The extension of the generalization of the sine family to the ALR distribution

Introduction

Modern statistical challenges require distributions capable of accommodating the complex data behaviors inherent in high-dimensional, skewed, or censored datasets. Data distribution and fit are essential for extracting actionable insights in fields such as reliability engineering and biomedical research. While classical probability models support traditional analyses, their rigidity often fails to account for non-uniform risk rates or thick-tailed phenomena [1] .This limitation has spurred the development of generalized families, ranging from transformationbased frameworks the exp-G family by [2] , the beta-G family by [3] , the gamma-G family by [4] , the RB-G family by [5] , the TX-G family by [6] , the Weibull-G family by [7] , the sine-G family by [8] , and the generalized odd-order Gamma-G family by [9] . Among these, the Alpha Leg-G family(ALG) [10] stands out for its ability to unify skewness and tail-weight adjustments through its cumulative distribution function (CDF): , 𝛼 > 0

This paper generalizes the sine family and suggests a new family of distributions.The sine family, density function (PDF), cumulative distribution function (CDF), survival function, and hazard rate are briefly described. The composite Alpha Leg-Rayleigh distribution will be generalized to the sine family in this paper, and a new distribution with superior features to the original distribution will be derived. To demonstrate the superiority of this distribution, we will also provide a simulation study of the true and estimated parameters.

Preliminaries

This section presents the fundamental concepts and mathematical formulations required for constructing the proposed distribution. Specifically, it reviews the ALG generating technique, the Sine family of distributions, and the Rayleigh distribution, which serves as the baseline model. These preliminary results provide the theoretical foundation for deriving the proposed ALG-Sine-Rayleigh (ALGSR) distribution and facilitate the development of its statistical properties in the subsequent sections.

Sine -G Family Distribution

Any baseline distribution can be used to create a new life distribution using the sine-G family [13] . If the baseline distribution's PDF and CDF are represented by (𝑧) and 𝐹(𝑧), respectively, then the new CDF and PDF, represented by 𝐹(𝑧) and 𝑓(𝑧), are as follows:

The Alpha Leg-Rayleigh (ALR) distribution By applying the same methods employed in the Alpha Leg-G (ALG) family to the Rayleigh distribution, the Alpha Leg-Rayleigh (ALR) distribution can be obtained. In particular, by substituting the Rayleigh distribution R for the underlying G distribution in the ALG framework, the resulting distribution adapts to the characteristics of the Rayleigh distribution while preserving the flexibility and qualities of the original construction. having this substitution, data having Rayleigh behavior can be modeled by the Alpha-Leg-Rayleigh distribution, offering a flexible expansion for various uses.

When the Rayleigh function is entered into formula (1), we obtain:

Equation ( 4) represents the new cumulative distribution function of ALR distribution. Now, we derive equation ( 4) in order to extract the density function of ALR distribution.

𝑓 𝐴𝐿𝑅 (𝑥; 𝜎

The survival function 𝑆 𝐴𝐿𝑅 (𝑥; 𝜎; 𝛼) of 𝑋~𝐴𝐿𝑅(𝛼, 𝜎) is defined as follows:

And the hazard function ℎ 𝐴𝐿𝑅 (𝑥; 𝜎; 𝛼) of the (ALR) distribution is defined as follows:

Construction of the ALG-Sine-Rayleigh Distribution

The construction of new probability distributions by integrating distribution-generating techniques with well-established baseline distributions has become an active area of research in statistical modeling. This strategy enhances model flexibility and enables the resulting distributions to capture a broader range of data characteristics. In this section, the ALG generator is combined with the Sine family using the Rayleigh distribution as the baseline distribution to introduce a new model, referred to as the ALG-Sine-Rayleigh (ALGSR) distribution. ) ,

Equation ( 9) provides the PDF of the SALRD, which may be obtained by deriving the aforementioned equation.

Certainly, the PDF in equation ( 9) holds the conditions of being PDF, as follows:

Parameters Estimation of the SALRD

In this section, the model parameters of SALRD are estimated, by the MLE method. Let T 1 , T 2 , …, T n be a random sample of size (n) from the SALRD. Then the likelihood function of SALRD is obtained by substituting the PDF into Equation ( 9)

The function of log-likelihood can be expressed as:

) By differentiating the log-likelihood function partially with respect to the unknown parameters (𝜎, α) and equate to zero, respectively, we obtain the following nonlinear equation.

It is impossible to solve equations ( 8) and ( 9) analytically since they are nonlinear. Therefore, to determine the estimated values of the coefficients (𝜎, α), numerical techniques such the Newton-Raphson method are employed.

To find the results of the numerical analysis and the results of the simulation data, we will use the R program.

Simulation data

To evaluate the performance of the proposed estimation procedure under different practical conditions, a comprehensive Monte Carlo simulation study was conducted. The simulation design considered several combinations of the model parameters, with 𝛼 = 0.5, 1, 2 and 3, and 𝜎 = 0.5, 1, and 2. These values were selected to represent a wide range of parameter settings, including small, moderate, and relatively large values, thereby providing a comprehensive assessment of the estimator's behavior under different distributional characteristics. Furthermore, different sample sizes (𝑛 = 30, 50, 100, 200, 500, and 1000) were employed to investigate the estimator's performance across both small and large samples. Small sample sizes were included to reflect situations frequently encountered in practical applications where data availability is limited, whereas larger sample sizes were considered to examine the asymptotic behavior of the estimation method as the amount of information increases. Each simulation scenario was replicated 1000 times to ensure stable and reliable empirical measures while reducing the effect of random sampling variation.

For each combination of parameter values and sample sizes, 1000 independent random samples were generated from the proposed SALRD distribution. The parameters were estimated using the Maximum Likelihood Estimation (MLE) method, and a set of Monte Carlo performance criteria was calculated for each scenario. These criteria include the mean estimate, Bias, Relative Bias (RB), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Average Standard Error (ASE), Coverage Probability (CP), Average Length (AL) of the corresponding 95% confidence intervals, and the Convergence Rate (CR). Collectively, these measures enable a comprehensive assessment of the accuracy, precision, efficiency, interval estimation performance, and numerical stability of the proposed estimation procedure.

Results

Tables 1 summarize the Monte Carlo simulation results for different combinations of the true parameter values and sample sizes.

The discussion focuses on examining the behavior of the proposed estimators as the sample size increases and assessing their statistical performance using the selected evaluation criteria. Particular attention is given to the consistency, estimation accuracy, variability, confidence interval performance, and convergence characteristics of the Maximum Likelihood Estimation method under various simulation settings. The results indicate that the accuracy of the parameter estimates improves progressively as the sample size increases. In particular, the estimated parameter values become closer to their corresponding true values, while the Bias and Relative Bias decrease steadily. Similarly, both the MSE and RMSE exhibit a noticeable downward trend with increasing sample size, reflecting an improvement in estimation precision. Furthermore, the Average Standard Error (ASE) decreases as the sample size grows, indicating greater stability and reduced variability of the estimators. The Coverage Probability (CP) remains close to the nominal confidence level of 95% for moderat e and large sample sizes, suggesting that the constructed confidence intervals possess satisfactory coverage properties. In addition, the Average Length (AL) of the confidence intervals decreases with increasing sample size, demonstrating that the confidence intervals become more precise while maintaining adequate coverage. Finally, the convergence rate of the optimization algorithm is consistently high across the simulation scenarios, confirming the numerical stability and reliability of the Maximum Likelihood Estimation procedure for estimating the parameters of the proposed SALRD distribution.

Conclusion

The Monte Carlo simulation study demonstrated the performance of the Maximum Likelihood Estimation (MLE) method for estimating the parameters of the proposed SALRD distribution under a variety of parameter combinations and sample sizes. The simulation results provided a comprehensive evaluation of the statistical properties of the estimators using several performance measures, including the Mean, Bias, Relative Bias (RB), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Average Standard Error (ASE), Coverage Probability (CP), Average Length (AL), and Convergence Rate (CR).

Overall, the findings indicate that the estimation procedure exhibits satisfactory performance, with noticeable improvements as the sample size increases. The estimators become more accurate and stable, while the estimation errors and variability decrease progressively. Moreover, the confidence intervals show appropriate coverage probabilities and reduced lengths for moderate and large sample sizes, confirming the reliability of the estimation procedure. The high convergence rate observed throughout the simulation study further demonstrates the numerical stability and effectiveness of the optimization algorithm. These results support the suitability of the Maximum Likelihood Estimation method for estimating the parameters of the proposed SALRD distribution and provide evidence of its practical applicability in statistical modeling.

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