Joint Cumulative Distribution Function Research Articles

New tests to detect outliers in the Pareto distribution

In this article, we investigate outlier observations in the Pareto distribution, by introducing two new tests: the generalized likelihood ratio and the uniformly most powerful tests for the outlier parameter of this distribution. Prior to this, we outline the necessary prerequisites for our study, including a model for outliers, the density function of the Pareto distribution in the presence of k outliers which is obtained from the same distribution, etc. Furthermore, we present joint probability density functions (conditional) and joint cumulative distribution functions (conditional) through several Lemmas and Corollaries. Then, using simulation study, we compare the power of our introduced tests against previously established for detecting outliers. Finally, we provide real examples to demonstrate the performance of these tests.

A New Bivariate Survival Model: The Marshall-Olkin Bivariate Exponentiated Lomax Distribution with Modeling Bivariate Football Scoring Data

This paper focuses on applying the Marshall-Olkin approach to generate a new bivariate distribution. The distribution is called the bivariate exponentiated Lomax distribution, and its marginal distribution is the exponentiated Lomax distribution. Numerous attributes are examined, including the joint reliability and hazard functions, the bivariate probability density function, and its marginals. The joint probability density function and joint cumulative distribution function can be stated analytically. Different contour plots of the joint probability density function and joint reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are given. The unknown parameters and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are estimated using the maximum likelihood method. Also, the Bayesian technique is applied to derive the Bayes estimators and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution. In addition, maximum likelihood and Bayesian two-sample prediction are considered to predict a future observation from a future sample of the bivariate exponentiated Lomax distribution. A simulation study is presented to investigate the theoretical findings derived in this paper and to evaluate the performance of the maximum likelihood and Bayes estimates and predictors. Furthermore, the real data set used in this paper comprises the scoring times from 42 American Football League matches that took place over three consecutive independent weekends in 1986. The results of utilizing the real data set approve the practicality and flexibility of the bivariate exponentiated Lomax distribution in real-world situations, and the bivariate exponentiated Lomax distribution is suitable for modeling this bivariate data set.

Discrete Joint Random Variables in Fréchet-Weibull Distribution: A Comprehensive Mathematical Framework with Simulations, Goodness-of-Fit Analysis, and Informed Decision-Making

This paper introduces a novel four-parameter discrete bivariate distribution, termed the bivariate discretized Fréchet–Weibull distribution (BDFWD), with marginals derived from the discretized Fréchet–Weibull distribution. Several statistical and reliability properties are thoroughly examined, including the joint cumulative distribution function, joint probability mass function, joint survival function, bivariate hazard rate function, and bivariate reversed hazard rate function, all presented in straightforward forms. Additionally, properties such as moments and their related concepts, the stress–strength model, total positivity of order 2, positive quadrant dependence, and the median are examined. The BDFWD is capable of modeling asymmetric dispersion data across various forms of hazard rate shapes and kurtosis. Following the introduction of the mathematical and statistical frameworks of the BDFWD, the maximum likelihood estimation approach is employed to estimate the model parameters. A simulation study is also conducted to investigate the behavior of the generated estimators. To demonstrate the capability and flexibility of the BDFWD, three distinct datasets are analyzed from various fields, including football score records, recurrence times to infection for kidney dialysis patients, and student marks from two internal examination statistical papers. The study confirms that the BDFWD outperforms competitive distributions in terms of efficiency across various discrete data applications.

A vine-copulas based multi-sensor fusion structural damage monitoring method and its application in dam engineering

A vine-copulas based multi-sensor fusion structural damage monitoring method and its application in dam engineering

ON IMPROVED GAUSSIAN CORRELATION INEQUALITIES FOR SYMMETRICAL -RECTANGLES EXTENDED TO CERTAIN MULTIVARIATE GAMMA DISTRIBUTIONS AND SOME FURTHER PROBABILITY INEQUALITIES

The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint cumulative distribution function (cdf), which is MTP2 (multivariate totally positive of order 2). Inequalities of the given type hold at least for all MTP2-cdfs on or with everywhere positive smooth densities. In particular, at least some infinitely divisible multivariate chi-square distributions (gamma distributions in the sense of Krishnamoorthy and Parthasarathy) with any positive real “degree of freedom” are shown to be MTP2. Moreover, further numerically calculable probability inequalities for a broader class of multivariate gamma distributions are derived. A different improvement for inequalities of the GCI-type, and of a similar type with three instead of two groups of components with more special correlation structures is also obtained. The main idea behind these inequalities is to find for a given correlation matrix with positive correlations, a further correlation matrix with smaller correlations whose inverse is an M-matrix and where the corresponding multivariate gamma distribution function is numerically available.

Bivariate cubic normal distribution for non-Gaussian problems

Bivariate cubic normal distribution for non-Gaussian problems

Recurrence relations for the joint distribution of the sum and maximum of independent random variables

Abstract In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: (i) continuous and (ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz’s integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology.

Bivariate Lomax Distribution Based on Clayton Copula: Estimation and Prediction

Lomax distribution has been studied by many statisticians due to its important role in reliability modeling and lifetime testing. The bivariate Lomax distribution is an important lifetime distribution in survival analysis. In this article, a bivariate Lomax distribution is constructed based on Clayton copula. The joint probability density function and the joint cumulative distribution function are derived in closed forms. Some properties of this bivariate distribution are discussed. The maximum likelihood and Bayes estimators for the unknown parameters are derived. Also, the maximum likelihood and Bayesian two-sample prediction for the future observations are obtained. The performance of the proposed bivariate distribution is examined using a simulation study. Finally, one real data set under the proposed distribution is considered to illustrate its flexibility and applicability for real-life applications.

Probabilistic optimal power flow computation for power grid including correlated wind sources

AbstractThis paper sets out to develop an efficient probabilistic optimal power flow (POPF) algorithm to assess the influence of wind power on power grid. Given a set of wind data at multiple sites, their marginal distributions are fitted by a newly developed generalized Johnson system, whose parameters are specified by a percentile matching method. The correlation of wind speeds is characterized by a flexible Liouville copula, which allows to model the asymmetric dependence structure. In order to improve the efficiency for solving POPF problem, a lattice sampling method is developed to generate wind samples at multiple sites, and a logistic mixture model is proposed to fit distributions of POPF outputs. Finally, case studies are performed, the generalized Johnson system is compared with Weibull distribution and the original Johnson system for fitting wind samples, Liouville copula is compared against Archimedean copula for modelling correlated wind samples, and lattice sampling method is compared with Sobol sequence and Latin hypercube sampling for solving POPF problem on IEEE 118‐bus system, the results indicate the higher accuracy of the proposed methods for recovering the joint cumulative distribution function of correlated wind samples, as well as the higher efficiency for calculating statistical information of POPF outputs.

On the Bivariate Extension of the Extended Standard U-quadratic Distribution

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the method of compounding or pseudo family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape with the following properties: $(i)$ $X$ and $Y$ have bathtub shapes; $(ii)$ $X$ has constant and $Y$ has bathtub shapes; and $(iii)$ $X$ has inverted bathtub and $Y$ has bathtub shapes. Moreover, some properties of the derived distribution such as the marginal distribution, conditional distributions, conditional moments, product and ratio moments are derived. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. Also, a simulation study is carried out to evaluate the behavior of the estimates of the parameters. Moreover, we derive a new bivariate Kumaraswamy distribution and use it to simulate bivariate data with $X$ and $Y$ having bathtub shapes. Furthermore, A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the said Bivarite Cubic Transmuted Uniform distribution. The result shows that the proposed Bivariate eSU distribution provides a better fit for the said simulated dataset as compared with the Bivariate CTU distribution.

On bivariate compound exponentiated survival function of the beta distribution: estimation and prediction

A bivariate distribution is a probability distribution that describes the joint behavior of two random variables. It provides information about the simultaneous variation of two variables, allowing us to analyze their relationship and dependencies. This article discussed the bivariate compound exponentiated survival function of the beta distribution. the joint cumulative distribution function and the joint probability density function were found in closed forms. Several characteristics of this distribution have been discussed. The maximum likelihood (ML) estimators of the parameters and two sample ML predictions of the future observations are derived. The Bayes estimators (BEs) of the parameters based on the squared error loss function and two sample Bayesian predictions of the future observations are presented. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, two data sets are considered in the framework of the proposed distributions to demonstrate their flexibility for real-life applications.

Универсальный метод Монте-Карло для процессов Леви и его экстремумов

The article proposes a universal approach to constructing Monte Carlo methods for pricing options with payoffs depending on the joint distribution of the final position of the Lévy process X_T and its infimum J_T (or supremum S_T). We derive approximate formulas for the conditional cumulative distribution functions of the Lévy process "P"(X_T<x|S_T=y) (P(X_T<x|J_T=y)), which are expressed through the partial derivative with respect to y of the joint cumulative distribution function "P"(X_T<x,S_T<y) (P(X_T<x,J_T<y)) and the density of the infimum (or supremum) at the final moment of time. By applying the Laplace transform to the joint cumulative distribution function of the Lévy process and its extremum, we use the approximate Wiener–Hopf factorization to represent the image of its partial derivative. By inverting the Laplace transform using the Gaver–Stehfest algorithm, we find the desired conditional cumulative distribution function. The developed algorithm for simulating the joint position of the Lévy process and its extremum at a given point in time consists of two key stages. At the first stage, we simulate the extremum value of the L´evy process based on the approximation of its cumulative distribution function "P"(S_T<x) (or P(J_T<x)). In the second step, we simulate the final value of the Lévy process based on the approximation of the conditional cumulative distribution function of the final position of the Lévy process relative to its extremum. The universality of the Monte Carlo method we developed lies in the implementation of a uniform approach for a wide class of Lévy processes, in contrast to classical approaches, when simulations are essentially based on the features of the probability distribution associated with the simulated random process or its extrema. In our approach, it is enough to know the characteristic exponent of the Lévy process. The most time-consuming computational unit for simulating a random variable based on a known cumulative distribution function can be effectively implemented using neural networks and accelerated through parallel computing. Thus, on the one hand, the approach we propose is suitable for a wide class of Lévy models, on the other hand, it can be combined with machine learning methods.

Probabilistic Optimal Power Flow Computation Using Stratified Quadrature Rule

In this paper, the probabilistic optimal power flow (POPF) problem is modeled as a black box uncertainty quantification problem. If distribution functions of POPF inputs are unknown, only samples are available, the multivariate Gaussian mixture model is suggested to recover the joint cumulative distribution function of POPF inputs, and a probability selection method is introduced to relate POPF inputs to an independent standard normal vector. Then, moment matching equations are proposed to characterize the uncertainty effects of POPF inputs on outputs, whereby some properties of the POPF problem are revealed, and a stratified quadrature rule (SQR) is proposed for POPF computation by combining a Hadamard matrix based quadrature rule and a discrete sine transformation matrix based quadrature rule. This newly developed SQR has a computational burden linear with the number of POPF inputs, and performs well in estimating means and standard deviations POPF outputs. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the accuracy and efficiency of the proposed method.

Nonparametric Multivariate Probability Density Forecast in Smart Grids With Deep Learning

This paper proposes a nonparametric multivariate density forecast model based on deep learning. It not only offers the whole marginal distribution of each random variable in forecasting targets, but also reveals the future correlation between them. Differing from existing multivariate density forecast models, the proposed method requires no a priori hypotheses on the forecasted joint probability distribution of forecasting targets. In addition, based on the universal approximation capability of neural networks, the real joint cumulative distribution functions of forecasting targets are well-approximated by a special positive-weighted deep neural network in the proposed method. Numerical tests from different scenarios were implemented under a comprehensive verification framework for evaluation, including the very short-term forecast of the wind speed, wind power, and the day-ahead forecast of the aggregated electricity load. Testing results corroborate the superiority of the proposed method over current multivariate density forecast models considering the accordance with reality, prediction interval width, and correlations between different random variables.

Drought Assessment Using Two-Variate Modelling

Drought is a natural hazard that mainly threatens water resources and food security. Due to climate change, an increase in drought events that may adversely affect various sectors is expected. Therefore, a reliable analysis of drought events is important. However, there is a deficiency in research works which deals with two-variate drought assessment and monitoring. This paper focused on drought assessment based on modelling the effective precipitation and runoff variables simultaneously. Drought indices are widely utilized to assess droughts in many regions around the world. The copula method was used to modelling a two-variate drought index based on the effective precipitation and runoff variables in northern Iraq as a case study. It was found that Frank is the optimal copula function to model the joint cumulative distribution function of effective precipitation and runoff. To assess the performance of the proposed two-variate index, it was compared with the widely used standardized precipitation index SPI and Standardized Runoff Index SRI. Where the Pearson correlation coefficients between 2-variate index and SPI, SRI were 0.78 and 86, respectively, for 3-month time scale, and were 0.83 and 82, respectively, for 6-month time scale. The results revealed that the proposed index is applicable and can comprehensively and effectively characterize drought events.

Environmental contours based on imprecise probability distributions

Environmental contours based on imprecise probability distributions

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.

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.

Parallel-connected timers and deadbands for reducing the number of false and missed alarms

Parallel-connected timers and deadbands for reducing the number of false and missed alarms

Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process

In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time T, it is known that the one-point height function fluctuations are of order $$T^{1/3}$$ and governed by the Tracy–Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $$T^{2/3}$$ and vertically by $$T^{1/3}$$ , converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin–Bufetov–Wheeler between the stochastic six-vertex model and the ascending Hall–Littlewood process (a certain measure on plane partitions). Using the Macdonald difference operators, we obtain formulas for two-point observables for the ascending Hall–Littlewood process, which for the six-vertex model give access to the joint cumulative distribution function for its height function. A careful asymptotic analysis of these observables gives the two-point convergence result under certain restrictions on the parameters of the model.