Bivariate Extension Research Articles

Statistical Properties of a Generalization Erlang Truncated Exponential Distribution with Applications and Its Bivariate Extension

Statistical Properties of a Generalization Erlang Truncated Exponential Distribution with Applications and Its Bivariate Extension

New Kantorovich-type Szász–Mirakjan Operators

In this paper, we present a Kantorovich-type Szász–Mirakjan operators. Initially, we establish the recurrence relationship for the moments of these operators and provide the central moments up to the fourth degree. Subsequently, we analyze the local approximation properties of these operators using Peetre’s K-function. We investigate the rate of convergence, by utilizing the ordinary modulus of continuity and Lipschitz-type maximal functions. Additionally, we prove weighted approximation theorems and Voronoskaja-type theorems specific to these new operators. Following this, we introduce bivariate extension of these operators and investigate some approximation properties. Lastly, we include several numerical illustrative examples.

Properties, quantile regression, and application of bounded exponentiated Weibull distribution to COVID-19 data of mortality and survival rates

Well-known continuous distributions such as Beta and Kumaraswamy distribution are useful for modeling the datasets which are based on unit interval [0,1]. But every distribution is not always useful for all types of data sets, rather it depends on the shapes of data as well. In this research, a three-parameter new distribution named bounded exponentiated Weibull (BEW) distribution is defined to model the data set with the support of unit interval [0,1]. Some fundamental distributional properties for the BEW distribution have been investigated. For modeling dependence between measures in a dataset, a bivariate extension of the BEW distribution is developed, and graphical shapes for the bivariate BEW distribution have been shown. Several estimation methods have been discussed to estimate the parameters of the BEW distribution and to check the performance of the estimator, a Monte Carlo simulation study has been done. Afterward, the applications of the BEW distribution are illustrated using COVID-19 data sets. The proposed distribution shows a better fit than many well-known distributions. Lastly, a quantile regression model from bounded exponentiated Weibull distribution is developed, and its graphical shapes for the probability density function (PDF) and hazard function have been shown.

A new bathtub and increasing failure rate model: An extension of the Mustapha type II distribution

This article introduces a new three-parameter lifetime model with an increasing and bathtub failure rate functions as an extension of the Mustapha type II distribution (MuII). The model can be very useful in statistical studies, reliability, computer sciences and engineering. Various mathematical and statistical properties of the distribution are discussed, such as moments, mean deviations, Bonferroni and Lorenz curves, entropy, order statistic, and extreme value distributions. Moreover, we consider the bivariate extension of the new model. Statistical inferences by the maximum likelihood method are discussed and assess by simulation studies. Applications of the proposed model to two right-skewed data are presented for illustration. The new model provides a better fit than some other existing distribution as measured by some model selection criteria and goodness of fits statistics.

Bivariate Extension of Weighted Dynamic Survival Extropy and Its Quantile Approach

This paper considers shift-dependent survival extropy in a bivariate setup and studies its properties. We also look into the problem of extending the same measure for conditionally specified models. It is shown that the proposed measure uniquely determines the distribution. Also, we established nonparametric estimators for the conditionally specified model, and the asymptotic properties of the proposed estimator are established under some regularity conditions. The effect of the proposed estimator is illustrated using simulated and real data sets. We have also proposed a quantile-based measure which uniquely determines the distribution. Also, we have obtained the real-life application of the proposed measure for a quantile function whose distribution function does not exist.

A new improved form of the Lomax model: Its bivariate extension and an application in the financial sector

The Lomax model, also known as Pareto Type-II, has broad feasibility, especially in financing. This work introduces a new generalization of the Lomax model called the arc-sine exponentiation Lomax, which helps consider economic phenomena. The arc-sine exponentiation Lomax distribution captures a variety of shapes of density and hazard functions. The estimators of the proposed model’s parameters are derived using the maximum likelihood method. In a simulation study, the accuracy and efficacy of estimators are evaluated by computing their mean square errors and biases.Furthermore, a bivariate extension of the arc-sine exponentiation Lomax model is also introduced. The bivariate extension is introduced using Farlie–Gumble–Morgenstern copula approach. The new bivariate model is called Farlie–Gumble–Morgenstern arc-sine exponentiation Lomax distribution. Finally, a data set of thirty-two observations representing the export of goods demonstrates the arc-sine exponentiation Lomax model. The best-fitting results of the arc-sine exponentiatial Lomax are compared with some prominent extensions of the Lomax distribution.

Modified Weighted Rayleigh Distribution and Its Bivariate Extension

In this paper, a new version of weighted Rayleigh distribution is constructed and studied. The statistical properties of the new distribution including the behavior of hazard and reversed hazard functions, moments, the central moments, moment generating function, mean, variance, coefficient of skewness, coefficient of kurtosis, median, mode, quantiles, stochastic ordering, exact information matrix and order statistics are also obtained, a simulation study and real data applications are performed. Furthermore, a bivariate extension of the new distribution called the bivariate modified Rayleigh (BMWR) distribution is introduced. The proposed bivariate distribution is of type Farlie–Gumbel–Morgenstern (FGM) copula. The BMWR distribution has modified weighted Rayleigh marginal distributions. The joint cumulative distribution function, the joint survival function, the joint probability density function, the joint hazard rate function and the statistical properties of the BMWR distribution are also derived.

Modified Weighted Uniform Distribution And Its Bivariate Extension

In this paper a new version of weighted uniform distribution is constructed and studied. The statistical properties of the new distribution including the behavior of hazard and reversed hazard functions, moments, the central moments, moment generating function, mean, variance, coefficient of skewness, coefficient of kurtosis, median, mode, quantile , stochastic ordering and order statistics are also obtained, a simulation study and real data applications are performed. Moreover, a bivariate extension of the new distribution named the bivariate modified uniform (BMWU) distribution is introduced. The proposed bivariate distribution is of type Farlie–Gumbel–Morgenstern (FGM) copula. The BMWU distribution has modified weighted uniform marginal distributions. The joint cumulative distribution function, the joint survival function, the joint probability density function, the joint hazard rate function and the statistical properties of the BMWU distribution are also derived.

New estimation methods for extremal bivariate return curves

AbstractIn the multivariate setting, estimates of extremal risk measures are important in many contexts, such as environmental planning and structural engineering. In this paper, we propose new estimation methods for extremal bivariate return curves, a risk measure that is the natural bivariate extension to a return level. Unlike several existing techniques, our estimates are based on bivariate extreme value models that can capture both key forms of extremal dependence. We devise tools for validating return curve estimates, as well as representing their uncertainty, and compare a selection of curve estimation techniques through simulation studies. We apply the methodology to two metocean data sets, with diagnostics indicating generally good performance.

The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data

The usefulness of (probability) distributions in the field of biomedical science cannot be underestimated. Hence, several distributions have been used in this field to perform statistical analyses and make inferences. In this study, we develop the arctan power (AP) distribution and illustrate its application using biomedical data. The distribution is flexible in the sense that its probability density function exhibits characteristics such as left-skewedness, right-skewedness, and J and reversed-J shapes. The characteristic of the corresponding hazard rate function also suggests that the distribution is capable of modeling data with monotonic and non-monotonic failure rates. A bivariate extension of the AP distribution is also created to model the interdependence of two random variables or pairs of data. The application reveals that the AP distribution provides a better fit to the biomedical data than other existing distributions. The parameters of the distribution can also be fairly accurately estimated using a Bayesian approach, which is also elaborated. To end the study, the quantile and modal regression models based on the AP distribution provided better fits to the biomedical data than other existing regression models.

A complementary, two-method spherical approach to direction-based archeomagnetic dating

We present a two-method spherical approach to archeomagnetic dating based on directional variations of the geomagnetic field after vector treatment of all data, including individual in situ structure-level data used to calculate the reference variation curve and the archeomagnetic direction to be dated. In this paper, the reference curve for France was determined from a compilation of data acquired from kiln structures using a sliding window technique in which the varying durations and time shifts between windows are fixed according to the temporal distribution of individual reference data, as well as the bivariate extension of Fisher's statistics. The first dating method involves identifying the time interval(s) in which the direction to be dated is closest to the mean directions that define the reference curve. The angles between the direction to be dated and reference curve directions, and the Fisher probability density function allow us to derive a coincidence probability density curve, from which it is possible to estimate a 95% probability level-based dating interval. The second dating method involves the determination of an archeomagnetic date in which the direction to be dated is statistically identical to a dated reference curve direction at the 95% confidence level. This approach is much more restrictive than the previous method because it requires an excellent agreement between the test and reference directions to obtain a dating result, while the first method is only based on relative proximity. Using examples of archeomagnetic dating, we show that these two methods are complementarity and should be applied jointly to account for some of the limitations inherent in archeomagnetic dating, particularly due to the dispersion of the reference structure-level data.

A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data

The primary objective of this article was to introduce a new probabilistic model for the discussion and analysis of random covariates. The introduced model was derived based on the Marshall–Olkin shock model. After proposing the mathematical form of the new bivariate model, some of its distributional properties, including joint probability distribution, joint reliability distribution, joint reversed (hazard) rate distribution, marginal probability density function, conditional probability density function, moments, and distributions for both Y=max{X1,X2} and W=min{X1,X2}, were investigated. This novel model can be applied to discuss and evaluate symmetric and asymmetric data under various kinds of dispersion. Moreover, it can be used as a probability approach to analyze different shapes of hazard rates. The maximum likelihood approach was utilized for estimating the parameters of the bivariate model. A simulation study was carried out to assess the performance of the parameters, and it was noted that the maximum likelihood technique can be used to generate consistent estimators. Finally, two real datasets were analyzed to illustrate the notability of the novel bivariate distribution, and it was found that the suggested distribution provided a better fit than the competitive bivariate models.

Some Structural Properties of the Generalized Kumaraswamy (GKw) $q_T-X$ Class of Distributions

Ampadu [1] introduced quantile generated probability distributions as a new way to generate continuous distributions. Combining this idea with the two-parameter Kumaraswamy (Kw) distribution, for example, see Wikipedia contributors [15], this paper introduces a so-called (GKw) $q_T-X$ class of distributions, and obtains some of their structural properties. Practicality of sub-models of this new class of distributions is shown to be effective in modeling real life data. Practicality to various disciplines is proposed as further investigation. A bivariate extension of this new class of distribution is also proposed, and the reader is asked to investigate its properties and applications.

Approximation properties of bivariate extension of blending type operators

The present article is in the continuation of our previous work [26], where we have improved the order of approximation of ??Bernstein P?lt?nea operators. In the given note, we study the bivariate extension of first order modification of these operators and their approximation properties such as convergence, error of approximation in terms of complete and partial modulus of continuity and their asymptotic formula. We present numerical examples to show the convergence of functions of two variables with the help of MATLsoftware. Also, we construct the GBS operators associated to the bivariate extension and present their approximation behavior.

Bivariate q-extended Weibull morgenstern family and correlation coefficient formulas for some of its sub-models

<abstract><p>A bivariate extension of a flexible univariate family is proposed. The new family is called bivariate q-extended Weibull Morgenstern family of distributions which can be constructed based on the Farlie-Gumbel-Morgenstern (FGM) copula technique. After introducing the new family, four sub-models are discussed in detail from the theoretical and numerical coefficient of correlation point of view with pointing to the effect of the $ q $ parameter.</p></abstract>

A new transmuted family of distributions: Properties and estimation with applications

In this paper, a new transmuted general family of distributions is introduced and studied. It offers an original alternative to the former transmuted family by revisiting its mathematical structure, opening a new perspective on modeling. We derive expressions for the moments, the incomplete moments, the moment generating function, entropy measures, and order statistics as main properties. Further, we introduce a bivariate extension of the new family. Some members of the family are introduced, involving Burr, Gompertz, Weibull and gamma distributions. We discuss the different methods of estimation of the model parameters and a simulation study is performed for one of the special models to check the asymptotic behavior of the estimates under all estimation methods. Further, the interest of the family is illustrated by fitting two real-life data sets to the mentioned sub-models.

Bivariate Lupaş-Durrmeyer type operators involving Pólya distribution

In this paper, we consider a bivariate extension of blending type approximation by Lupa?-Durrmeyer type operators involving P?lya Distribution. We illustrate the convergence rate of these type operators using Peetre?s K-functional, modulus of smoothness and for functions in a Lipschitz type space.

A bivariate extension of the Omega distribution for two-dimensional proportional data

Abstract When data generating mechanism generates two correlated data sets both defined on the unit interval, a bivariate probabilistic distribution defined on the unit square is needed for modelling the data. For this purpose, we give a Marshall-Olkin type bivariate extension of an omega distribution in this paper. This is in fact a bivariate unit-exponentiated-half-logistic distribution. We study its mathematical properties in detail. The distribution contains neither an exponential term nor any special function which complicates the computations. Maximum likelihood estimation method and its large sample inference are considered for model parameters. Alternatively, we also propose an expectation- maximization algorithm to compute the estimates. To see the performances of the proposed estimators and validate the theoretical results obtained for estimation, we present the results of a simulation study. Data fitting demonstrations show its applicability in modelling random proportions.

New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression

Probability distributions are very useful in modeling lifetime datasets. However, no specific distribution is suitable for all kinds of datasets. In this study, the bounded truncated Cauchy power exponential distribution is proposed for modeling datasets on the unit interval. The probability density function exhibits desirable shapes, such as left-skewed, right-skewed, reversed J, and bathtub shapes, whereas the hazard rate function displays J and bathtub shapes. For the purpose of modeling dependence between measures in a dataset, a bivariate extension of the proposed distribution is developed. The bivariate probability density function displays monotonic and non-monotonic shapes, making it suitable for modeling complex bivariate relations. Subsequently, the applications of the distribution are illustrated using COVID-19 data. The results revealed that the new distribution provides a better fit to the datasets compared to other existing distributions. Finally, a new quantile regression model is developed and its application demonstrated. The generated quantile regression model offers a decent fit to the data, according to the residual analysis.

On the Bivariate Composite Gumbel–Pareto Distribution

In this paper, we propose a bivariate extension of univariate composite (two-spliced) distributions defined by a bivariate Pareto distribution for values larger than some thresholds and by a bivariate Gumbel distribution on the complementary domain. The purpose of this distribution is to capture the behavior of bivariate data consisting of mainly small and medium values but also of some extreme values. Some properties of the proposed distribution are presented. Further, two estimation procedures are discussed and illustrated on simulated data and on a real data set consisting of a bivariate sample of claims from an auto insurance portfolio. In addition, the risk of loss in this insurance portfolio is estimated by Monte Carlo simulation.