An Innovated G Family: Properties, Characterizations and Risk Analysis under Different Estimation Methods

The idea of symmetry, which is used to describe the shape of a probability distribution, is a key concept in the theory of probability. The use of symmetric and asymmetric distributions is common in statistical inference, decision-making, and probability calculations. This article introduces a novel asymmetric model for assessing risks under a skewed claims dataset. The new distribution is also employed for both censored and uncensored validation testing. Four estimation methods, maximum likelihood, ordinary least squares, L-Moment, and Anderson Darling, were used for the risk assessment and analysis. To explain the exposure to risk within actuarial claims data, we introduced five crucial indicators, namely value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess losses. A numerical and graphical analysis is presented to assess the actuarial risk. Furthermore, the article discusses a newly developed Rao Robson Nikulin statistic for censored and uncensored validation testing. The validation testing also involved the insurance claims dataset.

In this article, we have introduced a new probability distribution having three parameters using half Cauchy family of distribution named half Cauchy extended exponential distribution. The statistical properties and characteristics of the proposed distribution like the hazard rate function (HRF), cumulative hazard function, and the probability density function (PDF), and the cumulative distribution function (CDF), quantile function and the skewness, kurtosis are provided. The parameters of the proposed distribution are estimated using the Cramer-Von-Mises (CVM) least-square estimation (LSE), and maximum likelihood estimators (MLE) methods. A real data set is analyzed to test the goodness-of-fit of the proposed distribution. It is found that the half Cauchy extended exponential distribution performed well as compared to some competing distributions.

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), Kolmogorov-Smirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.

In this paper, a new three-parameter lifetime distribution, alpha power transformed inverse Lomax (APTIL) distribution, is proposed. The APTIL distribution is more flexible than inverse Lomax distribution. We derived some mathematical properties including moments, moment generating function, quantile function, mode, stress strength reliability, and order statistics. Characterization related to hazard rate function is also derived. The model parameters are estimated using eight estimation methods including maximum likelihood, least squares, weighted least squares, percentile, Cramer–von Mises, maximum product of spacing, Anderson–Darling, and right-tail Anderson–Darling. Numerical results are calculated to compare the performance of these estimation methods. Finally, we used three real-life datasets to show the flexibility of the APTIL distribution.

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.

Increasing the amount of data with complex dynamics requires the constant updating of statistical distributions. This study aimed to introduce a new three-parameter distribution, named the new exponentiated Weibull (NEW) distribution, by applying the logarithmic transformation to the exponentiated Weibull distribution. The exponentiated Weibull distribution is a powerful generalization of the Weibull distribution that includes several classical distributions as special cases—Weibull, exponential, Rayleigh, and exponentiated exponential—which make it capable of capturing diverse forms of hazard functions. By combining the advantages of the logarithmic transformation and exponentiated Weibull, the new distribution offers great flexibility in modeling different forms of hazard functions, including increasing, J-shaped, reverse-J-shaped, and bathtub-shaped functions. Some mathematical properties of the NEW distribution were studied. Moreover, four different methods of estimation—the maximum likelihood (ML), least squares (LS), Cramer–Von Mises (CVM), and percentile (PE) methods—were employed to estimate the distribution parameters. To assess the performance of the estimates, three simulation studies were conducted, showing the benefit of the ML method, followed by the PE method, in estimating the model parameters. Additionally, five datasets were used to evaluate the effectiveness of the new distribution in fitting real data. Compared with some Weibull-type extensions, the results demonstrate the superiority of the new distribution in modeling various forms of real data and provide evidence for the applicability of the new distribution.

In this study, we construct a family of distributions that does not include additional parameter to any baseline distribution. This new family is a composition of two families of distributions, that is, the Kavya-Manoharan (KM) family and the Dinesh-Umesh-Sanjay (DUS) family and it is named the Kavya-Manoharan Dinesh-Umesh-Sanjay (KM-DUS) family. Further, an innovative two-parameter lifetime distribution being the KM-DUS Rayleigh-inverted Weibull (KM-DUS-RIW) distribution was proposed. In this new distribution, the Rayleigh-Inverted Weibull (RIW) was used as the baseline distribution. The new distribution is an improvement of the RIW in that it guarantees more flexibility and tractability with no increase in parameters hence offering a parsimonious but powerful tool for modeling reliability and survival data. The primary statistical characteristics of this KM-DUS-RIW distribution such as its probability density function, cumulative distribution function, quantile function, moments, and order statistic, have been derived and discussed in great detail. The methods named Maximum likelihood (ML), Least Squares (LS), Weighted Least Squares (WLS), Maximum Product Spacing (MPS), Cramér-von Mises (CVM), Anderson-Darling (AD), Right-Tailed Anderson-Darling (RTAD), Percentile (PERC) as well as Bayesian methods with different loss functions such as Squared Error, LINEX, and General Entropy are utilized for estimating the model parameters were explored. Of these, MPS consistently yields the most efficient estimates across various sample sizes. The practicality of the proposed distribution is exemplified by using real entomological data with adult progeny counts of Stegobium paniceum L. under choice and non-choice tests as application cases. Besides, the model is compared using the log-likelihood, AIC, BIC, HQIC, and several goodness-of-fit statistics, which show the KM-DUS-RIW model fits better than the competing available models. These findings attest to the KM-DUS-RIW distribution being a promising potential alternative to modeling complex lifetime data.

The continuous development of flexible probability distributions has become essential for accurately describing complex stochastic phenomena. In this study, a new theoretical based on the New Sine Type II Generalized Topp-Leone distribution is investigated. The model is constructed by integrating the sine trigonometric transformation with the Type II Topp-Leone generator applied to an arbitrary baseline distribution. This combination introduces additional shape parameters that enhance the adaptability of the resulting family of distributions. Several mathematical properties of the proposed model are derived. These include explicit expressions for the cumulative distribution function (cdf), probability density function (pdf), survival function, and hazard rate function. In addition, analytical forms of the quantile function, moments, moment generating function, and order statistics are established. The author’s adopted two estimation techniques in this work: the maximum likelihood estimation (MLE) and the maximum product of spacing (MPLS) estimation methods. The structural characteristics of the distribution reveal that it is capable of producing a wide range of density and hazard rate shapes depending on the selected baseline model and parameter values. The theoretical results presented in this work provide a general framework for constructing new members of the family and offer a foundation for further statistical developments involving estimation techniques and practical implementations. Plots of the pdf, cdf as well as the validity check on the pdf of the new model indicate that the model is adequate capable of handling all the intricate associated with real-life datasets.

In this paper, a new distribution of the four-parameter lifetime model called the Marshall–Olkin Gompertz is proposed on the basis of the Gompertz distribution. It is a generalization of the Marshall–Olkin Gompertz distribution having constant failure rate and can also be constant, decreasing, increasing unimodal and bathtub-shaped depending on its parameters. Some mathematical properties of this model such as the probability density function, cumulative distribution function, hazard rate function, central moments, moments of order statistics, Renyi and Shannon entropies and quantile function are derived. In addition, the maximum likelihood of its parameters method is estimated and this new distribution compared with some Gompertz distribution generalizations by means of a set of real data.

The important problem of the ratio of gamma and beta distributed random variables is considered. Six motivating applications (from efficiency modeling, income modeling, clinical trials, hydrology, reliability and modeling of infectious diseases) are discussed. Exact expressions are derived for the probability density function, cumulative distribution function, hazard rate function, shape characteristics, moments, factorial moments, variance, skewness, kurtosis, conditional moments, L moments, characteristic function, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, percentiles, order statistics and the asymptotic distribution of the extreme values. Estimation procedures by the methods of moments and maximum likelihood are provided and their performances compared by simulation. For maximum likelihood estimation, the Fisher information matrix is derived and the case of censoring is considered. Finally, an application is discussed for efficiency of warning-time systems.

This paper considers various estimation methods to estimate the unknown parameters of the DUS Inverse Weibull (DIW) distribution using the maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM) and the Anderson-Darling (AD) estimators. A Monte-Carlo simulation study is conducted to determine the most preferable estimators in terms of their efficiencies. Furthermore, the distribution of the error terms in the simple linear regression is assumed to be DIW to show the implementation of it to the linear models. We also carry out a simulation study for comparing the performances of the estimators of the unknown regression parameters.

Financialization has contributed to economic growth but has caused scandals, misselling, rogue trading, tax evasion, and market speculation. To a certain extent, it has also created problems in social and economic instability. It is an important aspect of Enterprise Security, Privacy, and Risk (ESPR), particularly in risk research and analysis. In order to minimize the damaging impacts caused by the lack of regulatory compliance, governance, ethical responsibilities, and trust, we propose a Business Integrity Modeling and Analysis (BIMA) framework to unify business integrity with performance using big data predictive analytics and business intelligence. Comprehensive services include modeling risk and asset prices, and consequently, aligning them with business strategies, making our services, according to market trend analysis, both transparent and fair. The BIMA framework uses Monte Carlo simulation, the Black–Scholes–Merton model, and the Heston model for performing financial, operational, and liquidity risk analysis and present outputs in the form of analytics and visualization. Our results and analysis demonstrate supplier bankruptcy modeling, risk pricing, high-frequency pricing simulations, London Interbank Offered Rate (LIBOR) rate simulation, and speculation detection results to provide a variety of critical risk analysis. Our approaches to tackle problems caused by financial services and the operational risk clearly demonstrate that the BIMA framework, as the outputs of our data analytics research, can effectively combine integrity and risk analysis together with overall business performance and can contribute to operational risk research.

In this work, we provide an extended one parameter distribution into a new two parameter life model and called it the New Extended “A” (NEA) distribution. This new model provides the best description of the data sets in different sectors. Both of the probability distribution function (PDF) and the cumulative distribution function (CDF) of the NEA distribution are derived from one of the new families. The suggested distribution’s properties including moments, quantile function, survival function, hazard rate function, odds function and mode are deduced and acquired in explicit forms using Taylor expansion to simplify the integrals. The Rényi entropy of NEA distribution is derived and the maximum likelihood method for estimating its parameters is deduced. Numerical simulations are used to study the behavior of the parameters with different sizes of samples. According to the simulation results of this work, the maximum likelihood estimators are advised to be a suitable estimation method for estimating the parameters of the NEA distribution. We show the importance and flexibility of the NEA distribution by comparing it to some competing distributions using five real data sets from different sectors in behavioral sciences, ecophysiological, industry and health. Our new model is shown to be better than all competing distributions. Received: October 6, 2023Accepted: November 15, 2023

Analyzing survival (life-testing) data and drawing inferences about them is a part of engineering and health sciences. So far, various statistical tools, e.g., survival (reliability) function (s f ), probability density function (pd f ), and hazard rate function (HR) were available among decision-making scientists to handle time-to-event data (complete or censored). But because functions (pd f ) estimators were interval (window) based, they mostly gave qualitative ideas having pictorial representation resembling step functions, ordinate remain constant when abscissa vary over an interval, thereby giving incomplete information. However, it can be sorted out with the use of kernel estimates of the above mentioned functions, resulting into smooth estimators. Moreover, the metric based on aging intensity function (AI) gives an alternative way of studying lifetime or clinical datasets as it is a quantitative measure (not interval-based), thereby depicting a broader view of a given data. In our study, we primarily focus on AI and HR functions estimated using four different kernels. We apply them to a case study of patients with primary malignant tumors of sternum (cf. Daniel and Cross, 2014) with the right-censored data. Our result shows that kernel estimates of HR and AI functions for patients with high grade tumor (HGT) are higher than for patients with low grade tumor (LGT), as expected. Thus, the study opens up a new direction for applying AI and HR functions in health sciences and engineering studies.

This article presents a new extension of the three-parameter log-logistic distribution, the so-called heavy-tailed log-logistic (HTLL) distribution. Some important mathematical properties of the new HTLL distribution are calculated. In addition, some numerical results of moments for the HTLL distribution are calculated. Extensive simulations were performed to investigate the estimation of the model parameters using many established approaches, including maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM), Anderson-Darling (AD), and right-tail Anderson-Darling (RTAD). The simulation results show that the AD approach has the highest efficiency among these approaches. The usefulness of the newly proposed model is demonstrated by analyzing two real data sets.