Abstract
We propose a trigonometric generalizer/generator of distributions utilizing the quantile function of modified standard Cauchy distribution and construct a logistic-based new G-class disbursing cotangent function. Significant mathematical characteristics and special models are derived. New mathematical transformations and extended models are also proposed. A two-parameter model logistic cotangent Weibull (LCW) is developed and discussed in detail. The beauty and importance of the proposed model are that its hazard rate exhibits all monotone and non-monotone shapes while the density exhibits unimodal and bimodal (symmetrical, right-skewed, and decreasing) shapes. For parametric estimation, the maximum likelihood approach is used, and simulation analysis is performed to ensure that the estimates are asymptotic. The importance of the proposed trigonometric generalizer, G class, and model is proved via two applications focused on survival and failure datasets whose results attested the distinct better fit, wider flexibility, and greater capability than existing and well-known competing models. The authors thought that the suggested class and models would appeal to a broader audience of professionals working in reliability analysis, actuarial and financial sciences, and lifetime data and analysis.
1. Introduction and Motivations
Generalizing a classical distribution is an old practice in distribution theory. The earliest work regarding generalizing a distribution was conducted by Pearson [1] using differential equations. Since 1985, the families (or classes) of distributions have been derived adopting the following famous methodologies: differential equation method, transformation techniques, compounding methodology, skewed distributions generation, parametric induction, quantile based approach, transformed (T)-transformer (X) mechanism, exponentiated T-X system, T-R\{Y\} approach, etc., It is possible to more readily manage data that is highly skewed with each new model that is developed because of its enhanced heavy-tailed, tractable core functions, and simplified simulation technique.
The literature review explores the following four critical points which prove the basis for this study. (i) In the statistical literature, a slew of new families and models for algebraically generalizers/distribution’s generators have been introduced in contrast of neglecting trigonometric equations (trigonometric generalizers). (ii) The interest in modelling directional and proportional data led applied researchers to develop and employ trigonometry function-based models capable of handling these datasets more smoothly and economically. (iii) The use of algebraic and trigonometric functions in mixture generalizers is still to be researched and investigated. (iv) Using the stated transformation, any current G class or model may be simply reversed in a subsequent version. The major motives are considered to be influenced and inspired by the generated results in terms of accuracy, adaptability, and goodness of fit (gof) which are included below:(i)To introduce a generator of distributions based on cotangent function (that is a combination of algebraic and trigonometric functions and generators concurrently).(ii)To introduce a new G-class called a new logistic-G class of distributions (LCG for short) in trigonometric scenario.(iii)There are several advantages to the suggested class, including its simplification, lack of non-identifiability, and lack of over-parametrization.(iv)Because of the injection of the cotangent function, the new CDF can increase flexibility, giving rise to new efficient and flexible models.(v)Non-generalized and non-exponentiated models, according to the literature, give insufficient gof.
The famous generalizers and corresponding G-families are presented in Table 1.
Moreover, the new generators and G-classes are introduced regarding several purposes. Among them, the recent and remarkable include the following. (A) To present a new family utilizing the additive model structure (see [6]). (B) New generator is defined with quantile function (see [7]). (C) To achieve better gof and more flexibility than existing classical models (see [8]). (D) Type I half logistic Burr XG family has been constructed by Algarni et al. [9]. (E) The bivariate Weibull-G family based on copula function using odd classes has been introduced by El-Sherpieny et al. [10]. (F) A significant amount of distribution families was proposed using parameter induction (inserting one or maybe more new parameters (s) to the baseline), for example, a new one was put forward by Cordeiro et al. [11]. (G) Adopting the T-X family methodology only (see [12]). (H) Presenting a flexible family which deals with both monotone and non-monotone hazard rate function (see [13]). (I) Using a flexible family to introduced flexible generalized Pareto distribution (see [14]). The whole real line interval distributions naturally come up when random variables should vary in the infinite real interval and several distributions such as logistic, normal, Laplace, t, Chen, and Gumbel distributions are supported on this interval. It has been used to describe the distribution of income and wealth in a fairly basic way. The logistic distribution has numerous uses in statistical analysis and has a form that resembles the normal distribution.
A previous study [15] introduced logistic distribution whose main functions (cdf and pdf) in a new format are given below:
Alzaatreh et al. [16] contributed transformed (T)-transformer (X) family of distributions (for short, T-X family) which expanded vision about generators of distributions , and in this article, via the T-X method, a new logistic-G class of statistical distributions is proposed adopting a cotangent-based trigonometric generator .
This paper is outlined as follows. Section 1 is about introduction and motivations while in Section 2, cotangent generator and LCG class are developed. In Section 3, some special models are presented while in Section 4, the characteristics of LCG class are deduced. In Section 5, a new model LCW along with significant properties is discussed and a simulation work is performed in Section 6. The importance of the new class and model is confirmed by two real-life applications using failure and survival datasets in Section 7. Finally, in Section 8, the conclusions are presented.
2. Development of Cotangent Generator
In this part, let us assume that is a random variable (r.v.) that follows the famous distribution, the standard Cauchy distribution, and its location parameter has value equal to while the other parameter which is called the scale parameter has value equal to ; then, its cdf and qf, respectively, are
Replacing by , , a new trigonometric generator based on cotangent function is achieved having support .
2.1. Genesis of LCG Class
By assuming that is considered as the pdf of a r.v. for and , it fulfills the T-X family’s following requirements. (i) Since , then . (ii) is differentiable and monotonically non-decreasing because cotangent function is differentiable and monotonically non-decreasing on . (iii) Since such that , then and . Similarly, as ; then, and .
Now, the main functions of LCG class in T-X format can be written as
2.1.1. LCG Class in Cotangent Scenario
Let be a logistic r.v. with cdf and pdf . Putting in (4), the cdf of new class is obtained and presented as
The pdf corresponding to (7) reduces to
2.1.2. LCG Class in Tangent Scenario
Putting in (4), the cdf of new class may be expressed as below:
The pdf corresponding to (9) reduces to
3. Special Models
Some special models of LCG class with their main functions and corresponding graphs are presented subsequently.
3.1. The Logistic Cot Exponential (LCE) Distribution
By assuming that the variable follows an exponential distribution, then we may be able to express the central functions for the LCE distribution in the form as follows:
3.2. The Logistic Cot Lindley (LCLi) Distribution
Assuming is a Lindley random variable, we can write the new one-parameter LCLi model that has the following cdf, pdf, and hazard function. Figure 1 demonstrates the graph plots of the LCE, and Figure 2 demonstrates the the graph plots of the LCLi.
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3.3. The Logistic Cot Gamma (LCGa) Distribution
Let be a gamma random variable; then, the LCGa model has the following main functions:
The plots of the LCGa are shown as graphs in Figure 3.
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3.4. The Logistic Cot Dagum (LCD) Distribution
By assuming that for the possibility that is a Dagum random variable, then the LCD distribution has the following primary functions:
The graphs in Figure 4 show the plots of the LCD.
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4. Mathematical Properties of LCG Class
Here, important properties of the new class are presented.
4.1. The Inverse Function for Both pdf and cdf (Quantile Function)
We present an additional property of which is qf:
Here, is the parental qf, whereas . So, we can write the quantile density function as follows:
4.2. Useful Reliability Functions
In this part of the paper, we will concentrate our efforts to introduce the most important reliability functions. First we will define the survival function (sf) , after that we write the equation of the hazard rate function (hrf) , and the reversed hazard rate function is as below, and the cumulative hazard rate function (chrf) and mills’ ratio are, respectively, given below.
4.3. The Density Function Analytical Formulas
By solving (18), we can get the roots of the equation, and we can call it the solutions of the density function of the new class; it is possible to provide an analytical description of the forms of the new density:
(18) can be multi-rooted equation. Let ; then,
4.4. The Hazard Function Analytical Formulas
In order to find the roots of (20), we must obtain the crucial points of the hrf .
As we can see, this equation has many solutions or we can say many roots are available for this equation. Suppose that . We have
Using any numerical software and (18) and (20), we can investigate local maximums and minimums, as well as inflexion points.
4.5. Linear Representation
The cdf of LCG class presented in (7) can be written as follows:
Using the relation may be easily formulated as
Through WolframAlpha, for. We can demonstrate the cdf (2.6) in this form after applying this series and exponent series , respectively.
Now, for the term , we will use the power series to expand it such that , , , etc,. are obtained by the aid of the highly speed MATHEMATICA software see Tahir [17]. Hence, and
We know that is considered as the exponentiated distribution function parameter in the power and
With the expansion of (8), the following formula may be obtained from the previously mentioned idea of exponentiated distributions:such that can be noted as the exponentiated density having a power parameter andand
4.6. Moments, Incomplete Moments, Mean Deviation, Skewness, and Kurtosis
Suppose that we have a random variable called that follows the the exp-G distribution that has a power parameter , i.e., having density . Now we gave two formulas for the th moment of which follows (27), and the first one can expressed as follows:
The th moment of can be written in a second form as it may be deduced from (30) in terms of the G qf like the following equations:where and . These integrals can be calculated numerically.
4.7. Weighted Moments
In this section, we introduce the equation of the weighted moments. So, we can write the probability weighted moment (PWM) of as
Then,
Putting the pdf of LCG class (given below)in (32), after applying the binomial expansion and exponential series, we get
Regarding , we can use power series expansion such that , , , etc. Similarly, for , we getwhere .
4.8. Generating Function
This section is devoted to introduce the moment generating function (mgf), and it can be written as follows:
4.9. Order Statistics
In this section, we will focus our attention on one of the most important properties which is the order statistics. We can easily write the form of order statistic density function as follows:such that .Considering the mathematical advancements described in PWM, becomessuch that is exp-G density possessing power parameter for .and
It should come as no surprise that the density of the LCG order statistics is a linear combination of exp-G densities; this is extremely evident, as revealed by (39), which is the main result to be demonstrated.
4.10. Entropy Measures
The Shannon entropy is defined as and for the LCG class, it may be formulated as the following equation:
Proof. Alzaatreh et al. [16] deduced Shannon entropy of T-X family. Since here , adopting the same methodology, we get aswhere is the mean of r.v. . Using (43), we can easily prove the Shannon entropy of the LCG class given in (42) where follows logistic distribution.
Rényi entropy issuch that and .We get the following result after performing the expansion by the aid of power series:After incorporating the result, the Rényi entropy will reduce toNow, we can write the final equation of the Rényi entropy as follows:where .
4.11. Some Transformations to Develop New G-Classes
(i)If the r.v standard logistic distribution, then = logistic cotangent-G class of distributions.(ii)If the r.v standard logistic distribution, then = logistic cotangent exponentiated-G class of distributions.(iii)If the r.v standard logistic distribution, then = generalized logistic cotangent-G class of distributions.(iv)If the r.v standard logistic distribution, then = generalized logistic cotangent exponentiated-G class of distributions.5. LCW Distribution
In this section, a two-parameter special model logistic cot Weibull (LCW) with its properties is presented.
5.1. Methodology
Taking , as the cdf of Weibull distribution while as the corresponding density, it follows the two-parameter LCWm we can write three of cdf, and its corresponding pdf and associated with its hazard function, as formulated below respectively, are:
The graphs represented in Figure 5 demonstrate the plots of the LCW.
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5.2. Reliability Functions
We have
5.3. Quantile Function
This section is devoted to demonstrate the quantile function equation of LCW which iswhere is the parent qf and .
5.4. Residual and Reverse Residual Life
The residual life has several uses in probability and statistics and risk assessment. Suppose that represents a unit’s lifespan and with ; then, the r.v. , for a fixed , is known as time since failure. The residual lifetime of LCW r.v. is denoted by and is defined as
Additionally, the reversed hazard rate function is written as
5.5. Stochastic Ordering
Stochastic ordering is a tool for analysing the constitutional properties of stochastic structures that are intricate. There are several forms of stochastic orderings that can be used to sort random variables according to their distinguished properties. Suppose and are independent random variables with cdfs and , respectively; then, is said to be smaller than iff it satisfies the following.(i)Stochastic order (S (K)) if for all .(ii)Hazard rate order (S (K)) if for all .(iii)Mean residual life order (S (K)) if for all .(iv)Likelihood ratio order (S (K)) if decrease in .
The LCW distribution is ordered according to the strongest “likelihood ratio” ordering, as demonstrated in the following theorem. The versatility of two-parameter LCW distribution is demonstrated. Let follow LCW and follow LCW . Then, the likelihood ratio iswhere and . Again,
If and , then , and hence , and .
5.6. Stress-Strength Reliability
In this section, we will introduce and define one of the most important properties of any distribution which is the reliability function ; this function may be represented as
Suppose both and are LCW independent random variables with parameters and and fixed scale parameter . Then,
After applying the binomial and exponent series expansion, then substituting and , the above equation reduces to
Solving complicated integration is very hard but with the aid of super computers and advanced mathematical software, we easily find the value of the hard integral introduced above (Table 2).
5.7. Submodels of LCW
5.8. Estimation
One of the most famous estimators is the maximum likelihood equation as we can easily obtain and write the log-likelihood function regarding the distribution’s parameters of the distribution under consideration by getting the logarithms function for the likelihood function . So as a result, we may formulate the log-likelihood function as follows:
Consider the following formulae to be the score vector's components , or in other words the derivative of the vector with respect to the two parameters:
The MLEs may be derived by putting the equations in the previous sentence to zero and solving them concurrently (see, for instance, [18]).
6. Results Deduced from the Simulation Work
In this phase of the study, we employed Monte Carlo simulation to assess the distribution’s effectiveness all across the estimation procedure. The MLEs of the model parameters of the LCW distribution are evaluated. For each sample size , , this simulation study is repeated times. The parametric values are (I) , , (II) , , (III): , , and (IV) , . The outputs related to the biases of the MLEs, mean square errors (MSEs), lower bounds, and upper bounds are displayed in Tables 3 and 4. As the sample size increases, in general, the biases, MSEs, L.bounds, and U.bounds of decrease while the CPs of the confidence intervals are quite close to the 95% nominal levels which indicate that the MLEs have good performance for estimating the parameters of the LCW distribution; also, we will use the conducted results to find the upper and lower bounds for the estimates for the parameters of the model. Tables 3 and 4 contain a summary of all simulation outcomes.
7. Applications and Data Analysis
Two different applications to actual datasets were shown by us in order to demonstrate the utility of the distribution that was suggested (LCW). The criteria of goodness of fit proved that it can be used in place of famous two, three, and four-parameter models and many others. All calculations are performed using the R script Adequacy Model. Moreover, the proposed LCW is compared with McDonald Weibull (McW), Exponentiated Kumaraswamy Weibull (EKumW), Burr Weibull (BurrW), Beta Weibull (BW), Gompertz Weibull (GoW), Exponentiated Odd Weibull (EOW), Topp-Leone Weibull (TLW), Odd Log Logistic Weibull (OLLW), Marshall Olkin Weibull (MOW), and many other Weibull based models.
7.1. Application 1: Failure Time Data
The first data correspond to the failure time of 50 components (per 1000 h) which were collected from Murthy et al. [19]. The dataset can be found easily in [19]. We avoid adding the data in the paper as they can be easily accessed. We have provided some statistics on the data used to make the reader comfortable in reading the paper. The summary statistics for this dataset are as follows: = 50, = 1.4140, = 3.3422, = 4.18, = 0.2075, = 4.4988, skewness = 1.38, and kurtosis = 0.92.
We can easily recognize that Figure 6 represents the graphical representation of the histogram, TTT plot, box plot, and kernel density for failure time data. Figure 7 represents the comparative cdf and pdf of LCW and other models using failure time data.
Table 5 provides the MLEs of the parameters while Table 6 provides the values of AIC, CAIC, BIC, HQIC, , , K-S, and values for each model. On the basis of the statistics given in these tables, the best fit model is LCW and has the potential to fit right-skewed data with the increasing failure rate.
7.2. Application 2: Survival Time Data
The upcoming data are for the mortality periods, measured in weeks, of 33 individuals with acute myelogenous leukemia which were used by Feigl and Zelen [20]. The dataset is given below. 65, 156, 100, 134, 16, 108, 121, 4, 39, 143, 56, 26, 22, 1, 1, 5, 65, 56, 65, 17, 7, 16, 22, 3, 4, 2, 8, 4, 3, 30, 4, 43.
The summary statistics for this data set are as follows: = 33, = 22.00, = 42.06, = 46.94, = 4.0, = 65.00, skewness = 1.07, and kurtosis = -0.16. Figure 8 represents the histogram, TTT plot, box plot, and kernel density for survival time data. Figure 9 represents the comparative cdf and pdf of LCW and other models using survival time data.
Table 7 provides the MLEs of the parameters, and Table 8 contains the values of AIC, CAIC, BIC, HQIC, , , K-S, and P values for each model. On the basis of the statistics presented in these tables, the best fitted model is LCW and has the potential to fit right-skewed data with increasing failure rate.
8. Concluding Remarks
A novel logistic-G family of distributions is developed, which employs trigonometric and algebraic generalizers based on cotangent functions. This class has been shown to be more adaptable and useful in a variety of practical applications, particularly survival, dependability, and failure modelling. Furthermore, a two-parameter model (LCW) with various density shapes, as well as hazard rate different shapes, is developed. This work also derives and presents many statistical and mathematical properties of the proposed family.
In parametric estimating, the maximum likelihood method is used, and a Monte Carlo simulation analysis is used to determine whether or not the estimates are suitable. To ascertain which distribution is most suitable for modelling the real datasets, we employ a number of goodness-of-fit measures that decide which one is the superior one among all its competitors. We show that, even with a higher number of parameters, this suggested distribution consistently delivers superior fits than other existing and competing Weibull models. We believe that the suggested class and related models would find wider applicability in sectors such as dependability and survival studies, hydrology, geology, and others.
9. Future Work
In the upcoming work, we will apply the proposed distribution and the new family of distribution to censored sample scheme. We will try different kinds of censoring schemes like type-I and type-II censored sample and we will generate random censored samples from the new distribution. We can extend our work to apply the proposed model to accelerated life test with different types such as constant and partially constant and maybe progressive stress accelerated life tests. At last, we will use different optimality criteria to the censored samples generated from the proposed model.
Data Availability
The data that were utilised to support the conclusions of this research may be found inside the paper itself.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work (grant code: 22UQU4290034DSR01).