A New Generator of Probability Models: The Exponentiated Sine-G Family for Lifetime Studies

In this article, we propose the exponentiated sine-generated family of distributions. Some important properties are demonstrated, such as the series representation of the probability density function, quantile function, moments, stress-strength reliability, and Rényi entropy. A particular member, called the exponentiated sine Weibull distribution, is highlighted; we analyze its skewness and kurtosis, moments, quantile function, residual mean and reversed mean residual life functions, order statistics, and extreme value distributions. Maximum likelihood estimation and Bayes estimation under the square error loss function are considered. Simulation studies are used to assess the techniques, and their performance gives satisfactory results as discussed by the mean square error, confidence intervals, and coverage probabilities of the estimates. The stress-strength reliability parameter of the exponentiated sine Weibull model is derived and estimated by the maximum likelihood estimation method. Also, nonparametric bootstrap techniques are used to approximate the confidence interval of the reliability parameter. A simulation is conducted to examine the mean square error, standard deviations, confidence intervals, and coverage probabilities of the reliability parameter. Finally, three real applications of the exponentiated sine Weibull model are provided. One of them considers stress-strength data.

Let us now perform a basic quantile analysis of the ESG family. Firstly, we can use the quantile function (QF) of the distribution to analyze its skewness and kurtosis; it also serves as a means of parameter estimation. Moreover, it is used for generating random data that follows a specified distribution. The QF aided in the computations of some distribution properties and goodness-of-fit measures. After some developments, the QF of the ESG family is given by Q(p) = Q(p; ξ) = G −1 2 π arcsin(p 1/α ) ; ξ , 0 < p < 1, where G −1 (x; ξ) is the QF related to G(x; ξ). In particular, the median is Obviously, the definition of G(x; ξ) is crucial to the modeling capabilities of the ESG family.
A precise example will be presented later.

Useful Expansion
In this part, we present the PDF of the ESG family in a series expansion form. To achieve this aim, some preliminary results are required. First, we recall that the sine and cosine series expansions are given by sin x = ∑ ∞ n=0 (−1) n x 2n+1 /(2n + 1)! and cos x = ∑ ∞ n=0 (−1) n x 2n /(2n)!. We will also need the power series raised to the power of integer as given by [15]. It is recalled in the following lemma. Lemma 1 ([15]). For a given power series of the form ∑ ∞ k=0 a k x k , where n denotes a positive integer, we have where c 0 = a n 0 , c m = [1/(ma 0 )] ∑ m k=1 (kn − m + k)a k c m−k for m ≥ 1.
The following lemma is for future technical purposes.

Lemma 2. We have
Proof. We can prove that A n is a convergent alternating series via standard convergence criteria, and the value of A n can be verified from https://www.wolframalpha.com/widgets/ gallery/view.jsp?id=d983db47634e1936eb568b79884012ac (accessed on 18 August 2021) or https://www.emathhelp.net/calculators/calculus-2/series-calculator/ (accessed on 18 August 2021).
A series expansion of the exponentiated version of a sine term is provided below.
Proof. We can write the sine series expansion by separating the first term as By considering Equation (4), we can conclude that ∑ ∞ n=1 [(−1) n (π/2) 2n /(2n + 1)!]G 2n (x; ξ) < 1, since 0 < G 2n (x; ξ) < 1 for all x. Let n = i + 1, and i ≥ 0, then, n ≥ 1. Therefore, where a i = (−1) i+1 (π/2) 2i+2 /(2i + 3)!. By the generalized binomial expansion and some algebra, we get Since l ≥ 1 is an integer, we can apply Equation (3) to the series with the power l, thus, The stated result is obtained. Now, we can apply Lemma 3 and the series expansion cos ((π/2)G( to represent the PDF in Equation (2) as Most of the important properties of the ESG family can be represented in a closed form analytically. The PDF is now a series of PDFs of the exponentiated baseline distribution.

Moments and Entropy
In this subsection, moments and Rényi entropy of the ESG family are derived. Let X be a random variable (RV) with distribution belonging to the ESG family. Then, the rth moment of X is traditionally obtained in the following way: µ r = E(X r ) = ∞ −∞ x r f (x)dx. By virtue of Equation (5), the following series expansion holds: Specific moments can be determined by setting r = 1, 2, 3, . . . . Entropy, on the other hand, is a measure of a system's disorder or randomness. The Rényi entropy of X is defined by Thus, to determine I R(ρ) , it is enough to compute ∞ −∞ f ρ (x)dx, and to plug it into the previous logarithmic-integral expression. To begin, we have For an expansion of the exponentiated cosine term, the following lemma is useful.
On the other hand, with a slight adaptation of Lemma 3, we have where V i,l = ( ρ(α−1) l )(π/2) ρ(α−1) b i and b i is similar to that of Lemma 3. Thus, by substituting Lemma 4 and Equation (9) in Equation (7), we can compute the integral of f ρ (x) to get the Rényi entropy.
The Shannon entropy can be expressed in a similar series expansion manner.

Stress-Strength Reliability
The system performance parameter, known as the stress-strength parameter, is crucial in the context of a system's mechanical reliability. In practice, a good design ensures that the strength exceeds the anticipated stress. If a component has stress modeled by an RV X and is subjected to strength modeled by an RV Y, the stress-strength parameter defined by R = P(Y < X) determines the system performance. The system will fail if and only if the applied stress exceeds the system's strength. On the other hand, R can be used to compare two populations. Engineering, biological sciences, and finance all have uses for stress-strength reliability parameters. See [16] (Chapter 7), among others.
The parameter R was studied by many authors in literature through various viewpoints. Assuming that X and Y are independent, the following distributions have been considered: generalized logistic distribution (see [17]), normal distribution (see [18,19]), exponential distribution with the common location parameter (see [20]), generalized exponential distribution (see [21]), generalized exponential Poisson distribution (see [22]), Poisson-odd generalized exponential distribution (see [23]), beta-Erlang truncated exponential distribution (see [24]), Poisson-generalized half logistic distribution (see [25]), Poisson half logistic distribution (see [26]), and Weibull (W) distribution (see [27][28][29]), among others Here, we study the expression of R in the setting of the ESG family. Suppose that X has the PDF f (x; α 1 , ξ), Y has the CDF F(y; α 2 , ξ), and X and Y are assumed to be independent. Then, the stress-strength reliability parameter is then defined as follows: Thus, We see that R has some manageable form that can make it easy for statistical purposes, and so on. In one of the next sections, we discuss the behavior of R with some particular distribution of the ESG family for illustration.

The ES Weibull Distribution
In this section, we concentrate our attention on one special distribution of the ESG family, called the ESW distribution or ESW(α, β, λ) distribution. In this case, the baseline in Equation (1) is the W distribution with CDF and PDF given as respectively. Therefore, the CDF of the ESW distribution has the following expression: If β = 1, the ESW distribution is reduced to the ES exponential (ESE) distribution. The corresponding probability density, survival, and hazard rate functions are expressed by and h(x) = h(x; α, β, λ) = (π/2)αβλx β−1 e −λx β cos (π/2) 1 − e −λx β sin (π/2) 1 − e −λx β α−1 respectively. Figures 1 and 2 show the plot of the PDF given by Equation (12) and HRF specified in Equation (14). In particular, Figure 2 shows that the failure rate is flexible enough to accommodate monotonic (increasing and decreasing) shapes, and also bathtub and upside-down bathtub shapes.     The decreasing behavior of the PDF is formulated in the following result.
Theorem 1. The PDF of the ESW distribution given by Equation (12) is a decreasing function for α ≤ 1 and β ≤ 1.

Proof.
We have When α ≤ 1 and β ≤ 1, all the terms in the sum are negative, implying that {log[ f (x)]} ≤ 0, so log[ f (x)] is decreasing, and the same for f (x). This ends the proof.

Proposition 1.
The asymptotic behavior of F(x) in Equation (11), f (x) in Equation (12) and s(x) in Equation (13) are as follows: • As x → 0, we have (15) and and Proof. Standard equivalency function findings are used in the proof. For this reason, the details are omitted.

Quantile and Moments
The QF of the ESW distribution can be derived from Equation (11) as The median of the ESW distribution is The skewness and kurtosis properties of the ESW distribution can be determined using the Bowley's skewness and Moor's kurtosis, among other quantile based measures. They are defined by respectively. Equation (17) shows that both B and M are independent of λ. The moments of the ESW distribution can be computed from µ r = ∞ −∞ x r f (x)dx. By letting u = sin (π/2) 1 − e −λx β , this integral is reduced to It is not known if the integral in Equation (18) can be derived analytically, but it can be computed numerically by software such as R. Further, to obtain the moments of the ESW distribution in a series form, we can use Equation (6), and the following lemma.

Proof.
In an expanded form, we have By setting u = λ(ω 2 + s)x β , we get The stated result is obtained.
In fact, most of the features of the ESW distribution may be obtained in a series form using this lemma. In particular, Equations (5) and (19) can be used to express the moments of the ESW distribution in a series of the following form: k and D (2) i,k,l are given in Equation (5) and ψ(ω 1 , ω 2 , ω 3 ) follows from Lemma 5.

Moments of Residual Life
Residual life functions are involved in a variety of fields, including engineering, quality control, and life testing. They also aid in the determination of the asymptotic distributions of order statistics. For an RV X with the ESW distribution, the related mean residual life is defined by . Now, we derive some asymptotic properties of ζ(t) for sufficiently large t, and asymptotic properties of ζ(t) for very small t. Theorem 2. Let X be an RV with the ESW distribution. Then, for t → ∞, the mean residual life of X satisfies where Γ(a, b) denotes the incomplete (upper version) gamma function.
Proof. From the asymptotic behavior of s(t) for t → ∞ in Equation (16), we have By applying the change of variable u = 2λ(x + t) β , and after some algebra, we get The stated equivalence is obtained.
Theorem 3. Let X be an RV with the ESW distribution. Then, for t → 0, the reversed mean residual life of X satisfies Proof. From the asymptotic behavior of F(t) for t → 0 in Equation (15), we havē The desired outcome is demonstrated.

Order Statistics and Asymptotic
We now focus on the order statistics from the ESW distribution, and some of their asymptotic properties. Let X 1 , X 2 , . . . , X n , n ≥ 1, be a sample of independent RVs following the ESW distribution. Let us consider X 1:n , X 2:n , . . . , X n:n the ordered versions of X 1 , X 2 , . . . , X n . Then, for any j = 1, 2, . . . , n, the PDF of X j:n is given as Therefore, by expressing F(x) as in Equation (11), and f (x) as in Equation (12) into Equation (20), we have Also, we can express f j:n (x) as a series expansion depending on PDFs of the ESW distribution with parameters α(j + m), β and λ as The asymptotic distributions for the minimum order statistic (X 1:n ) can be derived as follows. For detail see [30] (Chapter 8).
Theorem 4. Let X 1 , X 2 , . . . , X n , n ≥ 1, be a sample of independent RVs following the ESW distribution. Let X 1:n = inf(X 1 , . . . , X n ) and B * n = (X 1:n − a * n )/b * n . Then, for any x > 0, we have We recognize the CDF of the W distribution with parameters 1 and αβ. The normalizing constant can be derived from Equation (17) by following Theorem 8.3.6 of [30]. That is, we take a * n = 0 and b * n = Q(1/n).
Proof. We aim to apply Theorem 8.3.6 of [30]. By Equation (15), we have It follows from Theorem 8.3.6 of [30] the desired result. This ends the proof.

Parameter Estimation
In this part, we use maximum likelihood and Bayes estimation approaches to estimate the new model parameters. The maximum likelihood estimation of the ESG family is discussed in the general case, while the Bayes estimation is discussed for the ESW distribution only. The estimation techniques are examined by simulation studies.
The expression of the reliability parameter R of the ESW distribution is derived under a common scale parameter from two independent RVs with possibly different ESW distributions. Subsequently, the maximum likelihood estimation of R is discussed, and the nonparametric bootstrap confidence interval (CI) is used to determine the approximate CI of R, and finally assessed by simulation studies.

Maximum Likelihood Estimation
The method of maximum likelihood can be used to estimate the parameters of the models coming from the ESG family. To begin, let X 1 , X 2 , . . . , X n , n ≥ 1, be a sample of independent RVs whose distribution belongs to the ESG family. We denote by x 1 , x 2 , . . . , x n the corresponding observations. Let θ = (α, ξ) T be a vector of parameters. Then, the maximum likelihood estimate (MLE) vector of θ, sayθ = (α,ξ) T , is obtained by the maximization of the log-likelihood function given by with respect to θ. This maximization can be achieved by solving the following nonlinear system of equations given as and . Now, we can analyze the existence and uniqueness of the MLE of the parameter α. For more studies on the existence and uniqueness of the MLEs for various models, one can see [25,31,32], among others.
Proposition 2. Let D(α; ξ) represent the right-hand side of Equation (21), and suppose that ξ are true values of the parameters. Then, D(α; ξ) = 0 has a unique real root.
By applying the plug-in technique, we can use the estimates of the model parameters to provide estimates of the unknown distribution functions (PDF, SF, etc.).
For an in-depth statistical treatment on the parameters (interval estimations, test procedures. . . ), the observed information matrix is required. Here, it is given by . . , ξ m ) T , and writeξ = (ξ 2 ,ξ 3 , . . . ,ξ m ) T . Then, the approximate distribution of the random version ofθ is N m (θ, I(θ) −1 ) under the usual regularity conditions. The asymptotic CI for each parameter θ can be determine using 100 (1 where T rr is defined by the square root of the rth diagonal component of I(θ) −1 ), for r = 1, 2, . . . m and Ω /2 is the quantile (1 − /2) of the N 1 (0, 1) distribution. As a mathematical complement, the components of I(θ) are

Bayes Estimation
In this subsection, we discuss the Bayes estimation of the parameters of the ESW distribution. In this setting, let θ = (α, β, λ) T . The Bayes estimate (BE) vectorθ = (α,β,λ) T is constructed using the posterior distributions given the sample data. The procedure is briefly described below. Let N be the number of iterations and K be the number of the burn in. Then the square error loss (SEL) function for the assumed prior distribution is given as S(θ, θ) = (θ − θ) 2 , and is minimized by the posterior mean, asθ The highest posterior density (HPD) credible interval forθ is determined using the package HDInterval (see [33]) in the R software. let X 1 , X 2 , . . . , X n , n ≥ 1, be a sample of independent RVs with the ESW distribution, and x 1 , x 2 , . . . , x n be the observations. We recall that the unknown parameters are α, β, and λ. Adopting the Bayesian paradigm, we suppose that α, β, and λ are independent RVs that follow the gamma distribution with PDF defined by respectively, i.e., p 1 (x), p 2 (x), and p 3 (x) are the prior PDFs. Let L(θ|data) be the likelihood function defined by Then, the joint posterior PDF of θ|data is specified by where g(data; θ) = L(θ|data)p 1 (α)p 2 (β)p 3 (λ) is the joint PDF associated with the data. The marginal posterior PDF of α, β, and λ can be derived from Equation (22) as The related posterior distributions are not from well-known distributions. As a result, we can obtain samples from posterior distributions using the Metropolis-Hastings (MH) method and the Gibbs sampling technique, as detailed below (see [34][35][36]) (the normal distribution is utilized as a suitable distribution for the MH method): 1. Start by initial guess (α (0) , β (0) , λ (0) ), 2. Set t = 1, 3. Apply the MH algorithm to generate α (t) from π 1 (α), 4. Apply the MH algorithm to generate λ (t) from π 3 (λ), 5. Apply the MH algorithm to generate β (t) from π 2 (β), 6. Set t = t + 1, 7. Repeat step 3 to 6, T times.

Simulation Study I
The performance of maximum likelihood and Bayes estimations are evaluated using simulation results. We generate 1000 samples of size n = (50, 60, 70, . . . , 300) from the ESW distribution for some selected parameter values using Equation (17). The bias, mean square error (MSE), the average length of CI (ALCI), and coverage probability (CP) of the estimates are computed using R3.5.3 developed by [38]. For the MLEs, we use the package called maxLik developed by [39] in R-software by maxBFGS. This package allows us to obtain the MLEs and information matrix, and the package matlib elaborated by [40] can be used to get the inverse of the information matrix to compute the confidence interval. The resulting simulations are given in Figures 4-7. For the Bayes estimation, we used N = 3000 number of iterations and the first 30% as a burn-in sample. We notice that the estimation behaves well when the all hyperparameters are greater than one and unimodal gamma PDF. It is indicated that the MLE and BE perform consistently, the bias, MSE, and the ALCI decrease as the sample size increases; the CP for each parameter is approaching 0.95 in all cases.

Estimation of the Stress-Strength Reliability from the ESW Distribution
In this part, the expression of a reliability parameter of the ESW distribution is derived when the parameter λ is common. The maximum likelihood estimation of R is obtained. The nonparametric percentile bootstrap (Bp) and Student's bootstrap (Bt) are used to approximate the CI of R. Finally, we assessed the estimates through simulation studies.
In the setting of the ESW distribution, let X be an RV having the PDF f (x; α 1 , β 1 , λ) and Y be an RV with the CDF F(x; α 2 , β 2 , λ). We assume that X and Y are independent. Then, the stress-strength reliability parameter is given by Now, by considering the changes of variables u = (π/2) 1 − e −λx β 1 and The parameter R above has no closed form expression, but it can be easily calculated using mathematical software such as R and Matlab, among others. Moreover, when β 1 = β 2 , the expression of R can be deduced from Equation (10).

Maximum Likelihood Estimation of R
Assume that X 1 , X 2 , . . . , X n , n ≥ 1, is a sample of independent RVs whose distribution is the ESW(α 1 , β 1 , λ) distribution, and Y 1 , Y 2 , . . . , Y m , m ≥ 1, is a sample of independent RVs whose distribution is the ESW(α 2 , β 2 , λ) distribution. We work with the observations of these samples, denotes by x 1 , x 2 , . . . , x n and y 1 , y 2 , . . . , y m , respectively. Let θ = (α 1 , α 2 , β 1 , β 2 , λ) T be the vector of unknown parameters,θ = (α 1 ,α 2 ,β 1 ,β 2 ,λ) T be the MLE of θ, andR be the estimate of R. Then, the log likelihood function is given as It can be maximized with respect to θ to obtainθ. The above equation can equivalently be maximized by the solution of the system of equations given below: Once the MLEθ is obtained, we can computeR from Equation (23).

Simulation Study II
Here, a simulation is conducted to study the performance of the MLE and bootstrap CI of R. A simulated sample size of 1000 is obtained via Equation (17) for various sample sizes and parameters from the ESW(α 1 , β 1 , λ) and ESW(α 2 , β 2 , λ) distributions. Let n and m be these sample sizes, respectively. We consider various cases of (n, m) as (20,20), (30,20), (40,40), (40,60) and (60, 60). In each case, the MLEs of R is obtained and the standard deviation (SD), and bias and MSE of R are considered. A 95% CI of R is obtained using the nonparametric percentile Bp and Bt CIs. The bootstrap is obtained using B = 1000 replications. The result of the simulation studies is given in Table 1. It shows that, as the two sample sizes increase, the SDs and MSEs decrease; the estimated value of R converges to its actual value; the ALCI in the two bootstrap methods decreases as the sample size increases. Also, the CP goes to the nominal level of 0.95.

Application
Using three real data applications, we demonstrate the performance of the ESW model in comparison to its submodels and several other existing models. Two data sets for stressstrength reliability assessments make up one application. We estimate all the competing model's parameters by the maximum likelihood method. Also, the parameters of the ESW model are further estimated by the Bayes techniques. We traditionally compare the fitted models using the Akaike information criterion (AIC), Bayesian information criterion (BIC), and consistent Akaike information criterion (CAIC), all are defined via the estimated log-likelihood (L) as a main ingredient. Further, the goodness-of-fit statistics known as the Kolmogorov-Smirnov (KS), Anderson-Darling (AD), and Cramér-von Mises (CvM) are considered. As usual, the data are better represented by the model with the least value of these metrics than the other models. In the case of the MLEs and BEs of the parameters of the ESW model, we compare them via the KS, AD, and CvM. The competing models are generalized exponential (GE) model (see [42]), generalized Rayleigh (GR) model (see [43]), generalized exponential Poisson (GEP) model (see [44]), half logistic Poisson (HLP) model (see [45]), exponentiated Nadarajah-Haghighi (ENH) model (see [46,47]), generalized inverse Weibull (GIW) model (see [48]), and W model.

Real Data Application I
This data set is constituted by the total milk production from the first birth of 107 cows of the SINDI race. The data can be found in [49]. They were also analyzed by [10]. Concretely, the data set is: 0.4365, 0.4260, 0.5140, 0.6907, 0.7471, 0.2605, 0.6196, 0.8781,  0.4990, 0.6058, 0.6891, 0.5770, 0.5394, 0.1479, 0.2356, 0.6012, 0.1525, 0.5483, 0.6927, 0.7261,  0.3323, 0.0671, 0.2361, 0.4800, 0.5707, 0.7131, 0.5853, 0.6768, 0.5350, 0.4151, 0.6789, 0.4576,  0.3259, 0.2303, 0.7687, 0.4371, 0.3383, 0.6114, 0.3480, 0.4564, 0.7804, 0.3406, 0.4823, 0.5912,  0.5744, 0.5481, 0.1131, 0.7290, 0.0168, 0.5529, 0.4530, 0.3891, 0.4752, 0.3134, 0.3175, 0.1167 Figure 8 is the total time on test (TTT) plot of the data set. This shows that data exhibiting an increasing HRF. The ESW model is a capable candidate to accommodate this kind of HRF. The numerical values of the considered statistical measures for each model are computed and presented in Table 2. The results show that the ESW model provides a better fit than the other competing models. The Bayes estimation method shows a better fit in terms of KS, while the maximum likelihood method shows a better fit in terms of AD and CvM. Both of these techniques are sufficient choices for parameter estimation of the ESW model. Figure 9 illustrates the plots of the histogram with the fitted PDF of the ESW model and the empirical SF with the fitted SF of the ESW model. Figure 10 displays the plots of the HRF of the ESW model for the data; empirical cumulative HRF (CHRF) with the fitted CHRF of the ESW model, and quantile-quantile (QQ) plot of the ESW model. The obtained fits are totally satisfying. Figure 11 illustrates the profile log-likelihood of each parameter for the given data set. The uniqueness of the estimates can be seen. To show the performance of the BEs, Figure 12 shows the iterations obtained from the MH algorithm and the Gibbs sampling technique for each parameter, and Figure 13 shows the posterior PDFs of each parameter based on the iterations.
We end this part by indicating the observed information matrix of the ESW model:

Real Data Application II
This data set is provided by [50]. It consists of the remission times (in months) of a random sample of 128 bladder cancer patients, also studied by [25,51] Figure 14 displays the TTT plot of the data set. It shows that the data exhibit an upside-down-bathtub HRF, and the ESW model is a capable candidate to accommodate such an HRF. The numerical values of these measures for all the models are computed and given in Table 3; the results show that the ESW model captures the information of the data better than the other competing models. Figure 15 gives the plot of the histogram with the fitted PDF of the ESW model and empirical SF with the fitted SF of the ESW model. Figure 16 shows the plot of the HRF of the ESW model, empirical CHRF with the fitted CHRF of the ESW model, and QQ plot of the ESW model. Figure 17 is the plots of the profile log-likelihood of each parameter. To illustrate the Bayes estimates performance, Figure 18 shows the iterations obtained from the MH algorithm and the Gibbs sampling technique for each parameter, and Figure 19 shows the posterior PDFs of each parameter based on the iterations obtained.

Real Data Application III
In this subsection, we illustrate the performance of the ESW model in the stress strength reliability studies by using real data sets. Also, we demonstrate the application of the proposed estimation techniques in a practical scenario. We compute the value of R by maximum likelihood approach and the 95% nonparametric Bp and Bt CIs of R using B = 1000 replications. The KS is used to show how good the fit of the ESW model is for the two data sets.
The reference [52] provides the following data sets. One data set represents single fibers tested under tension at gauge lengths of 10 mm (data1), and the other represents impregnated tows of 1000 fibers tested at gauge lengths of 20 mm (data2), of sizes n = 63 and m = 69, respectively.
They are given as: Data1( The estimated values from the application are given in Table 4; it is quite clear that the ESW model fits the two data sets well as measured by KS (i.e., KS 1 for the ESW distribution underlying Data1(X) and KS 2 for the ESW distribution underlying Data2(X)), and its performance indicates that the ESW model can be considered as a good candidate in stress strength reliability analysis. Figure 20 displays the empirical and fitted CDF of the ESW model, and the PDF of the estimated bootstrap values of R for the data sets. Figure 21 shows the profile log likelihood of the estimated parameters. It proves that the obtained MLE is unique.

Conclusions
The exponentiated sine-generated family is a new family of distributions proposed in this paper. Some important mathematical and statistical properties were derived, such as the series representation of the probability density function, quantile function, moments, stress-strength reliability parameter, and Rényi entropy. A special member of the family, called the exponentiated sine Weibull (ESW) distribution, was derived and studied. We analyzed its skewness and kurtosis, mean residual and reversed mean residual life functions, order statistics, and extreme value distributions. The maximum likelihood estimation and Bayes estimation under the square error loss function of the ESW model were discussed. They were assessed using simulation studies via the mean square error and confidence interval of the estimates, and they work well. The expression of a reliability parameter of the ESW distribution is derived when the scale parameter is common. The maximum likelihood is used to estimate this special parameter, and nonparametric boot-strap techniques were considered for the confidence interval. The estimation was assessed by simulation studies and worked well by examining the mean square error, standard deviations, confidence intervals, and coverage probability. In the end, three applications of the ESW model to real data were provided for illustration, in which the ESW model outperforms some other existing models in terms of fitting and is demonstrated as a good candidate for stress-strength parameter analysis. The potential for new applications of the ESW model in the applied field is huge, opening the door to new statistical analysis in applied science dealing with important lifetime data. Also, another specific member of the new family can be investigated for further applied perspectives.
Author Contributions: All authors equally contribute to the manuscript. All authors have read and agreed to the published version of the manuscript.