Continuous Univariate Distributions Research Articles

A New Family of Distributions with an Application to Exponentially Distribution

This paper introduces a new continuous distribution family called the Alpha logarithm family, which is a new modelling strategy for fitting data subject to univariate continuous distributions. This is achieved by introducing an additional parameter for greater flexibility using a single-parameter Natural logarithm transformation which can enhance some of the modeling capabilities of some Parental Continuous Distributions: This technique was applied to the exponential distribution to obtain a new two-parameter distribution, and the changes that occurred in the exponential distribution were observed. The general properties and functions of the new distribution were also derived and studied, and the estimators of the two parameters were derived. The efficiency of the estimators is verified through the simulation study. The new distribution is also applied to two sets of real data to prove the benefit of the new transformation, and we show that the proposed model is better than the asymptotic distributions with which it was compared on the selected data.

Lomax tangent generalized family of distributions: Characteristics, simulations, and applications on hydrological-strength data

This article proposes and discusses a novel approach for generating trigonometric G-families using hybrid generalizers of distributions. The proposed generalizer is constructed by utilizing the tangent trigonometric function and distribution function of base model G(x). The newly proposed family of uni-variate continuous distributions is named the “Lomax Tangent Generalized Family of Distributions (LT-G)” and structural-mathematical-statistical properties are derived. Some special and sub-models of the proposed family are also presented.A Weibull-based model, ‘The Lomax Tangent Weibull (LT-W) Distribution,” is discussed and the plots of density (pdf) and hazard (hrf) functions are also explained. Model parameter estimates are estimated by employing the maximum likelihood estimation (MLE) procedure. The accuracy of the MLEs is evaluated through Monte Carlo simulation. Last but not least, to demonstrate the flexibility and potential of the proposed distribution, two actual hydrological and strength data sets are analyzed. The obtained results are compared with well-known, competitive, and related existing distributions.

A new univariate continuous distribution with applications in reliability

In this article, the odd Lomax Gompertz distribution has been introduced, which is derived by modifying the Gompertz distribution to serve as a baseline model in the odd generalized Lomax distribution. The newly proposed model offers enhanced flexibility and provides a promising alternative for modeling lifetime data. This study seeks to establish a solid theoretical foundation for its application through the exploration of several properties, such as non-central moments, stochastic orderings quantile function, and entropy measure, for the new model. Additionally, by conducting simulation analysis, the performance of the various estimation methods is being assessed, which enables the identification of the most reliable approach for estimating the unknown parameters of the newly developed model. The simulation analysis of the two-risk metrics, namely, value at risk and expected shortfall, revealed the ability of the distribution to capture diverse failure rate patterns, which makes it particularly relevant for assessing financial risks. Finally, the suggested model is practiced to two real-life datasets to provide the compelling evidence of superior flexibility and practical versatility compared to existing models in the literature.

The T-Exponentiated Exponential{Frechet} Family of Distributions: Theory and Applications

This article introduces a new family of Generalized Exponentiated Exponential distribution. Using the T-R{Y} framework, a new family of T-Exponentiated Exponential{Y} distributions named T-Exponentiated Exponential{Frechet} family of distributions is proposed. Some general properties of the family such as hazard rate function, quantile function, non-central moment, mode, mean absolute deviations and Shannon’s entropy are discussed. A new continuous univariate probability distribution which is a member of the T-Exponentiated Exponential{Frechet} family of distributions is introduced. From the general properties of the family, expressions are derived for some specific properties of the new distribution. To show the usefulness of the T-Exponentiated Exponential{Frechet} family of distributions, the new distribution is fitted to two real life data sets and the results are compared with the results of some other existing distributions.

Explicit Expressions for Most Common Entropies.

Entropies are useful measures of variation. However, explicit expressions for entropies available in the literature are limited. In this paper, we provide a comprehensive collection of explicit expressions for four of the most common entropies for over sixty continuous univariate distributions. Most of the derived expressions are new. The explicit expressions involve known special functions.

Distribution-free Phase II triple EWMA control chart for joint monitoring the process location and scale parameters

ABSTRACT Distribution-free or nonparametric control charts are used for monitoring the process parameters when there is a lack of knowledge about the underlying distribution. In this paper, we investigate a single distribution-free triple exponentially weighted moving average control chart based on the Lepage statistic (referred as TL chart) for simultaneously monitoring shifts in the unknown location and scale parameters of a univariate continuous distribution. The design and implementation of the proposed chart are discussed using time-varying and steady-state control limits for the zero-state case. The run-length distribution of the TL chart is evaluated by performing Monte Carlo simulations. The performance of the proposed chart is compared to those of the existing EWMA-Lepage (EL) and DEWMA-Lepage (DL) charts. It is observed that the TL chart with a time-varying control limit is superior to its competitors, especially for small to moderate shifts in the process parameters. We also provide a real example from a manufacturing process to illustrate the application of the proposed chart.

Uncertainty quantification in low voltage distribution grids: Comparing Monte Carlo and general polynomial chaos approaches

Changes in load and distributed generation in low voltage distribution systems (LVDS) have made the individual consumer offtake highly uncertain for the system operator. In order to accurately determine hosting capacity of such systems, congestion related stochastic indices, e.g. probability of undervoltage and overvoltage, must be calculated. These require either too many deterministic calculations (simulation-based methods) or the use of analytical probabilistic power flow. Numerical simulation-based methods often use a Monte Carlo (MC) based approach due to its simplicity. Recently, analytical methods such as general polynomial chaos (gPC) expansion have gained increasing interest. This paper develops a non-intrusive gPC formulation for congestion determination in LVDS and illustrates its effectiveness compared to the MC methods. Both MC and gPC methods are compared for computational time, and accuracy, for a set of realistic feeders with high photovoltaic (PV) penetration. The PV power injection uncertainty is characterized by a univariate continuous distribution. The paper illustrates the merits of using the general polynomial chaos expansion with degree 2 and Sobol sequences based testing points for the probabilistic assessment of LVDS with many uncertainty sources. • Monte Carlo, Quasi-Monte Carlo and general Polynomial Chaos are compared. • The comparisons are made for low voltage distribution system applications. • The different variant of general Polynomial Chaos is compared for the tail and mean events to obtain the best variant. • The influence of the number of uncertainty sources in choosing the right approach is evaluated. • Analysis helps to select the most appropriate probabilistic power flow tool for distribution systems.

On a prior based on the Wasserstein information matrix

We introduce a prior for the parameters of univariate continuous distributions, based on the Wasserstein information matrix, which is invariant under reparameterisations. We discuss the links between the proposed prior with information geometry. We present sufficient conditions for the propriety of the posterior distribution for general classes of models. We present a simulation study that shows that the induced posteriors have good frequentist properties.

On distributions with fixed marginals maximizing the joint or the prior default probability, estimation, and related results

Motivated by (random) lifetimes of electronic components or financial institutions we study the problem of maximizing the probability that (i) a random variable X is not smaller than another random object Y and (ii) that X and Y coincide within the class of all random variables X,Y with given univariate continuous distribution functions F and G, respectively. We show that the maximization problems correspond to finding copulas maximizing the mass of the endograph Γ≤(T)={(x,y)∈[0,1]2:y≤T(x)} and the graph Γ(T)={(x,T(x)):x∈[0,1]} of T=G∘F−, respectively. After providing simple, copula-based proofs for the existence of copulas attaining the two maxima m¯T and w¯T we generalize the obtained results to the case of general (not necessarily monotonic) transformations T:[0,1]→[0,1] and derive simple and easily calculable formulas for m¯T and w¯T involving the distribution function FT of T (interpreted as random variable on [0,1]). The latter are then used to characterize all non-decreasing transformations T:[0,1]→[0,1] for which m¯T and w¯T coincide. A strongly consistent estimator for m¯T is derived and proven to be asymptotically normal under very mild regularity conditions. Several examples and graphics illustrate the main results and falsify some seemingly natural conjectures, an application of some of the obtained results to the seemingly unrelated topic of relative effects indicates the importance of the tackled questions.

Bivariate Discrete Poisson-Lindley Distributions.

Two families of bivariate discrete Poisson–Lindley distributions are introduced. The first is derived by mixing the common parameter in a bivariate Poisson distribution by different models of univariate continuous Lindley distributions. The second is obtained by generalizing a bivariate binomial distribution with respect to its exponent when it follows any of five different univariate discrete Poisson–Lindley distributions with one or two parameters. The use of probability-generating functions is mainly employed to derive some general properties for both families and specific characteristics for each one of their members. We obtain expressions for probabilities, moments, conditional distributions, regression functions, as well as characterizations for certain bivariate models and their marginals. An attractive property of all bivariate individual models is that they contain only two or three parameters, and one of them is readily estimated by simple ratios of their sample means. This feature, and since all marginal distributions are over-dispersed, strongly suggests their potential use to describe bivariate dependent count data in many different areas.

Uniqueness of principal points with respect to p-order distance for a class of univariate continuous distribution

In this paper, the uniqueness of principal points with respect to p-order distance is studied. When the density function of univariate bounded continuous distribution fluctuates in a specified range, the principal points are unique. It is completely different from the strongly unimodal condition.

On a Special Weighted Version of the Odd Weibull-Generated Class of Distributions

In recent advances in distribution theory, the Weibull distribution has often been used to generate new classes of univariate continuous distributions. They find many applications in important disciplines such as medicine, biology, engineering, economics, informatics, and finance; their usefulness is synonymous with success. In this study, a new Weibull-generated-type class is presented, called the weighted odd Weibull generated class. Its definition is based on a cumulative distribution function, which combines a specific weighted odd function with the cumulative distribution function of the Weibull distribution. This weighted function was chosen to make the new class a real alternative in the first-order stochastic sense to two of the most famous existing Weibull generated classes: the Weibull-G and Weibull-H classes. Its mathematical properties are provided, leading to the study of various probabilistic functions and measures of interest. In a consequent part of the study, the focus is on a special three-parameter survival distribution of the new class defined with the standard exponential distribution as a reference. The exploratory analysis reveals a high level of adaptability of the corresponding probability density and hazard rate functions; the curves of the probability density function can be decreasing, reversed N shaped, and unimodal with heterogeneous skewness and tail weight properties, and the curves of the hazard rate function demonstrate increasing, decreasing, almost constant, and bathtub shapes. These qualities are often required for diverse data fitting purposes. In light of the above, the corresponding data fitting methodology has been developed; we estimate the model parameters via the likelihood function maximization method, the efficiency of which is proven by a detailed simulation study. Then, the new model is applied to engineering and environmental data, surpassing several generalizations or extensions of the exponential model, including some derived from established Weibull-generated classes; the Weibull-G and Weibull-H classes are considered. Standard criteria give credit to the proposed model; for the considered data, it is considered the best.

Computing highest density regions for continuous univariate distributions with known probability functions

We examine the problem of computing the highest density region (HDR) in a computational context where the user has access to a density function and quantile function for the distribution (e.g., in the statistical language R). We examine several common classes of continuous univariate distributions based on the shape of the density function; this includes monotone densities, quasi-concave and quasi-convex densities, and general multimodal densities. In each case we show how the user can compute the HDR from the quantile and density functions by framing the problem as a nonlinear optimisation problem. We implement these methods in R to obtain general functions to compute HDRs for classes of distributions, and for commonly used families of distributions. We compare our method to existing R packages for computing HDRs and we show that our method performs favourably in terms of both accuracy and average speed.

On the Log-Logistic Distribution and Its Generalizations: A Survey

In this paper, we present a review on the log-logistic distribution and some of its recent generalizations. We cite more than twenty distributions obtained by different generating families of univariate continuous distributions or compounding methods on the log-logistic distribution. We reviewed some log-logistic mathematical properties, including the eight different functions used to define lifetime distributions. These results were used to obtain the properties of some log-logistic generalizations from linear representations. A real-life data application is presented to compare some of the surveyed distributions.

On some properties of Rayleigh-Cauchy distribution

We introduce a three-parameter univariate continuous distribution called the Raleigh-Cauchy Distribution (RCD) to extend the two-parameter Cauchy distribution using T-R{Y} families. Some structural properties of this new distribution such as reliability measures and hazard function, mixture representation, order statistics and asymptotic behavior of RCD are obtained. The plots of the density and hazard rate of four cases of the new distribution were constructed for ease of understanding of its shapes under different parameter combinations. The method of maximum likelihood estimation was proposed in estimating its parameters. The new distribution provides consistently better fit than the other four models selected from the literature.

On a new modeling strategy: The logarithmically-exponential class of distributions

<abstract> In this paper, a promising modeling strategy for data fitting is derived from a new general class of univariate continuous distributions. This class is governed by an original logarithmically-exponential one-parameter transformation which has the ability to enhance some modeling capabilities of any parental distribution. In relation to the current literature, it appears to be a "limit case" of the well-established truncated generalized Fréchet generated class. In addition, it offers a natural alternative to the famous odd inverse exponential generated class. Some special distributions are presented, with particular interest in a novel heavy-tailed three-parameter distribution based on the Lomax distribution. Functional equivalences, modes analysis, stochastic ordering, functional expansions, moment measures, information measures and reliability measures are derived. From generic or real data, our modeling strategy is based on the new class combined with the maximum likelihood approach. We apply this strategy to the introduced modified Lomax model. The efficiency of the three parameter estimates is validated by a simulation study. Subsequently, two referenced real data sets are adjusted according to the rules of the art; the first one containing environmental data and the second one, financial data. In particular, we show that the proposed model is preferable to four concurrents also derived from the Lomax model, including the odd inverse exponential Lomax model. </abstract>

A general class of new continuous mixture distribution and application

A generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an investigation is conducted on a distribution and its generalization. Several available generalizations of the distribution are reviewed and recent trends in the construction of generalized classes with a generalized mixing parameter are discussed. To check the suitability and comparability, real data set have been used.

Weibull-H-NHE distribution with properties and applications on COVID-19 dataset

A new three parameters univariate continuous distribution called Weibull-H-NHE distribution on COVID-19 data with Properties and Applications on COVID-19 data using Weibull-Hclass of distribution is introduced here. Some major properties of model with application on covid-19 data are presented in study. Different curves like probability density, hazard rate, and empirical distribution etc. are included in the study. Parameters are estimated by using method of least square. Model validation is tested by calculating different information criteria values as well as using Q-Q and P-P plots. The applicability of the model is tested on three sets of real data sets. Model application is checked by considering their real data set. All the calculations are done using the R language programming.

A general class of new continuous mixture distribution and application

A generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an investigation is conducted on a distribution and its generalization. Several available generalizations of the distribution are reviewed and recent trends in the construction of generalized classes with a generalized mixing parameter are discussed. To check the suitability and comparability, real data set have been used.

The Gamma Kumaraswamy-G family of distributions: theory, inference and applications

Abstract In this paper, we introduce a new family of univariate continuous distributions called the Gamma Kumaraswamy-generated family of distributions. Most of its properties are studied in detail, including skewness, kurtosis, analytical comportments of the main functions, moments, stochastic ordering and order statistics. The next part of the paper focuses on a particular member of the family with four parameters, called the gamma Kumaraswamy exponential distribution. Among its advantages, the following should be mentioned: the corresponding probability density function can have symmetrical, left-skewed, right-skewed and reversed-J shapes, while the corresponding hazard rate function can have (nearly) constant, increasing, decreasing, upside-down bathtub, and bathtub shapes. Subsequently, the inference on the gamma Kumaraswamy exponential model is performed. The method of maximum likelihood is applied to estimate the model parameters. In order to demonstrate the importance of the new model, analyses on two practical data sets were carried out. The results proved more favourable for the studied model than for any of the other eight competitive models.