A New Sine-Based Probabilistic Approach: Theory and Monte Carlo Simulation with Reliability Application

Heydari, Tahereh; Zare, Karim; Shokri, Soheil; Khodadadi, Zahra; Almaspoor, Zahra

doi:https://doi.org/10.1155/2024/9593193

Journal of Mathematics

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Abstract Introduction Numerical Simulation Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 9593193 | https://doi.org/10.1155/2024/9593193

A New Sine-Based Probabilistic Approach: Theory and Monte Carlo Simulation with Reliability Application

Tahereh Heydari,¹Karim Zare,¹Soheil Shokri,²Zahra Khodadadi,¹and Zahra Almaspoor³

Academic Editor: Markos Koutras

Received24 Oct 2023

Revised09 Dec 2023

Accepted19 Dec 2023

Published22 Jan 2024

Abstract

Data modeling is a very crucial stage for decision making in applied sectors. Probability distributions are considered important tools for decision making. So far, numerous probability distributions have been developed and implemented. Most of these distributions are developed by introducing from one to eight additional parameters. Sometimes, the addition of new parameters leads to re-parameterization problems. To avoid such issues, we introduce a novel probabilistic approach. The proposed approach may be termed as a new weighted sine- method. The beauty and key advantage of the new weighted sine- method are that it has no additional parameters. Through using the new weighted sine- method, a new weighted sine-Weibull distribution is introduced, which is a modification of the Weibull distribution. The estimators of the new model are also derived. Furthermore, a simulation study is carried out to evaluate the estimators of the new weighted sine-Weibull distribution. Finally, a practical application from the reliability sector is considered to evaluate the new weighted sine-Weibull distribution. Based on certain decision tools, it is observed that the proposed model is the best competing distribution for applying it in the reliability sector.

1. Introduction

In practical scenarios, the Weibull distribution, its special cases such as Rayleigh and exponential distributions, and other existing probability models such as Gamma and Beta distributions are frequently implemented for data modeling. These probability distributions have greater applicability in applied sectors, especially, in the engineering sector. Among the mentioned distributions, the Weibull model holds a special place. For some established studies and practical applications of the Weibull distribution, see Basu et al. [1]; Bhattacharya and Bhattacharjee [2]; and Azad et al. [3].

Suppose has the Weibull distribution; then, its CDF is

The probability density function (PDF) corresponding to equation (1) is expressed by

In numerous applied areas, due to the complex form of data, the Weibull distribution does not give sufficiently good results (i.e., best fitting). Therefore, to capture the complex form of data, numerous updated and flexible modifications of equation (1) have been studied. For example, Vanem and Fazeres-Ferradosa [4] studied the truncated translated Weibull model, Alotaibi et al. [5] considered the alpha power Weibull distribution to deal with the engineering datasets, Thach [6] proposed a three-component additive Weibull distribution and applied it for modeling the reliability datasets, Abd El-Monsef et al. [7] analyzed a reliability dataset using the Poisson modified Weibull model, Dessalegn et al. [8] introduced the modified Weibull model for the bamboo fibrous dataset, Rehman et al. [9] analyzed the failure data by using another modified Weibull model, and Li et al. [10] developed a three-parameter Weibull distribution for modeling the fracture datasets.

In the recent time, researchers have given great devotion to proposing new statistical methodologies, and then they considered the special cases of the proposed methods (see Hassan et al. [11]; Eghwerido et al. [12]; Altun et al. [13]; Nakamura et al. [14]; and Kundu [15]). Most of these special cases are introduced by incorporating one to eight additional parameters [16].

In the most recent time, researchers are focusing on using trigonometric functions to develop new statistical methodologies (see Mahmood et al. [17]; Rajkumar and Sakthivel [18]; and Al-Babtain et al. [19], among others).

Recently, Kumar et al. [20] explored a modern approach using a trigonometric function, especially, a sine function. The CDF of their proposed approach based on the sine function iswith PDF expressed bywhere .

In this paper, we also introduce another approach using the sine function to generate flexible probability models, namely, a new weighted sine- (NWS-) method. The beauty of the NWS- family is that it does not involve extra parameters. The NWS- method can be considered a weighted version of the CDF provided in equation (2).

Definition 1. The CDF and PDF of the NWS-G method are, respectively, expressed byandThe survival function (SF) of the NWS- method isThe hazard function (HF) of the NWS- method isThe cumulative hazard function (CHF) of the NWS- method isSection 2 is designed for calculating the mathematical properties of the NWS- distributions. In Section 3, we use the approach defined in equation (5) to study a novel modification of the Weibull distribution. Certain estimation methods are discussed in Section 4. Section 5 provides a comprehensive and exhaustive simulation study. Section 6 is reserved for carrying out the core aim of illustrating the NWS-Weibull distribution. Finally, certain remarks about this work are presented in Section 7.

2. Mathematical Properties

We derive different mathematical properties of the NWS- distributions, involving the quantile function (QF), moment, moment generating function (MGF), and probability-weighted moments (PWMs).

2.1. The QF

Let follow the NWS- distributions, and its QF is derived by inverting equation (5), expressed by

Equation (10) shows that the QF of the NWS- distributions is not in straightforward form (i.e., explicit form). Thus, for obtaining random numbers from any member of the NWS- family, we would need to use numerical methods by incorporating computer software.

2.2. The Moment

In this subsection, we derive the moment of the NWS- family. Suppose follows the NWS- family with PDF in equation (6); then, the moment of , expressed by , is derived aswhere represents the support of . Using equation (6) in equation (11), we obtain

Use the series , given by

Let in equation (13). Then, from equation (13), we get

Using equation (14) in equation (12), we get

Use the series , given by

Let and in equation (16). Then, from equation (16), we get

Using equation (17) in equation (15), we getwhere

From expression in equation (19), we can see that is the exponentiated version of with exponentiated parameter .

The moment can further be used to derive certain characteristics for any special model of the NWS- family.

2.3. The MGF

In this subsection, we derive the MGF of the NWS- family. Let follow the NWS- family with PDF in equation (6); then, the MGF of , expressed by , is derived as

Using equation (6) in equation (20), we obtain

On substituting, we get

2.4. The PWMs

In this subsection, we derive the PWMs of the NWS- family. Suppose follow the NWS- family with PDF in equation (6); then, the PWMs of , expressed by , are derived as

Using equation (6) in equation (23), we obtainwhere

From the expression in equation (25), we can see that is the exponentiated version of with exponentiated parameter .

3. A NWS-Weibull Distribution

Here, we define certain key functions of the NWS-Weibull distribution. In this regard, we first define the CDF of the NWS-Weibull distribution by using equation (1) in equation (5), as given byand PDF

Figure 1 illustrates some visual illustrations of of the NWS-Weibull distribution. The pictorial illustrations of are obtained for and . Figure 1 pictorially describes that of the NWS-Weibull distribution can capture four useful shapes. The possible shapes of include decreasing shape (magenta outline), right-skewed shape (blue outline), left-skewed shape (red outline), and symmetrical form (green outline).

Furthermore, some pictorial illustrations of of the NWS-Weibull distribution are given in Figure 2. The pictorial visualization of is obtained for and . Figure 2 vividly describes that of the NWS-Weibull distribution can capture three useful shapes. The possible shapes of include decreasing behavior (magenta outline), unimodal shape (blue outline), and increasing form (red outline).

Moreover, the SF, HF, and CHF of the NWS-Weibull model are, consequently, expressed byrespectively.

4. Estimation Methods and Simulation Studies

In this section, we implement numerous estimation methods to obtain the estimators of the NWS-Weibull distribution. These methods are based on the maximization or minimization of an objective function.

4.1. Maximum Likelihood Estimation

Here, we derive the estimators (i.e., maximum likelihood estimators (MLEs)) of the NWS-Weibull distribution by maximizing the log-likelihood function (LLF) of the NWS-Weibull distribution. Corresponding to in equation (27), the LLF is given by

Maximizing with respect to and , we get the estimators and , respectively. The simulation results based on the maximum likelihood estimation method are expressed by in Tables 1–6.

4.2. Anderson–Darling (AD) Estimation

The Anderson–Darling estimators (ADEs) of the NWS-Weibull distribution can be obtained by optimizingor

Corresponding to the AD estimation method, the simulation results are expressed by in Tables 1–6.

4.3. Cramér–von Mises (CVM) Estimation

In this subsection, we discuss the Cramér–von Mises (CVM) estimation approach to obtain the Cramér–von Mises estimators (CVMEs) of the NWS-Weibull distribution. and of the NWS-Weibull distribution can be obtained by optimizingor

Corresponding to the CVM estimation method, the simulation results are expressed by in Tables 1–6.

4.4. The Maximum Product of Spacing (MPS) Estimation

The maximum product estimators (MPEs) of the NWS-Weibull distribution can be obtained by optimizing

Corresponding to the MPS estimation method, the simulation results are expressed by in Tables 1–6.

4.5. Ordinary Least-Squares Estimation

In this subsection, we discuss the approach of the ordinary least-squares (OLS) estimation to obtain the ordinary least-square estimators (OLSEs) of the NWS-Weibull distribution. The and of the NWS-Weibull distribution can be derived by optimizingor

Corresponding to the OLS estimation method, the simulation results are expressed by in Tables 1–6.

4.6. Percentile (PC) Estimation

The percentile estimators (PCEs) of the NWS-Weibull distribution can be obtained by optimizingor

Corresponding to the PC estimation method, the simulation results are expressed by in Tables 1–6.

4.7. Right-Tailed Anderson–Darling Estimation

In this subsection, we discuss the right-tailed Anderson–Darling (RAD) estimation approach to obtain the right-tailed Anderson–Darling estimators (RADEs) of the NWS-Weibull distribution. and of the NWS-Weibull distribution can be obtained by optimizing the following expression:or

Corresponding to the RAD estimation method, the simulation results are expressed by in Tables 1–6.

4.8. Weighted Least-Squares Estimation

Here, we discuss the weighted least-squares (WLS) estimation approach to obtain the weighted least-square estimators (WLSEs) of the NWS-Weibull distribution. and of the NWS-Weibull distribution can be obtained by optimizing the following expression:or

Corresponding to the WLS estimation method, the simulation results are expressed by in Tables 1–6.

4.9. Left-Tailed Anderson–Darling (LAD) Estimation

The left-tailed Anderson–Darling estimators (LADEs) of the NWS-Weibull distribution are obtained by optimizingor

Corresponding to the WLS estimation method, the simulation results are expressed by in Tables 1–6.

4.10. Minimum Spacing Absolute Distance Estimation

In this subsection, we discuss the minimum spacing absolute distance (MSAD) estimation approach to obtain the minimum spacing absolute distance estimators of the NWS-Weibull distribution. and of the NWS-Weibull distribution can be obtained by optimizing the following expression:

Corresponding to the MSAD estimation method, the simulation results are expressed by in Tables 1–6.

4.11. Minimum Spacing Absolute-Log Distance Estimation

The minimum spacing absolute-log distance estimators of the NWS-Weibull distribution can be obtained by optimizing

Corresponding to the MSAD estimation method, the simulation results are expressed by in Tables 1–6.

5. Numerical Simulation

We examine the behaviors of and of the NWS-Weibull distribution, which is obtained by utilizing different estimation approaches in Section 3. The performances of and through the simulation results for a sample of size are assessed. The evaluation of and is done using different criteria. These criteria are given by(i)Absolute bias (AB):(ii)Mean squared error (MSE):(iii)Mean absolute relative error (MARE):

The key goals of carrying out the simulation study are to (i) evaluate the behaviors of the estimators and (ii) find out the best estimation method for estimating the parameters of the NWS-Weibull distribution.

For the NWS-Weibull distribution, the simulation results are obtained by using the statistical package of version along with the function.

The simulation results are shown in Tables 1–5. The numerical numbers, presented in superscripts in Tables 1–5, represent the ranks of the estimation methods. Furthermore, the partial ranks and total ranks for the implemented estimation methods are presented in Table 6.

Tables 1–6 reveal that as increases, we can easily reach the following conclusions:(i)The consistency attribute is demonstrated by each estimator.(ii)The AB of all estimators reduces.(iii)The MSE of all estimators reduces.(iv)The MARE of all estimators reduces.(v)The maximum product of the spacing estimation method is the best estimation strategy. We thus encourage researchers to use this approach if they have datasets from the NWS-Weibull distribution.

6. Data Modeling

This section offers the practical importance (using numerical tools and visual illustration) of the NWS-G method by considering the NWS-Weibull distribution. To carry out the practical illustration of the NWS-Weibull distribution, we choose a practical dataset taken from the engineering sector. It has one hundred observations and describes the breaking stress of carbon fibers (measured in Gba) (see Nichols and Padgett [21]). Researchers who previously considered this dataset include Barreto-Souza et al. [22] and Oseghale and Akomolafe [23]. The dataset is given in Table 7. Figure 3 describes the descriptive plots of the data.

(a)

(b)

(c)

(d)

To establish the practical edge of the NWS-Weibull distribution over other models, we select three existing models as competitive models. The competitive models include the Weibull distribution, its known flexible extension, namely, the flexible Weibull extension (FW-Extension), weighted sine-Weibull (WS-Weibull), and exponentiated Weibull (E-Weibull) distribution. The SFs of these distributions are(i)Weibull distribution:(ii)FW-Extension distribution:(iii)WS-Weibull distribution:(iv)E-Weibull distribution:

For the considered dataset, the discrimination among the NWS-Weibull distribution and above selected competing distributions is made using seven statistical tools with the value. Out of these seven statistical tools, we have four statistical criteria (also called information criteria) and three statistical tests (i.e., goodness-of-fit tests) with the value.

The values of the selected information criteria are calculated using the formulas(i)Akaike information criterion (AIC):(ii)Bayesian information criterion (BIC):(iii)Consistent Akaike information criterion (CAIC):(iv)Hannan–Quinn information criterion (HQIC):

In the above formulas of the information criteria, represents the sample size and denotes the model parameters. Furthermore, the selected statistical test values are computed using the formulas(i)Anderson–Darling (AD) test:(ii)Cramér–von Mises (CM) test:(iii)Kolmogorov–Smirnov (KS) test:

In the above formulas of the statistical tests, represents the sample size, represents the CDF of the fitting model, represents the empirical CDF, and represents the distance between and . It is important to keep in view that a particular model having the smallest values of the above statistical tools is taken as the best suitable model among the class of the fitted/applied distributions.

The MLEs, information criteria, and statistical test values are obtained by implementing the library with the help of -function of version with (see Appendix).

After conducting the numerical study, Table 8 reports the values of , and of all the fitting competing probability models. Furthermore, the values of the information criteria and statistical tests for all distributions are, respectively, presented in Tables 9 and 10. Looking at the reported results in Tables 9 and 10, it is clearly seen that the NWS-Weibull model can be ranked first (first rank in the sense that it has the highest value and the lowest information criterion and statistical test values).

We see that Tables 9 and 10 clearly support the claim of the suitability of the NWS-Weibull distribution for the engineering data. After the numerical evaluation of the NWS-Weibull model, we also show the best-fitting assertion of the NWS-Weibull distribution for the carbon fiber data using visual illustrations. To meet the best-fitting assertion of the NWS-Weibull distribution visually, we consider the empirical CDF, estimated PDF, quantile-quantile (QQ), and Kaplan–Meier survival plots (see Figure 4). In regard to the obtained visual illustrations in Figure 4, the NWS-Weibull distribution is shown to closely fit the engineering dataset.

7. Concluding Remarks

A new distributional approach termed as new weighted sine- method was proposed and studied. It was obtained by incorporating the sine function. The NWS- method was introduced without using any additional parameters. Using the new weighted sine-G method, a novel generalized form of the Weibull distribution named NWS-Weibull distribution was studied. The estimators of the NWS-Weibull model were derived using eleven different estimation methods. Furthermore, Monte Carlo simulation studies were incorporated using these estimation methods. The advantage and practical significance of the new distribution were established by inspecting a practical application taken from the engineering sector. Based on the seven statistical measures and the value, it was numerically and visually shown that the NWS-Weibull distribution is a more apt model for the engineering dataset.

Appendix

In the below code, represents , represents , and x is used for . data = c (3.7, 2.74, 2.73, 2.5, 3.6, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.2, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.7, 2.03, 1.8, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65) ################ Proposed Model: PDF pdf_pm < −function (par, x) { a = par [1] d = par [2] (pi/2) a d (x^(a−1)) exp (−d x^a) cos((pi/2) (1 − exp (−d x^a))) exp(1 − (sin ((pi/2) (1 − exp (−d x^a))))) (1 − sin ((pi/2) (1 − exp (−d x^a)))) } ################ Proposed Model: CDF cdf_pm < −function (par, x) { a = par[1] d = par[2] sin((pi/2) (1 − exp (−d x^a))) exp (1 − sin((pi/2) (1 − exp (−d x^a)))) } set.seed (0) goodness.fit (pdf = pdf_pm, cdf = cdf_pm, starts = c (1.2, 1.2), data = data, method = “SANN,” domain = c (0, Inf), mle = NULL)

Data Availability

The dataset is provided within the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2024 Tahereh Heydari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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