Continuous Probability Distributions Research Articles

Spectral decomposition of [formula omitted] and Poincaré inequality on a compact interval – Application to kernel quadrature

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ∫abf(x)dμ(x)=∑i=1nwif(xi) where f belongs to H1(μ). Here, μ belongs to a class of continuous probability distributions on [a,b]⊂R and ∑i=1nwiδxi is a discrete probability distribution on [a,b]. We show that H1(μ) is a reproducing kernel Hilbert space with a continuous kernel K, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although K has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a T-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature.We derive several results for the Poincaré quadrature weights and the associated worst-case error. When μ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as b−a23n−1 for large n. By comparison with known results for H1(0,1), this shows that the Poincaré quadrature is asymptotically optimal. For a general μ, we provide an efficient numerical procedure, based on finite elements and linear programming. Numerical experiments provide useful insights: nodes are nearly evenly spaced, weights are close to the probability density at nodes, and the worst-case error is approximately O(n−1) for large n.

Comparison of probability distributions used for harnessing the wind energy potential: a case study from India

AbstractModeling wind speed data is the prime requirement for harnessing the wind energy potential at a given site. While the Weibull distribution is the most commonly employed distribution in the literature and in practice, numerous scientific articles have proposed various alternative continuous probability distributions to model the wind speed at their convenient sites. Fitting the best distribution model to the data enables the practitioners to estimate the wind power density more accurately, which is required for wind power generation. In this paper we comprehensively review fourteen continuous probability distributions, and investigate their fitting capacities at seventeen locations of India covering the east and west offshore corner as well as the mainland, which represents a large variety of climatological scenarios. A first main finding is that wind speed varies a lot inside India and that one should treat each site individually for optimizing wind power generation. A second finding is that the wide acceptance of the Weibull distribution should at least be questioned, as it struggles to represent wind regimes with heterogeneous data sets exhibiting multimodality, high levels of skewness and/or kurtosis. Our study reveals that mixture distributions are very good alternative candidates that can model difficult shapes and yet do not require too many parameters.

Skewness-seeking behavior and financial investments

AbstractRecent theoretical and empirical advancements highlight the pivotal role played by higher-order moments, such as skewness, in shaping financial decision-making. Nevertheless, contemporary experimental research predominantly relies on limited-outcome lotteries, an oversimplified representation distant from real-world investment dynamics. To bridge this research gap, we conducted a rigorously pre-registered experiment. Our study delves into individuals’ preferences for investment opportunities, examining the influence of skewness of continuous probability distributions of returns. We document an inclination towards positively skewed outcome distributions. Furthermore, we uncovered a substitution effect between risk appetite and the sign of skewness. Finally, we unveiled a robust positive correlation between skewness-seeking behavior and a propensity for speculative behavior. Simultaneously, a distinct negative correlation surfaced between skewness-seeking behavior and the perceived risk associated with positive skewness.

The Generalized Odd Maxwell-Kumaraswamy Distribution: Its Properties and Applications

This study introduces a novel family of continuous probability distributions using Alzaatreh‘s technique to explore the Generalized Odd Maxwell-Generated (GOM-G) distribution family. Within the Generalized Odd Maxwell family, the GOM-Kumaraswamy distribution is presented as extended form of the Kumaraswamy distribution. The investigation thoroughly examines the cumulative distribution function, probability density function, hazard, and survival functions, as well as mixture representations of the Generalized Odd Maxwell-Kumaraswamy distribution. The suggested distribution's structural properties, including moments, skewness, kurtosis, probability-weighted moments, entropies, stress-strength models, and order statistics, are derived. Model parameters are estimated through the maximum likelihood method, and simulation studies evaluate the performance of maximum likelihood estimations using the quantile function. Employing two real-life datasets, goodness-of-fit measures such as Akaike Information Criterion (AIC), Corrected Akaike Information Criterion, Bayesian Information Criterion (BIC), Hannan-Quinn Information Criterion (HQIC), and chi-square goodness-of-fit tests demonstrate the adaptability and flexibility of the GOM-Kumaraswamy distribution against competing distributions, including Kumaraswamy-Kumaraswamy and Kumaraswamy-Burr III. The results reveal that the GOM-Kumaraswamy distribution exhibits the lowest AIC, CAIC, BIC, and HQIC values and the highest goodness-of-fit values compared to other models, suggesting its superiority as the preferred fit for both dataset forms. This proposed distribution contributes to practical applications, unveiling its potential significance in modeling real-world phenomena within the domains of hydrology and engineering.

Maximum Entropy Criterion for Moment Indeterminacy of Probability Densities.

We deal with absolutely continuous probability distributions with finite all-positive integer-order moments. It is well known that any such distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). In this paper, we follow the maximum entropy approach and establish a new criterion for the M-indeterminacy of distributions on the positive half-line (Stieltjes case). Useful corollaries are derived for M-indeterminate distributions on the whole real line (Hamburger case). We show how the maximum entropy is related to the symmetry property and the M-indeterminacy.

Assessing the Impact of Copula Selection on Reliability Measures of Type P(X < Y) with Generalized Extreme Value Marginals

In reliability studies, we are interested in the behaviour of a system when it interacts with its surrounding environment. To assess the system’s behaviour in a reliability sense, we can take the system’s intrinsic quality as strength and the outcome of interactions as stress. Failure is observed whenever stress exceeds strength. Taking Y as a random variable representing the stress the system experiences and random variable X as its strength, the probability of not failing can be taken as a proxy for the reliability of the component and given as P(Y<X)=1−P(X<Y). This way, in the present paper, it is considered that X and Y follow generalized extreme value distributions, which represent a family of continuous probability distributions that have been extensively applied in engineering and economic contexts. Our contribution deals with a more general scenario where stress and strength are not independent and copulas are used to model the dependence between the involved random variables. In such modelling framework, we explored the proper selection of copula models characterizing the dependence structure. The Gumbel–Hougaard, Frank, and Clayton copulas were used for modelling bivariate data sets. In each case, information criteria were considered to compare the modelling capabilities of each copula. Two economic applications, as well as an engineering one, on real data sets are discussed. Overall, an easy-to-use methodological framework is described, allowing practitioners to apply it to their own research projects.

A novel generalized Weibull Poisson G classof continuous probabilisticdistributions with some copulas, properties and applications to real-life datasets.

The current study introduces and examines copula-coupled probability distributions. It explains their mathematical features and shows how they work with real datasets. Researchers, statisticians, and practitioners can use this study's findings to build models that capture complex multivariate data interactions for informed decision-making. The versatility of compound Gfamilies of continuous probability modelsallows them to mimic a wide range of events. These incidents can range from system failure duration to transaction losses to annual accident rates. Due to their versatility, compound families of continuous probability distributions are advantageous. They can simulate many events, even some not well represented by other probability distributions. Additionally, these compound families are easy to use. These compound families can also show random variable interdependencies. This work focuses on the construction and analysis of the novel generalized Weibull Poisson-G family. Combining the zero-truncated-Poisson G family and the generalized Weibull G family creates the compound G family. This family's statistics are mathematically analysed. This study uses Clayton, Archimedean-Ali-Mikhail-Haq, Renyi's entropy, Farlie, Gumbel, Morgenstern, and their modified variations spanning four minor types to design new bivariate type G families. The single-parameter Lomax model is highlighted. Two practical examples demonstrate the importance of the new family.

THE COMPARISON SOME MODELS (GAMMA, SLASHED QUASI-GAMMA, THREE PARAMETERS QUASI GAMMA, TWO PARAMETER GAMMA-EXPONENTIAL, AND MODIFIED LOG-LOGISTIC DISTRIBUTIONS) FOR DAILY TEMPERATURE DATA

With the increase in hot temperature events in recent years, there is growing interest in measuring the frequency of recurring temperature events. One of the basis for assessing the frequency of recurrence of temperature events is the probability distribution of temperature events, therefore temperature data is needed to produce statistical modeling, especially in determining the best probability distribution. The study intended to estimate the best-fitted probability model for the daily temperature at the Pekanbaru station in Indonesia from 2010 to 2020 using several statistical analyses. Five continuous probability distributions such as Gamma (GM), Slashed Quasi-Gamma (SQG), Three Parameters Quasi Gamma (TPQG), Two Parameter Gamma-Exponential (TPGE), and Modified Log-Logistic (MLL) distributions were fitted for these tasks using the maximum likelihood technique. To determine the model’s fit to the temperature data, several goodness-of-fit tests were applied, including the graphical methos test (density plot) and Numerical criteria method test (AIC, BIC, and -Log Likelihood). The GM and MLL distribution are found to be the best-fitted probability distribution based on goodness-of-fit tests for the daily temperature data at the Pekanbaru station

Skew Lomax Distribution, Parameter Estimation, Its Properties, and Applications

In this paper, a three-parameter distribution called the ‘Skew-Lomax’ distribution is proposed, which is the special case of the Azzalini distribution to generalize the Lomax distribution. The Lomax distribution is also called Pareto type II distribution, which is a heavy-tailed continuous probability distribution for a non-negative random variable. The statistical properties of the proposed Skew-Lomax distribution, including mean, variance,  moments about the origin, cumulative distribution function, hazard rate function, quantile function, and the random number generation have been derived. Also, the method of maximum likelihood and the method of moment to estimate the parameters of this distribution have been proposed. Three real data sets have been used to illustrate the usefulness, flexibility, and application of the proposed distribution. The coefficient of determination, chi-square test statistics, and the sum of the square of error depicts that the proposed model is more flexible than the Lomax distribution.

A New Life-Time Model with Bathtub and Inverted Bathtub Failure Rate Function

In this paper, a three parameter model called Zubair- Kumaraswamy (Z-Kum) distribution is proposed. The extension was done using Zubair G-Family [1] of continuous probability distribution to extend the well knownKumaraswamy distribution to make it more flexible in modeling and predicting real world phenomenon.  Some basic structural properties of the new distributions like cdf, pdf, quantile functions, moments, moment generating functions, characteristics functions and order statistics was obtained. Survival function, hazard function, reversed hazard rate function and a cumulative hazard rate function was also obtained. Behavior of the hazard rate plot exhibit increase, decrease, Bathtub and inverted Bathtub shape. Maximum likelihood estimate was used to estimate the Z-Kum distribution parameters. Monte Carlo simulation was carried out to evaluate the performance of MLE method adopted. Result of the simulation studies indicates that MLE is good for the estimation of our distribution parameters. To compare the proposed model with the other fitted existing models, analytical measure of goodness of fit of some information criterion was considered using three real life data sets. From the results obtained, it is evident that our proposed model gives better fit than the other competing models and therefore, our proposed model provide greater flexibility in modeling real world phenomenon.

Modelling Mixed Traffic Time Headway Distribution Induced by Bus Rapid Transit (BRT) in Cape Town, South Africa

Background: The safety and capacity of roadways are immensely influenced by time headways, a major traffic flow characteristic. Time headways are frequently utilized in various aspects of traffic and transportation engineering studies, including capacity analyses, studies of safety, modelling of lane-changing and car-following behaviour, and level of service assessment. Time headway, measured in seconds, refers to the distance between two subsequent passing cars moving over a single spot on the road. This paper reports the statistical modelling of the time headway distribution of a mixed traffic flow scenario caused by BRT via statistical headway distribution models. The paper posits that the introduction of BRT dedicated lane and its adopted design configuration induces anomalous traffic flows on its adjoining lanes with attendant time headway differentials, irrespective of the type of adopted design configuration for the corridor. The objectives of this study were to fit the BRT-induced time headway distribution to probability distribution models and determine the most appropriate distribution model that fits the data derived using the goodness of fit tests. Consequently, this contribution, therefore, fills the gap in research with respect to determining the precise model that fits the time headway distribution of mixed traffic flow scenarios induced by BRT. Methods: The time headway distribution data collected at four designated road segments on a multilane provincial route R27 caused by BRT dedicated lane in Cape Town, South Africa, were fitted to five probability distribution models viz, Lognormal, Inverse Gaussian, Log-logistic, Generalized Extreme Value (GEV), and Burr using MATLAB software. The fitted headway data were with respect to the traffic flows on the adjoining lanes to BRT or traffic flow ‘without BRT’. Precisely, the empirical data were collected over a three-month period using an Automatic Traffic Counter (ATC). Using ModelRisk software, the goodness of fit of the probability models was assessed by the Akaike Information Criterion (AIC), the Schwartz or Bayesian Information Criterion (SIC or BIC), the Hannan Quinn Information Criterion (HQIC), and the loglikelihood (LLH) model performance criteria respectively. The distribution model with the lowest AIC criterion and largest loglikelihood (LLH) values describe the model that best fits the headway data across the four sites. Results: Results showed that the models fitted the headway data well at each site by visual inspection of the attendant Probability Density Function (PDF) and the Probability plots (P-P). However, the Burr distribution provided the overall best fit based on the AIC and LLH values at 95% confidence and 0.05 significance levels across the four sites. At sites 02, 03, and 04, it ranked first with the lowest AIC values of 4025.99, 2595.56, 3815.36, and corresponding largest LLH values of -2008.98, -1293.76, and -1903.66, respectively, while the lognormal distribution performed best at site 01, with AIC value of 4445.14 and LLH value of -2220.57, closely followed by the Burr distribution with AIC and LLH values of 4453.05 and -2222.51, respectively. The P-values, which ranged between 0.65 and 0.80, showed the likelihood of the occurrence of the data sets under the null hypothesis. Hence the null hypothesis was accepted. Conclusion: The study concluded that the introduction of BRT dedicated lanes affects the adjoining lanes’ mixed traffic time headways’ distribution, and the headways are continuously distributed, hence fitting continuous probability distributions.

NEW GENERALIZED ODD FRÉCHET-ODD EXPONENTIAL-G FAMILY OF DISTRIBUTION WITH STATISTICAL PROPERTIES AND APPLICATIONS

A new lifetime continuous probability distribution called the new Generalized Odd Fréchet-Odd-Exponential-G Family of Distribution is developed using the principle of Alzaatreh. The developed distribution is flexible for studying positive real-life datasets. The statistical properties related to this family are obtained. The parameters of the family were estimated by using a technique of maximum likelihood. A NewGeneralized Odd Fréchet-Odd-Exponential-Weibull model is introduced. This distribution was fitted with a set of lifetime data. A Monte Carlo simulation is applied to test the consistency of the estimated parameters of this distribution in terms of their bias and mean squared error with a comparison of M.L.E and the maximum product spacing (MPS).The outcome of the Monte Carlo simulation shows that the M.L.E method is the best technique for estimating the parameter of the New Generalized Odd Frechet-Odd-Exponential-Weibull distribution and the New Generalized Odd Frechet-Odd-Exponential-Rayleigh distribution than the M.PS method. The outcomes of the application on the data set produce a higher flexibility than some of the competing distributions. The distributions serve as a viable alternative to other distributions available in the literature for modelling positive data.

Phase-type stress-strength reliability models under progressive type-II right censoring

. The study of stress-strength reliability estimation based on phase-type distribution helps to gather results on estimation of stress-strength reliability with any probability distribution that is defined on the non negative real numbers as any discrete or continuous probability distributions on the positive real line can be represented as phase-type. The matrix representation of the parameters of phase-type distributions helps in their flexible evaluation and easy manipulation. Also in many of the experimental studies, it is very convenient and useful to apply progressive type-II right censoring mechanism in the process of data collection. In this article, we consider the estimation of stress-strength reliability (R) based on phase-type distribution under progressive type-II right censoring mechanism. Both stress and strength random variables are assumed to follow either continuous phase-type or discrete phase-type distribution. We have developed the algorithm for computing Maximum likelihood estimate (MLE) of R based on the expectation maximization (EM) method and the Bayes estimate of R using Markov Chain Monte Carlo technique. A detailed numerical illustration using simulated data/ real data sets are carried out.

Modeling Real-life Data Sets with a Novel G Family of Continuous Probability Distributions: Statistical Properties, and Copulas

This work presents a novel two-parameter G family of continuous probability distributions with compounded parameters. To determine and examine the pertinent mathematical properties, calculations are performed. In one of the special sections, the standard inverse-Rayleigh baseline model is mathematically and statistically emphasized. We generated a number of bivariate and multivariate distributions using the copula method. These new distributions will aid in the modelling of bivariate and multivariate data. The applicability and flexibility of the new compounded two-parameters-G family are demonstrated through three applications to real-life data sets. These examples demonstrate the applicability of the family.

Asymptotic approximations of expectations of power means

In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general nonnegative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behavior depending on sample size contributes to interesting connections with the power mean of probability distribution.

Simultaneous Confidence Intervals for All Pairwise Differences between Means of Weibull Distributions

The Weibull distribution is a continuous probability distribution that finds wide application in various fields for analyzing real-world data. Specifically, wind speed data often adhere to the Weibull distribution. In our study, our aim is to compare the mean wind speed datasets from different areas in Thailand. To achieve this, we proposed simultaneous confidence intervals for all pairwise differences between the means of Weibull distributions. The generalized confidence interval (GCI), method of variance estimates recovery (MOVER), and a Bayesian approach, utilizing both gamma and uniform prior distributions, are proposed to construct simultaneous confidence intervals. Through simulations, we find that the Bayesian highest posterior density (HPD) interval using a gamma prior distribution demonstrates satisfactory performance, while the GCI proves to be a viable alternative. Finally, we applied these proposed approaches to real wind speed data in northeastern and southern Thailand to illustrate their effectiveness and practicality.

Boltzmann-Shannon interaction entropy: A normalized measure for continuous variables with an application as a subsample quality metric.

The recent introduction of geometric partition entropy offered an alternative to differential Shannon entropy for the quantification of uncertainty as estimated from a sample drawn from a one-dimensional bounded continuous probability distribution. In addition to being a fresh perspective for the basis of continuous information theory, this new approach provided several improvements over traditional entropy estimators including its effectiveness on sparse samples and a proper incorporation of the impact from extreme outliers. However, a complimentary relationship exists between the new geometric approach and the basic form of its frequency-based predecessor that is leveraged here to define an entropy measure with no bias toward the sample size. This stable normalized measure is named the Boltzmann-Shannon interaction entropy (BSIE)) as it is defined in terms of a standard divergence between the measure-based and frequency-based distributions that can be associated with the two historical figures. This parameter-free measure can be accurately estimated in a computationally efficient manner, and we illustrate its utility as a quality metric for subsampling in the context of nonlinear polynomial regression.

On Testing of Multiple Hypotheses of Continuous Probability Distributions Arranged into Two Groups

The optimal Neyman-Pearson procedure of detection is investigated for models characterized by four continuous probability distributions arranged into two groups considered as hypotheses. It is worthy to note that the case of three discrete probability distributions arranged in two groups was studied by Haroutunian and Yesayan in [1]. The Neyman-Pearson theorem holds immense importance when it comes to solving problems that demand decision making or conclusions to a higher accuracy.

NEW LIFE-TIME CONTINUOUS PROBABILITY DISTRIBUTION WITH FLEXIBLE FAILURE RATE FUNCTION

Recently, the area of distribution theory has been receiving increased interest in generating or defining new classes of continuous probability distributions by way of extending the existing distributions. The new generated distribution is expected to be more flexible and have wider acceptability in modeling and predicting real world data sets.  In this research, we proposed and study an extension of Ikum distribution using Zubai G- Family (2018) of distribution called Z-Ikum distribution. Expression of some basic structural properties of the new distribution such as cdf, pdf, quantile functions, moments, moment generating function, characteristics function and order statistics was derived. Survival function, hazard rate function, commutative hazard rate function and reversed hazard rate function was also discussed. Plots of the hazard rate function showincrease, decrease and bathtub shapes.Estimation of the proposed distribution parameters was carried out using MLE method. Performance of the parameter estimation was also evaluated via simulation studies. Result of the simulation studies indicates that our estimator is consistent. Three life data sets were used to evaluate the performance of our proposed distribution over some existing distributions. Result of the empherical study revealed that our proposed distribution performwell in modeling real life data than the competing distributions.

Generalization of graph network inferences in higher-order graphical models

AbstractProbabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.