Incomplete Moments Research Articles

Some Properties of the Cosine Lomax Distribution with Applications

This paper introduces a new trigonometric distribution, named the Cosine Lomax (CLM) distribution. This distribution is a combination of the Lomax distribution and the Cosine-G family of distributions. Among the many mathematical moments, moment-generating function, incomplete moments, and the quantile function. We try to determine the model parameters using the maximum likelihood estimation method. Simulation studies evaluate the effectiveness of maximum likelihood estimators using bias and root mean square error. We applied the CLM distribution to two real-world datasets and tested for consistency using the Akaike information criterion, the consistent AIC, the Hannan-Quinn information criterion, the Bayesian information criterion, the Kolmogorov-Smirnov p-value, Cramer Von-Mises, and Andersen-Darling. The CLM model fared better when compared to the following distributions: Lomax, Inverse Lomax, Inverse Weibull, Burr Type X, Rayleigh, and Exponential, as well as measures of data set fit.

The Hybrid Odd Exponential-Exponential Distribution: Statistical Properties and Applications

In this paper, we study a new statistical distribution called the hybrid odd exponential-exponential distribution (HOE-E distribution for short) we the aim of applying it to real data sets. This new distribution appears as a sub-model of the larger HOEE-Φ family in (Mahdi et al., 2024). We explore many mathematical and statistical properties of this distribution such as the series expansion of the PDF, the quantile function, moments and their generating function, incomplete moments, the shape of the PDF, and the shape of the hazard rate function, order statistics, and parameter estimation using the maximum likelihood estimation. We use this distribution to model a plasma concentration data set. Moreover, this we show that the suggested HOE-E distribution outperformance many other distributions based on various statistical criteria like AIC, CAIC, BIC, HQIC, W, and A.

Unit extended exponential distribution with applications

In this paper, the unit extended exponential distribution is presented. The extended exponential distribution is a two-parameter lifespan distribution that has been demonstrated to be very adaptable. Our goal is to convert this flexibility between the unit interval and the extended exponential distribution. Several different types of probability density functions, including decreasing, left-skewed, right-skewed, and unimodal, have been shown. However, the hazard rate function may take on various forms, including J-shaped, U-shaped, and bathtub shapes. A few statistical properties, such as moments, mean, variance, skewness, kurtosis, expectation-weighted moments, incomplete moments, conditional moments, mean deviation, Lorenz and Bonferroni curves, mean residual life, mean inactivity time, and various entropy measures, round out the theoretical section. The maximum likelihood technique is utilized to estimate model parameters using unit data. This study provides simulation data to evaluate the proposed strategy. Two applications utilizing actual data sets emphasize the significance of the new model compared to existing unit models.

Constant Stress-Partially Accelerated Life Tests of Vtub-Shaped Lifetime Distribution under Progressive Type II Censoring

In lifetime tests, the waiting time for items to fail may be long under usual use conditions, particularly when the products have high reliability. To reduce the cost of testing without sacrificing the quality of the data obtained, the products are exposed to higher stress levels than normal, which quickly causes early failures. Therefore, accelerated life testing is essential since it saves costs and time. This paper considers constant stress-partially accelerated life tests under progressive Type II censored samples. This is realized under the claim that the lifetime of products under usual use conditions follows Vtub-shaped lifetime distribution, which is also known as log-log distribution. The log–log distribution is highly significant and has several real-world applications since it has distinct shapes of its probability density function and hazard rate function. A graphical description of the log–log distribution is exhibited, including plots of the probability density function and hazard rate. The log–log density has different shapes, such as decreasing, unimodal, and approximately symmetric. Several mathematical properties, such as quantiles, probability weighted moments, incomplete moments, moments of residual life, and reversed residual life functions, and entropy of the log–log distribution, are discussed. In addition, the maximum likelihood and maximum product spacing methods are used to obtain the interval and point estimators of the acceleration factor, as well as the model parameters. A simulation study is employed to assess the implementation of the estimation approaches under censoring schemes and different sample sizes. Finally, to demonstrate the viability of the various approaches, two real data sets are investigated.

A novel asymmetric extension of power XLindley distribution: properties, inference and applications to engineering data

It is impossible to overstate the importance of using trigonometric functions appropriately in distribution theory. The main contribution of the research is to construct a flexible trigonometric extension of the power XLindley distribution. More specifically, we build an innovative two-parameter lifetime distribution known as the sine power XLindley distribution (SPXLD) using characteristics from the sine-generated family of distributions. As the main motivational fact, it provides an attractive alternative to the power Lindley, power XLindley, weighted Lindley, and extended power Lindley distributions; it may be better able to model lifetime phenomena presenting data of leptokurtic and platkurtic nature. In contrast to the increasing, decreasing, and reversed-j-shaped hazard rate function, the density exhibits asymmetric shapes with varying peakedness levels. Several significant characteristics are illustrated, including moments, the quantile function, the probability density function in series representation, the stress-strength reliability, and incomplete moments. To analyze the behavior of the suggested distribution, sixteen estimation techniques are applied, such as the maximum likelihood, percentiles, some methods of minimum distances, some methods based on minimum and maximum spacing distances, and the Kolmogorov method. After that, an extensive simulation study and the examination of two survival real datasets are used to show the viability, usefulness, and adaptability of the SPXLD. Relevant goodness of fit criteria demonstrates that the SPXLD fits several current distributions.

A new extension of the Gumbel distribution with biomedical data analysis

In the field of biomedical research, data characteristics often exhibit significant variability, challenging the applicability of classical Gumbel distribution for biomedical data modeling. To address this, this paper introduces a novel extension of the Gumbel model known as the odd beta prime Gumbel (OBP-Gum) model. Derived from the odd beta prime family, the new distribution exhibits greater kurtosis compared to the traditional Gumbel distribution. Importantly, the proposed distribution is designed to capture right-skewed, left-skewed, and nearly symmetric density functions, as well as increasing, decreasing, constant, and upside-down bathtub shapes for its hazard rate function, providing excellent curvature features for creating flexible statistical models for biomedical research. We derive the fundamental features of the OBP-Gum model, such as the quantile function, linear representations, moment generating function, moments, skewness, kurtosis, incomplete moments, and Rényi and Tsallis entropies. Parameter estimation for this new model is conducted using the maximum likelihood estimation method. A simulation study demonstrates the performance of the model parameters. The empirical findings, based on applications to two biomedical datasets, suggest that the OBP-Gum distribution outperforms existing models, particularly in handling extreme observations. Instead of relying on conventional models for decision-making, this research provides relevant stakeholders with an improved statistical distribution for more accurate biomedical data modeling.

A new extended Fréchet model with different estimation methods and applications

In this study, we introduce a new extension of the Fréchet distribution known as the new extended Fréchet (NE_Fr) model. The NE_Fr is created by combining the new extended X family of distributions and the Fréchet distribution. The NE_Fr has more flexibility than the classical Fréchet distribution and some generalizations of the Fréchet distribution. The probability density function of the NE_Fr can be decreasing, unimodal and right skewed shape but it's hazard rate function can be decreasing or up-side-down shape. Several mathematical properties of the new model were derived by calculating the quantile function, the ordinary moments, the incomplete moments, the moment generating function, the conditional moment, the Bonferroni curve and the Lorenz curve. Several entropy measures were developed for this purpose. The inferences from the NE_Fr distribution were investigated using several established methods such as maximum likelihood estimation, least squares and weighted least squares estimation, and Anderson-Darling estimation. The simulation results demonstrated the computational efficiency of these techniques. The proposed NE_Fr distribution was demonstrated to be useful by examining three actual data sets.

Advancing estimation techniques and their applications in engineering and medical data analysis through novel continuous and discrete linear failure rate distribution extension

In this paper, we present a new heavy-tailed distribution called the heavy-tailed linear failure rate (HTLFR) distribution. Various statistical properties of the proposed distribution are derived, including the quantile function, the median, the ordinary moments, the moment generating function, the incomplete moments and the conditional moments. Some actuarial measures such as value at risk, expected shortfall, tail value at risk, tail variance and tail variance premium are calculated. Three different methods of estimation such as the maximum likelihood method, the maximum product spacing method and the Bayesian method as well as some simulation results for the model parameters of the HTLFR distribution under complete samples are examined. The results of the real data set show that the proposed distribution has greater flexibility and has been empirically evaluated under complete data. Discrimination analysis was employed to ensure fairness, equity, and accuracy in decision-making processes for selecting the best model, comparing the proposed distribution with known distributions. A discrete analog of the HTLFR distribution is proposed.

The New Strange Generalized Rayleigh Family: Characteristics and Applications to COVID-19 Data

In this paper, we introduce a novel family of continuous distributions known as the Odd Generalized Rayleigh-G Family. Within this family, we present a special sub-model known as the odd Generalized Rayleigh Inverse Weibull (OGRIW) distribution. The OGRIW distribution is derived by combining the T-X family and the Generalized Rayleigh distribution. We provide a comprehensive expansion of the (PDF) and (CDF) for the OGRIW distribution. Additionally, we investigate several mathematical properties of the OGRIW distribution, including moments, moment-generating function, incomplete moments, quantile function, order statistics and Rényi entropy. To estimate the model parameters, we employ the maximum likelihood method, aiming to identify the parameter values that maximise the likelihood of the observed data. Finally, we apply the proposed OGRIW distribution to two real COVID-19 datasets from Mexico and Canada. The results of these applications demonstrate that the new distribution exhibits remarkable flexibility and outperforms other comparative distributions in terms of accurately modelling the COVID-19 data.

Statistical analysis of the inverse power Zeghdoudi model: Estimation, simulation and modeling to engineering and environmental data

Abstract In this research, we investigate a brand-new two-parameter distribution as a modification of the power Zeghdoudi distribution (PZD). Using the inverse transformation technique on the PZD, the produced distribution is called the inverted PZD (IPZD). Its usefulness in producing symmetric and asymmetric probability density functions makes it the perfect choice for lifetime phenomenon modeling. It is also appropriate for a range of real data since the relevant hazard rate function has one of the following shapes: increasing, decreasing, reverse j-shape or upside-down shape. Mode, quantiles, moments, geometric mean, inverse moments, incomplete moments, distribution of order statistics, Lorenz, Bonferroni, and Zenga curves are a few of the significant characteristics and aspects explored in our study along with some graphical representations. Twelve effective estimating techniques are used to determine the distribution parameters of the IPZD. These include the Kolmogorov, least squares (LS), a maximum product of spacing, Anderson-Darling (AD), maximum likelihood, minimum absolute spacing distance, right-tail AD, minimum absolute spacing-log distance, weighted LS, left-tailed AD, Cramér-von Mises, AD left-tail second-order. A Monte Carlo simulation is used to examine the effectiveness of the obtained estimates. The visual representation and numerical results show that the maximum likelihood estimation strategy regularly beats the other methods in terms of accuracy when estimating the relevant parameters. The usefulness of the recommended distribution for modelling data is illustrated and displayed visually using two real data sets through comparisons with other distributions.

Topp-Leone Exponentiated Pareto Distribution: Properties and Application to Covid-19 Data

This paper proposes a new Topp-Leone Exponentiated Pareto (TLEtP) distribution. The new distribution family is derived by expanding the Topp Leone-G family of distributions with additional positive shape parameters. The corresponding density and distribution functions are derived and shown. Some of the derived mathematical properties of the distribution include quantile function, ordinary and incomplete moments generating function (mgf), hazard function, survival function, odd function, probability weighted moment, and distribution of order statistic. The parameters of the distribution are estimated using Maximum Likelihood method. The proposed distribution’s validity is demonstrated by fitting two sets of real data and comparing the results with two existing same-family distributions, the Topp-Leone Pareto type I(TLPI) and Pareto (P), with the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC), respectively. The comparison of the proposed Topp-Leone Exponentiated Pareto (TLEtP) to the Topp-Leone Pareto type I(TLPI) and Pareto (P) distribution demonstrate that the TLEtP distribution offers a better fit for the data sets than the other two distributions.

Properties and Applications of A New Attractive Distribution

AbstractWe provide a new, flexible model called the Odd Kappa-Exponential (OK-E) distribution. The shape of its hazard rate function (hrf) might be constant, declining, growing, inverted-J, bathtub, or inverted-bathtub. The probability density function (pdf) and the cumulative distribution function (cdf) have both been expressed as linear expansions. Bonferroni and Lorenz curves, ordinary and incomplete moments, the quantile function, the mean residual life, the mean waiting time, and the entropy are all defined. The maximum likelihood method is used to estimate the values of the model’s unknown parameters. To verify the precision of the estimate, we ran a simulation study. The attractiveness and adaptability of the Odd Kappa-Exponential model were shown using four real-world examples from the fields of economics, engineering, and the environment.

Introducing the new arcsine-generator distribution family: An in-depth exploration with an illustrative example of the inverse weibull distribution for analyzing healthcare industry data

The study is about a novel Arcsin-function based generator of new families of distributions. We chose the inverse Weibull distribution as the reference distribution to see if the generator could be employed. This generator helps for developing a distribution called the novel Arcsin inverse Weibull. The main features of the suggested distribution have been taken into account. Some of the indicators used in this class include the density function, complete and incomplete moments, average deviation, and aging indicators. The model's parameters are determined using the maximum likelihood method in both simulations and data analysis. The effectiveness of the suggested model in the healthcare sector is demonstrated by analyzing five sets of data, revealing its superior fit compared to the traditional inverse sine model, which is associated with the inverse Weibull model.

Medical application with an extended Ailamujia inverted Weibull model: Properties, estimation and simulation

In this article, we propose a new lifetime model called Kavya-Manoharan Ailanujia inverted Weibull distribution, which is obtained from the Kavya-Manoharan (KM) transformation family and Ailanujia inverted Weibull distribution. Various statistical properties are obtained, involving moments, incomplete moments, conditional moments, mean deviations, mean residual lifetime and mean past lifetime and entropy. Moreover, the newly proposed distribution, characterized by two parameters, proves to be remarkably versatile, as demonstrated in the application results. Several statistical properties of the KMAIW distribution are deduced, and the estimation of its parameters is accomplished through three distinct techniques. To assess the effectiveness of these approaches, a comprehensive simulation study is conducted. The significance and applicability of the proposed model are underscored through the analysis of a real-world dataset. In comparative evaluations against well-known distributions featuring two, three, and four parameters, the suggested model consistently outperforms them all. This highlights the superior fitting capability and adaptability of the proposed model in comparison to established distributions with varying parameter complexities.

X-exponential-G Family of Distributions With Applications

A new family of continuous distributions called the X-exponential-G (XE-G) family is proposed. Explicit expressions are derived for the ordinary and incomplete moments, generating functions, mean deviation about the mean and median, Shannon and R\'{e}nyi entropies, and order statistics of this new family. Estimation of the parameters of the new family is done using the method of maximum likelihood. Assessment of the performance of the maximum likelihood estimates is carried out through a simulation study using the quantile function of the XE-G distribution. The usefulness of this new family is illustrated by modeling two real datasets.

Transformation of Rayleigh Distribution, Properties, and Application

The need to develop the theory of statistics and its properties follows from the fact that many types of data cannot be fitted by classical distributions. This fact invites many researchers to generate new distributions, find their properties, and implement a data set to find the best distribution that can fit the data better. In this paper, we propose special cases of Rayleigh distribution and their relationship to wellknown distributions like half-logistic distribution (HLD), generalized half-logistic distribution (GHLD), and exponentiated half-logistic distribution (EHLD). We have mainly discussed the relationship of a transformation technique of those special cases of Rayleigh distribution with different parameter values to the assigned distributions (HLD, GHLD, EHLD). We also show the mathematical statistical properties of such special cases like the rth moment, central moment, incomplete moments, the probability weighted moments, the stochastic ordering, and interval estimation within the proposed parameters. Consequently, such properties are derived to generate modern statistical characteristics related to the special cases of Rayleigh distribution. Moreover, we have set table for the calculations of particular cases with their derived moments that have previously found their theoretical representations. Finally, we set off some conclusions related to the results of this humble work.

A flexible model for bounded data with bathtub shaped hazard rate function and applications

<p>The unit new X-Lindley distribution, which is a novel one-parameter distribution on the unit interval, is presented in this study. It was developed by altering the new X-Lindley distribution using the exponential transformation. This new one-parameter distribution's fundamental features, including moments, incomplete moments, Lorenz and Bonferroni curves, Gini index, mode, extropy, Havrda and Charvat entropy, Rényi entropy, and Tsallis entropy, are explored. Additionally, it has bathtub-shaped hazard rate functions and monotonically increasing hazard rate functions with a single parameter. The new distribution is therefore sufficiently rich to model real data. Also, different estimation methods, such as maximum likelihood, least-squares, and weighted least-squares, are used to estimate the parameters of this model, and using a simulation research, their respective performances are evaluated. Finally, two real-life datasets are used to demonstrate the suggested model's competency.</p>

A Bounded Lifetime Distribution Specified by a Trigonometric Function: Properties, Regression Model, and Applications

Trigonometric functions have gained considerable attention in recent studies for developing new lifetime distributions. This is due to their parsimonious framework, mathematical tractability, and flexibility of the newly developed distributions. In this paper, the Sine‐generated family is used to create a new bounded lifetime distribution, known as Sine‐Marshall–Olkin Topp–Leone distribution, for modeling data defined on the unit interval. Some statistical properties of the new bounded distribution, including quantile function, ordinary moments, incomplete moments, moment‐generating functions, inequality measures, Rényi entropy, and probability weighted moments are derived. Seven methods of parameter estimation, including maximum likelihood, ordinary least squares, weighted least squares, moment product spacing, Anderson–Darling, and Cramér–von Mises estimators are used to estimate the parameters of the new distribution. The behavior of the estimators obtained from the estimation methods are investigated using Monte Carlo simulation studies. The results show that the estimation methods are asymptotically efficient and consistent. The flexibility of the new bounded distribution is examined via data fitting using two proportional datasets. The results of the fittings show that the new bounded distribution performs significantly better than the competing bounded lifetime distributions. Finally, Sine‐Marshall–Olkin Topp–Leone regression model is presented as an alternative to the beta and Kumaraswamy regression models.

Statistical Inference for Heavy‐Tailed Burr X Distribution with Applications

In this article, we present a new distribution, the so‐called heavy‐tailed Burr X (HTBX) distribution. It comes from the newly discovered heavy‐tailed (HT) family of distributions. A notable feature is that the associated probability density function can have a right‐skewed distribution that approximates symmetry, unimodality, and decreasing values, which makes it well suited for modeling various datasets. The mathematical properties of the new distribution are obtained by calculating the quantile function, ordinary moments, incomplete moments, moment generating function, conditional moment, mean deviation, Bonferroni curve, and Lorenz curve. Extensive simulation was performed to investigate the estimation of the model parameters using many established approaches, including maximum likelihood estimation, least squares estimation, weighted least squares estimation, Cramer–von Mises estimation, Anderson–Darling estimation, maximum product of spacing estimation, and percentile estimation. The simulation results showed the computational efficiency of these strategies and showed that the maximum likelihood strategy of estimation is the best strategy. The utility and importance of the newly proposed model are demonstrated by analyzing three real datasets. The HTBX distribution is compared to several well‐known extensions of the Burr distribution such as exponentiated Kavya‐Manoharan Burr X, Kavya‐Manoharan Burr X, Burr X, Kumaraswamy Rayleigh, Kumaraswamy Burr III, exponentiated Burr III, Burr III, Kumaraswamy Burr‐II, Rayleigh, and HT Rayleigh models by using different measures. The numerical results showed that the HTBX model fit the data better than the other competitive models.

Truncated Cauchy Power Kumaraswamy Generalized family of distributions: Theory and Applications

A new family called the Truncated Cauchy Power Kumaraswamy -G family of distributions is proposed. Some special models of this family are introduced. Statistical properties of the family such as expansion of density function, moments, incomplete moments, mean deviation, bonferroni and Lorenz curves are proposed. We discuss the method of maximum likelihood to estimate the model parameters and study its performance by simulation. Real data sets are modeled to illustrate the importance and exibility of the proposed model in comparison to some known ones yielded favourable results.