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 Figure2. This 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.Figure 2
The reliability and hazard rate functions of the CT-Lindley distribution with different values ofλ and θ.
(a)(b)(c)(d)


### 3.6. Generating Random Sample

The quantile function for the CT-Lindley distribution is obtained by solvingFx=u; see, for example, Rahman et al. [13] and further proceed as follows:(20)1θ+13e−3θxθ+1eθx−1−x+1θe2θxθ+12+eθxθ+1λθ+θx+1−2λθ+θx+12=u,where u∼U0,1 and the above can be further obtained as(21)x=−1θθ+1+We−θ−1θ+1j−1,with(22)j=12−232/3λ−133+Δ13+9Δ2−12λ+233Δ1+27Δ2−1/3,Δ1=−λ3λλλ+6−108u−1u−15+8,Δ2=λ22u−1,W=LambertWfunction.


## 3.1. 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. Therth raw moments of the CT-Lindley distribution take the following form:(8)μr′=Γr+13θrθ+136−r+2−2r+327θ+13+r333θ+4r2+9θ3θ+7+38rλ−3r+3×θ+1θ+12r+2θ+r+1λ−1−3λ4θ+12+r2+4θ+5r.Letr = 1 in (8); we have(9)μ=1108θθ+13108θ+12θ+2−3θ6θθ+4+19+19λ.The variance of CT-Lindley distribution is obtained as(10)σ2=111664θ2θ+16−11664θ+14θθ+4+2+36θ+12377+θ15089θ22θθ+6+225λ+3θ6θθ+4+19+192λ2.It is possible to obtain all other higher-order moments usingr>2 in equation (7).


## 3.2. Moment Generating Function

A moment is calculated by either a sum or an integral function of a distribution. The 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. This function is called a moment generating function, which is stated by the following theorem.Theorem 3.1.
The moment generating functionMxtis represented by the symbolMxtand has the following definition:(11)MXt=∑r=0∞trr!6−r+2Γr+13θrθ+13−2r+327θ+13+r333θ+4r2+9θ3θ+7+38rλ−3r+3θ+1θ+12r+2θ+r+1λ−1−3λ4θ+12+r2+4θ+5r.Proof 1.
The moment generating function is defined as(12)MXt=EetX,=∫0∞etXfxdx,where fx is given in (7). By using the series expansion of etX given by Zwillinger and Jeffrey [43], we have(13)MXt=∫0∞∑r=0∞trr!xrfxdx,=∑r=0∞trr!EXr.
By substituting (7) into (13), we have the moment generating function MXt.


## 3.3. Characteristic Function

The 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.The 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 functionϕxtis(14)ϕXt=∑r=0∞itrr!6−r+2Γr+13θrθ+13−2r+327θ+13+r333θ+4r2+9θ3θ+7+38rλ−3r+3θ+1θ+12r+2θ+r+1λ−1−3λ4θ+12+r2+4θ+5r,where i=−1is the imaginary unit andt∈ℝ.Proof 2.
The proof is simple.
Instead of calculating the density functions directly, characteristic functions are an alternate way for simplifying the probability distributions function. The manipulation of probability distributions, the formation of moments, or the shape of a group of points for any random variable, as well as the fitting of probability distributions to large samples of data, are all examples of practical applications of characteristic functions.


## 3.4. Quantile Function and Median

The quantile function for the CT-Lindley distribution is obtained by solvingFx=q; see, for example, Rahman et al. [16] and further proceed as follows:(15)1θ+13e−3θxθ+1eθx−1−x+1θ×e2θxθ+12+eθxθ+1λθ+θx+1−2λθ+θx+12=q,which can be written as(16)xq=−1θθ+1+We−θ−1θ+1y−1,where(17)y=12−232/3λ−13Δ1+9Δ23−12λ+233Δ1+27Δ2−1/3,Δ1=−λ3λλλ+6−108q−1q−15+8,Δ2=λ22q−1,W=LambertWfunction.Using (16), it is easy to obtain the first quartile Q1, second quartile Q2, or median and third quartile Q3 by setting q = 0.25, 0.50, and 0.75, respectively.


## 3.5. Reliability Analysis

The complement of a distribution function is known as the reliability function, and for CT-Lindley distribution, it is defined as follows:(18)Rt;θ,λ=1−1θ+13e−3θtθ+1eθt−1−t+1θ×e2θtθ+12+eθtθ+1λθ+θt+1−2λθ+θt+12.The 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(19)ht=e2θtθ2t+11+−1+e−2tθ6eθtθ+1θ+tθ+1−6θ+θt+12/θ+12λθ+θt+1−e2θtλ−1+3λeθtθ+θt+1/θ+1−2λθ+θt+12/θ+12.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 Figure2. This 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.Figure 2
The reliability and hazard rate functions of the CT-Lindley distribution with different values ofλ and θ.
(a)(b)(c)(d)


## 3.6. Generating Random Sample

The quantile function for the CT-Lindley distribution is obtained by solvingFx=u; see, for example, Rahman et al. [13] and further proceed as follows:(20)1θ+13e−3θxθ+1eθx−1−x+1θe2θxθ+12+eθxθ+1λθ+θx+1−2λθ+θx+12=u,where u∼U0,1 and the above can be further obtained as(21)x=−1θθ+1+We−θ−1θ+1j−1,with(22)j=12−232/3λ−133+Δ13+9Δ2−12λ+233Δ1+27Δ2−1/3,Δ1=−λ3λλλ+6−108u−1u−15+8,Δ2=λ22u−1,W=LambertWfunction.


## 4. Order Statistics

The probability density function of therth order statistic for the proposed cubic transmuted Lindley distribution is given as follows:(23)fr:nx=n!r−1!n−r!eθ−xx+1θ21+−1+e−2θx6θ+1eθxθ+θx+1−6θ+θx+12/θ+12λθ+1×e−3θxθ−x+1+θ+1eθx−1θ+1λeθxθ+θx+1−2λθ+θx+12+θ+12e2θxθ+1−3r−11−e−3θxθ−x+1+θ+1eθx−1θ+1λ×eθxθ+θx+1−2λθ+θx+12+θ+12e2θxθ+1−3n−r,where r=1,2,…,n and r=1 gives the lowest order statistic X1:n density function, which is(24)f1:nx=n!n−1!eθ−xx+1θ21+−1+e−2θx6θ+1eθxθ+θx+1−6θ+θx+12/θ+12λθ+1×1−e−3θxθ−x+1+θ+1eθx−1θ+1λeθxθ+θx+1−2λθ+θx+12+θ+12e2θxθ+1−3n−r.Also, for usingr=n in (23), the density function of the highest order statistic Xn:n is obtained by(25)fn:nx=n!r−1!eθ−xx+1θ21+−1+e−2θx6θ+1eθxθ+θx+1−6θ+θx+12/θ+12λθ+1×e−3θxθ−x+1+θ+1eθx−1θ+1λeθxθ+θx+1−2λθ+θx+12+θ+12e2θxθ+1−3r−1.Forλ=0, it has the density function of the rth order statistic for CT-Lindley distribution as follows; for more details, see Ghitany et al. [32].(26)fr:nx=n!r−1!n−r!.θ2x+1eθ−xθ+1.eθ−xθ−x+1+θ+1eθx−1θ+1r−1×1−eθ−xθ−x+1+θ+1eθx−1θ+1n−r.Thekth order moment of Xr:n for the proposed cubic transmuted Lindley distribution is obtained by using the following equation:(27)EXr:nk=∫0∞xrkfxr:nxdx.


## 5. Estimation Based on Complete Sample and Censored Sample

The estimation of model parameters for the CT-Lindley distribution has been done by the MLE method. For doing this, consider a random samplex1,x2,…,xn of size n from the proposed cubic transmuted Lindley distribution, which has the likelihood function as(28)L=∏i=1nθ2θ+1xi+1e−xiθ1+λe−2θxiθ+126θ+1×eθxiθ+θxi+1−6θ+θxi+12−1,x∈ℝ+,and the log-likelihood function is(29)l=2nlogθ−nlogθ+1+∑i=1nlogxi+1−θ∑i=1nxi+∑i=1nLog−6λe−θxiθ+θxi+1θ+1+5λ+1−6λ1−e−θxiθ+θxi+1θ+12.So, by differentiating and maximizing the log-likelihood function in (29), with respect to unknown parameters, θ and λ are obtained as follows:(30)∂l∂θ=θ+2nθθ+1−∑i=1nxi+−6λψ1−ψ2+5λ+1−1∑i=1nψ3+ψ4+ψ5−ψ6,and(31)∂l∂λ=∑i=1n−61−Ω2−6Ω+5−6λ1−Ω2−6λΩ+5λ+1,where(32)ψ1=1−e−θxiθ+θxi+1θ+12,ψ2=6λe−θxiθ+θxi+1θ+1,ψ3=6λe−θxiθ+1xiθ+θxi+1,ψ4=θ+θxi+1θ+1−xi+1−6λ2e−2θxiθ+1−1+θxi+θxi2,ψ5=1θ+12θ−θ2xi3−2θ2xi2−θ2xi−θxi2+θxi+2,ψ6=1θ+12θ2+2θ+2θxi+θ2xi2+2θ2xi+1,Ω=e−θxiθ+θxi+1θ+1.The maximum value is obtained by setting∂l/∂θ=0 and ∂l/∂λ=0, and solving the associated nonlinear system of equations gives the maximum likelihood estimate Θ^=θ^,λ^′ of Θ=θ,λ′. The 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](33)θ^/λ^∼ℕθ/λ,V^11V^12V^21V^22.The asymptotic variance-covariance matrix V of the estimatesθ^,λ^ is obtained by inverting the Hessian matrix. Approximate 1001−α two-sided confidence intervals for θ and λ are given by(34)θ^±Zα2V^11,λ^±Zα2V^22,where Zα is the α percentile of the standard normal distribution.


### 5.1. 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.Letx1,x2,…,xn 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(35)Lθ,λ=n!n−r!1−Fxr:n;θ,λn−r∏i=1rfxi:n;θ,λ.The data is made up on the observations:x1:n<x2:n<…<xr:n a, and the information that (n−r) objects persist further than the expiry of the period xr:n, where r represents the total number of uncensored items. Assume we have nCT-Lindley distribution observation, which is located in life testing. The 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 xr:n is random. The following formula (19) represents the likelihood function of the CT-Lindley distribution, based on Type-II censoring.(36)Lθ,λ=∏i=1rxi:n+11+λe−2θxi:nθ+126θ+1eθxi:nθ+θxi:n+1−6θ+θxi:n+12−1Rxr:n;θ,λn−rθ2rθ+1re−θ∑i=1rxi:n.The 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.


### 5.2. MPS under Censoring Sample

The general form for the MPS function under Type-II censored samples is given as(37)PSθ,λ=n!n−r!1−Fxr:n;θ,λn−r∏i=1r+1Fx1:n;θ,λFxi:n;θ,λ−Fxi−1:n;θ,λi=2,…,r1−Fxr:n;θ,λ.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 sampleθ,λ.


## 5.1. 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.Letx1,x2,…,xn 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(35)Lθ,λ=n!n−r!1−Fxr:n;θ,λn−r∏i=1rfxi:n;θ,λ.The data is made up on the observations:x1:n<x2:n<…<xr:n a, and the information that (n−r) objects persist further than the expiry of the period xr:n, where r represents the total number of uncensored items. Assume we have nCT-Lindley distribution observation, which is located in life testing. The 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 xr:n is random. The following formula (19) represents the likelihood function of the CT-Lindley distribution, based on Type-II censoring.(36)Lθ,λ=∏i=1rxi:n+11+λe−2θxi:nθ+126θ+1eθxi:nθ+θxi:n+1−6θ+θxi:n+12−1Rxr:n;θ,λn−rθ2rθ+1re−θ∑i=1rxi:n.The 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.


## 5.2. MPS under Censoring Sample

The general form for the MPS function under Type-II censored samples is given as(37)PSθ,λ=n!n−r!1−Fxr:n;θ,λn−r∏i=1r+1Fx1:n;θ,λFxi:n;θ,λ−Fxi−1:n;θ,λi=2,…,r1−Fxr:n;θ,λ.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 sampleθ,λ.


## 6. 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), wherex has CT-Lindley distributed for different parameters θ,λ as follows.In Table1, θ=0.5 and λ=−0.5 and 0.9.Table 1
MLE and MPS based on Type-II censored and complete sample whenθ=0.5.
θ=0.5MLEMPSλP (%)SchemeMeanBiasMSEMeanBiasMSE-0.570Type-II50θ0.50660.00660.00570.50260.00260.0057λ−0.44770.05230.1642−0.5366−0.03660.1530100θ0.50840.00840.00290.50660.00660.0028λ−0.47360.02640.0674−0.5149−0.01490.0630200θ0.50640.00640.00140.50550.00550.0014λ−0.48160.01840.0369−0.5019−0.00190.035490Type-II50θ0.4985−0.00150.00370.4932−0.00680.0035λ−0.47470.02530.1385−0.5398−0.03980.1249100θ0.50410.00410.00190.50130.00130.0018λ−0.48910.01090.0631−0.5208−0.02080.0597200θ0.50400.00400.00100.50250.00250.0010λ−0.49270.00730.0335−0.5082−0.00820.0326100Complete50θ0.4976−0.00240.00330.4867−0.01330.0033λ−0.47700.02300.1397−0.5323−0.03230.1247100θ0.50330.00330.00170.4964−0.00360.0016λ−0.49050.00950.0620−0.5144−0.01440.0583200θ0.50390.00390.00090.4998−0.00020.0009λ−0.49370.00630.0329−0.5044−0.00440.03200.970Type−II50θ0.4992−0.00080.00170.4948−0.00520.0037λ1.12910.22910.36270.8007−0.09930.1066100θ0.50240.00240.00100.4997−0.00030.0019λ0.99960.09960.09670.8364−0.06360.0413200θ0.50280.00280.00040.50200.00200.0004λ0.93200.03200.02800.8555−0.04450.018290Type-II50θ0.4950−0.00500.00160.4982−0.00180.0015λ1.03920.13920.16180.8013−0.09870.0729100θ0.50040.00040.00080.50190.00190.0008λ0.96990.06990.05840.8421−0.05790.0318200θ0.50200.00200.00040.50260.00260.0004λ0.92280.02280.01870.8594−0.04060.0140100Complete50θ0.4948−0.00520.00170.4981−0.00190.0015λ1.05320.15320.22700.7728−0.12720.0603100θ0.4999−0.00010.00080.50200.00200.0008λ0.95310.05310.04540.8205−0.07950.0263200θ0.50170.00170.00040.50280.00280.0004λ0.91990.01990.01540.8507−0.04930.0118In Table2, θ=2 and λ=−0.5 and 0.9.Table 2
MLE and MPS based on Type-II censored and complete sample whenθ=2.
θ=2MLEMPSλP (%)SchemeMeanBiasMSEMeanBiasMSE-0.570Type-II50θ2.03920.03920.12172.02050.02050.1186λ−0.44870.05130.1684−0.5323−0.03230.1478100θ2.04290.04290.06172.03340.03340.0599λ−0.47560.02440.0695−0.5160−0.01600.0649200θ2.03140.03140.02902.02650.02650.0286λ−0.48320.01680.0377−0.5030−0.00300.036290Type-II50θ1.9975−0.00250.07301.9708−0.02920.0699λ−0.47380.02620.1447−0.5365−0.03650.1295100θ2.02080.02080.03842.00650.00650.0369λ−0.49060.00940.0656−0.5209−0.02090.0619200θ2.01900.01900.01962.01140.01140.0191λ−0.49390.00610.0347−0.5086−0.00860.0337100Complete50θ1.9928−0.00720.06591.9399−0.06010.0640λ−0.47660.02340.1450−0.5269−0.02690.1292100θ2.01660.01660.03331.9830−0.01700.0317λ−0.49190.00810.0638−0.5125−0.01250.0603200θ2.01810.01810.01741.9983−0.00170.0167λ−0.49490.00510.0339−0.5037−0.00370.03300.970Type-II50θ1.9934−0.00660.04751.9930−0.00700.0347λ1.19450.29450.70940.8064−0.09360.0995100θ2.00680.00680.03232.00730.00730.0194λ1.00130.10130.11230.8380−0.06200.0388200θ2.01300.01300.00902.01050.01050.0089λ0.93240.03240.02860.8552−0.04480.018390Type-II50θ1.9732−0.02680.03861.9947−0.00530.0314λ1.04060.14060.31830.8016−0.09840.0743100θ2.00170.00170.01712.01060.01060.0169λ0.97190.07190.06210.8415−0.05850.0322200θ2.00910.00910.00782.01310.01310.0078λ0.92320.02320.01930.8589−0.04110.0143100Complete50θ1.9762−0.02380.03471.9943−0.00570.0311λ1.06000.16000.24190.7729−0.12710.0615100θ1.9997−0.00030.01712.01120.01120.0165λ0.95440.05440.04720.8199−0.08010.0267200θ2.00800.00800.00792.01400.01400.0077λ0.91990.01990.01590.8500−0.05000.0120Different 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. The optimum technique is the one that minimizes bias and mean squared error.From Tables1-2, we can conclude the following:(i)
Asn 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)
The MPS estimates have more relative efficiency than MLE.


### 6.1. Simulation Algorithm

In this part of the paper, we will discuss the simulation steps:(1)
The first step is to set the parameters’ starting values(2)
Second, generate a random sample of sizen 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(5)
Solve the likelihood equation and the MPS equations to get the estimates(6)
Do these stepsN = 1000 times, find the average values of the parameters every time, and compute any other measures related to the simulation


## 6.1. Simulation Algorithm

In this part of the paper, we will discuss the simulation steps:(1)
The first step is to set the parameters’ starting values(2)
Second, generate a random sample of sizen 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(5)
Solve the likelihood equation and the MPS equations to get the estimates(6)
Do these stepsN = 1000 times, find the average values of the parameters every time, and compute any other measures related to the simulation


## 7. 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. Three 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 Table3.Table 3
The summary statistics along with Skewness and Kurtosis of the data sets.
Data setMinQ1MedianMeanQ3MaxSkewnessKurtosisWheaton River0.1002.1259.5012.2020.1264.001.4422.725Extreme Wind Speed15.9225.8033.9041.4552.3494.340.885-0.009To 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.


### 7.1. The Wheaton River Data

The data from Bourguignon et al. [45] refer to the Wheaton River at Carcross, Yukon Territory, Canada, exceeding flood peaks (in m3/s). The data are comprised of 72 exceedances from 1958 to 1984, rounded to the nearest decimal place. This 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 log-likelihood values of the proposed model are shown in Table 4.Table 4
MLE of the parameters and respective SE with log-likelihood for selected distributions.
DistributionParametersEstimateSELog-likelihoodCubic transmuted Lindleyθ0.17310.0159− 252.207λ− 0.99990.2756Cubic ranked transmuted Lindleyθ0.18850010.0034− 253.723λ10.99990.4943λ2−0.99990.4943Transmuted Lindleyθ0.13860.0165− 263.115λ0.22180.2218Lindleyθ0.15300.0128− 264.202The 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.Figure 3
Estimated distribution functions for the selected models along with the proposed cubic transmuted Lindley distribution over the empirical distribution functions for the (a) Wheaton River and (b) Extreme Wind Speed data sets.
(a)(b)Figure 4
Estimated density functions for the selected models along with the proposed CT-Lindley distribution are plotted over for the (a) Wheaton River and (b) Extreme Wind Speed data sets.
(a)(b)The obtained values of the model selection criterion, shown in Table5, 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.Table 5
Model selection criteria for selected models.
Distribution−2 log-likelihoodAICAICcBICCubic transmuted Lindley504.415508.414508.588512.968Cubic ranked transmuted Lindley519.415525.9232526.588532.7532Transmuted Lindley526.230530.230530.404534.783Lindley528.405530.405530.462532.682


### 7.2. 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. This 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 describes the estimated distribution functions for the selected models along with the proposed cubic transmuted Lindley distribution over the empirical distribution functions for the (a) Wheaton River and (b) Extreme Wind Speed data sets.Table 6
MLE of the parameters and respective SE with log-likelihood for selected distributions.
DistributionParametersEstimateSELog-likelihoodCubic transmuted Lindleyθ0.04300.0034− 223.723λ0.99990.4943Ranked transmuted Lindleyθ0.09974190.0034− 243.123λ10.00000010.4943λ20.99999990.4943Transmuted Lindleyθ0.06440.0116− 224.415λ−0.99990.979Lindleyθ0.04712.1e − 05− 232.165The values of model selection criterion values are shown in Table7, which prove that the proposed CT-Lindley model provides better performance than other models in case of environmental data.Table 7
Model selection criteria for selected models.
Distribution−2 log-likelihoodAICAICcBICCubic transmuted Lindley447.447451.447451.691455.349Cubic ranked transmuted Lindley448.100493.582452.588499.436Transmuted Lindley448.831452.831453.076456.734Lindley464.330466.330466.410468.281


### 7.3. Fatigue Life

The data are given by Birnbaum and Saunders (19) on the fatigue life of 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle of 31,000 psi. The data are 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, and 173.40. The AIC for the proposed distribution is 230.0895, and the BIC is 232.3605, while for the cubic ranked transmuted Lindley, AIC is 232.4806, and BIC is 235.8871. As a result, we may conclude that the suggested distribution beats its rivals.


## 7.1. The Wheaton River Data

The data from Bourguignon et al. [45] refer to the Wheaton River at Carcross, Yukon Territory, Canada, exceeding flood peaks (in m3/s). The data are comprised of 72 exceedances from 1958 to 1984, rounded to the nearest decimal place. This 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 log-likelihood values of the proposed model are shown in Table 4.Table 4
MLE of the parameters and respective SE with log-likelihood for selected distributions.
DistributionParametersEstimateSELog-likelihoodCubic transmuted Lindleyθ0.17310.0159− 252.207λ− 0.99990.2756Cubic ranked transmuted Lindleyθ0.18850010.0034− 253.723λ10.99990.4943λ2−0.99990.4943Transmuted Lindleyθ0.13860.0165− 263.115λ0.22180.2218Lindleyθ0.15300.0128− 264.202The 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.Figure 3
Estimated distribution functions for the selected models along with the proposed cubic transmuted Lindley distribution over the empirical distribution functions for the (a) Wheaton River and (b) Extreme Wind Speed data sets.
(a)(b)Figure 4
Estimated density functions for the selected models along with the proposed CT-Lindley distribution are plotted over for the (a) Wheaton River and (b) Extreme Wind Speed data sets.
(a)(b)The obtained values of the model selection criterion, shown in Table5, 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.Table 5
Model selection criteria for selected models.
Distribution−2 log-likelihoodAICAICcBICCubic transmuted Lindley504.415508.414508.588512.968Cubic ranked transmuted Lindley519.415525.9232526.588532.7532Transmuted Lindley526.230530.230530.404534.783Lindley528.405530.405530.462532.682


## 7.2. 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. This 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 describes the estimated distribution functions for the selected models along with the proposed cubic transmuted Lindley distribution over the empirical distribution functions for the (a) Wheaton River and (b) Extreme Wind Speed data sets.Table 6
MLE of the parameters and respective SE with log-likelihood for selected distributions.
DistributionParametersEstimateSELog-likelihoodCubic transmuted Lindleyθ0.04300.0034− 223.723λ0.99990.4943Ranked transmuted Lindleyθ0.09974190.0034− 243.123λ10.00000010.4943λ20.99999990.4943Transmuted Lindleyθ0.06440.0116− 224.415λ−0.99990.979Lindleyθ0.04712.1e − 05− 232.165The values of model selection criterion values are shown in Table7, which prove that the proposed CT-Lindley model provides better performance than other models in case of environmental data.Table 7
Model selection criteria for selected models.
Distribution−2 log-likelihoodAICAICcBICCubic transmuted Lindley447.447451.447451.691455.349Cubic ranked transmuted Lindley448.100493.582452.588499.436Transmuted Lindley448.831452.831453.076456.734Lindley464.330466.330466.410468.281


## 7.3. Fatigue Life

The data are given by Birnbaum and Saunders (19) on the fatigue life of 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle of 31,000 psi. The data are 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, and 173.40. The AIC for the proposed distribution is 230.0895, and the BIC is 232.3605, while for the cubic ranked transmuted Lindley, AIC is 232.4806, and BIC is 235.8871. As a result, we may conclude that the suggested distribution beats its rivals.


## 8. Conclusions and Recommendations

The 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 real-life data sets that arise in various areas of life [47, 48].


## 9. 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.



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*Source: 3543517-2022-05-17.xml*