Continuous Random Variables Research Articles

Decision making with dynamic uncertain continuous information

Decision making is the ability to select the best alternative from a set of candidates based on their respective values. When the value depends on uncertain future events, this task becomes more complicated. The question is then whether to wait for more information before making a decision or to stop and make a decision based on uncertain information. This has been addressed in previous work, when the information (events) could be represented as discrete random variables. However, there are real world domains where this assumption is incorrect. Thus, in this paper, we propose a novel framework and algorithms designed to cope with the challenge posed when future events are represented as continuous random variables. More specifically, we define a mathematical representation to model the utility functions of the candidates and introduce optimal and approximate algorithms to compute the best time to stop, and make a decision in order to optimize the utility. We evaluate our model and algorithms theoretically and empirically, and measure their performance in terms of the gain they achieve and their runtime. Our experiment demonstrates that there is no significant difference between the quality of the decision reached by the two algorithms, while the runtime of the optimal algorithm is much higher than that of the approximate algorithm.

Strong deviation theorems for delayed sums of the nonnegative continuous random variables

In this paper, we extend the strong deviation theorems for functions of the nonnegative continuous random variables to the case of delayed sums of the nonnegative continuous random variables, represented by inequalities and described by asymptotic delayed log-likelihood ratio. As corollaries, we obtain several strong law of large numbers for delayed sums of functions of the nonnegative continuous random variables. The results of this manuscript extend and improve the corresponding results of some current literatures.

Chance constrained programming with some non-normal continuous random variables

Stochastic or probabilistic programming is a branch of mathematical programming that deals with some situations in which an optimal decision is desired under random uncertainty of some parameters. In this paper, we consider some chance constrained linear programming problems where the right hand side parameters of the chance-constraints follow some non-normal continuous distributions such as power function distribution, triangular distribution and trapezoidal distribution. To find the solution of the stated problems, we first convert the problems in to equivalent deterministic models. Then standard linear programming techniques are used to solve the equivalent deterministic models. Some numerical examples are presented to illustrate the methodology.

Independence test of a continuous random variable and a discrete random variable

In many cases, we are interested in identifying independence between variables. For continuous random variables, correlation coefficients are often used to describe the relationship between variables; however, correlation does not imply independence. For finite discrete random variables, we can use the Pearson chi-square test to find independency. For the mixed type of continuous and discrete random variables, we do not have a general type of independent test. In this study, we develop a independence test of a continuous random variable and a discrete random variable without assuming a specific distribution using kernel density estimation. We provide some statistical criteria to test independence under some special settings and apply the proposed independence test to Pima Indian diabetes data. Through simulations, we calculate false positive rates and true positive rates to compare the proposed test and Kolmogorov-Smirnov test.

Some estimations on continuous random variables for (k,s) −fractional integral operators

AbstractIn this work, we establish some new (k,s) −fractional integral inequalities of continuous random variables by using the (k,s) −Riemann-Liouville fractional integral operator.

English

Partial orderings and measures of information for continuous univariate random variables with special roles of Gaussian and uniform distributions are discussed. The information measures and measures of non-Gaussianity including third and fourth cumulants are generally used as projection indices in the projection pursuit approach for the independent component analysis. The connections between information, non-Gaussianity and statistical independence in the context of independent component analysis is discussed in detail.

Estimating the distribution of a stochastic sum of IID random variables

Abstract Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

On Theoretical Principle and Practical Applicability of Ranked Nodes Method for Constructing Conditional Probability Tables of Bayesian Networks

This paper provides new insight into the theoretical principle and the practical applicability of the ranked nodes method (RNM) that is used to construct conditional probability tables (CPTs) for Bayesian networks (BNs) by expert elicitation. RNM is designed for specific types of discrete random variables called ranked nodes that are common in real-world applications of BNs. Despite its active use in recent years, there remains ambiguity about the exact theoretical basis of RNM which can hamper its effective employment. In addition, there are a lack of studies about the general ability of CPTs generated with RNM to represent probabilistic relationships in real-world applications. In this paper, it is shown how the generation of probabilities with RNM is underpinned by a regression model of continuous random variables. Then, it is experimentally determined that in typical applications of RNM, one can generate in a matter of seconds CPTs whose elements reflect well probabilities given by the underlying regression model. Another experiment discovers that CPTs generated with RNM provide a good average fit to a large portion of various real-world CPTs investigated. This confirms the usefulness of RNM in practical applications. The results of the experiment also indicate that choices made by the user of RNM can considerably impact the ability of a generated CPT to represent a given probabilistic relationship. This paper then provides practical advice on the efficient use of RNM with regard to the user-controlled features explored in the experiment.

Co-Clustering of Ordinal Data via Latent Continuous Random Variables and Not Missing at Random Entries

ABSTRACTThis article is about the co-clustering of ordinal data. Such data are very common on e-commerce platforms where customers rank the products/services they bought. In more detail, we focus on arrays of ordinal (possibly missing) data involving two disjoint sets of individuals/objects corresponding to the rows/columns of the arrays. Typically, an observed entry (i, j) in the array is an ordinal score assigned by the individual/row i to the object/column j. A new generative model for arrays of ordinal data is introduced along with an inference algorithm for parameters estimation. The model accounts for not missing at random data and relies on latent continuous random variables. The fitting allows to simultaneously co-cluster the rows and columns of an array. The estimation of the model parameters is performed via a classification expectation maximization algorithm. A model selection criterion is formally obtained to select the number of row and column clusters. To show that our approach reaches and often outperforms the state of the art, we carry out numerical experiments on synthetic data. Finally, applications on real datasets highlight the model capacity to deal with very sparse arrays. Supplementary materials for this article are available online.

Bayesian nonparametric test for independence between random vectors

A nonparametric approach for testing independence among groups of continuous random variables is proposed. Gaussian-centered multivariate finite Polya tree priors are used to model the underlying probability distributions. Integrating out the random probability measure, a tractable empirical Bayes factor is derived and used as the test statistic. The Bayes factor is consistent in the sense that it tends to infinity under the alternative, and zero under the null. A p-value is then obtained through a permutation test based on the observed Bayes factor. Through a series of simulation studies, the performance of the proposed approach is examined and compared to several existing approaches based on the power of the test as well as the observed Bayes factor. Lastly, the proposed method is applied to a set of real data in ecology.

GENERATION OF VALUES FROM DISCRETE PROBABILITY DISTRIBUTIONS WITH THE USE OF CHAOTIC MAPS

The values of random variables are commonly used in the field of artificial intelligence. The literature shows plenty of methods, which allows us to generate them, for example, inverse cumulative density function method. Some of the ways are based on chaotic projection. The chaotic methods of generating random variables are concerned with mainly continuous random variables. This article presents the method of generating values from discrete probability distributions with the use of properly constructed piece-wise linear chaotic map. This method is based on a properly constructed discrete dynamical system with chaotic behavior. Successive probability values cover the unit interval and the corresponding random variable values are assigned to the determined subintervals. In the next step, a piece-wise linear map on the subintervals is constructed. In the course of iterations of the chaotic map, consecutive values from a given discrete distribution are derived. The method is presented on the example of Bernoulli distribution. Furthermore, an analysis of the discussed example is conducted and shows that the presented method is the fastest of all analyzed methods.

Distribution-free three- sample test for detecting trend in the pure location model

A three-sample distribution-free test for detecting non-decreasing ordered alternatives is proposed to alleviate some of the problems in the existing tests, including the risk of frequent detection of a trend when it is not actually present. The test is based on the natural U-statistic estimation of the parameter where are three independent continuous random variables. Actual levels of the test and power values in the case of equal variances for a variety of sample sizes and over a wide class of distributions under the null hypothesis and the alternative are obtained via simulation. The results indicate that the proposed test compares favorably with existing tests. In all simulations the nonparametric test provided relatively good power, accurate control over the size of the test, and better protection against false detection of a non-existing trend.

Probabalistic modelling of the results of economic activity as a function of random values

Introduction. Mathematical modeling of economic processes is necessary for the unambiguous formulation and solution of the problem. In the economic sphere this is the most important aspect of the activity of any enterprise, for which economic-mathematical modeling is the tool that allows to make adequate decisions. However, economic indicators that are factors of a model are usually random variables. An economic-mathematical model is proposed for calculating the probability distribution function of the result of economic activity on the basis of the known dependence of this result on factors influencing it and density of probability distribution of these factors.Methods. The formula was used to calculate the random variable probability distribution function, which is a function of other independent random variables. The method of estimation of basic numerical characteristics of the investigated functions of random variables is proposed: mathematical expectation that in the probabilistic sense is the average value of the result of functioning of the economic structure, as well as its variance. The upper bound of the variation of the effective feature is indicated. Results. The cases of linear and power functions of two independent variables are investigated. Different cases of two-dimensional domain of possible values of indicators, which are continuous random variables, are considered. The application of research results to production functions is considered. Examples of estimating the probability distribution function of a random variable are offered. Conclusions. The research results allow in the probabilistic sense to estimate the result of the economic structure activity on the basis of the probabilistic distributions of the values of the dependent variables. The prospect of further research is to apply indirect control over economic performance based on economic and mathematical modeling.

Inequalities Between First and Second Order Moments for Continuous Probability Distribution

In this research article we obtained some inequalities between moments of 1st and 2nd order for a continuous distribution over the interval [x, y], when infimum and supremum of the continuous probability distribution is taken into consideration. These inequalities have shown improvement and are better than those exist in literature. Inequalities also obtained for continuous random variables which vary in [x, y] interval, such that the probability density function (pdf) (t) become zero in [p, q]  [x, y].The improvement in inequalities have been shown graphically. Here in this paper we deduced some existing inequalities by using the inequalities obtained in Theorem 2.1 and Theorem 2.2.

Generalized expected value of continuous random variables and its applications

In this work, we develop a generalized expected value (GE) of a continuous random variable and propose a novel estimate of GE. Asymptotic normality of this estimator (namely “KWA”) is established and a consistent estimate of the asymptotic variance is given. The relationships between KWA and other existing estimators of central tendency such as sample mean, median, trimmed mean and Hodges-Lehmann estimate are studied. The estimator is also applied to construct one-sample hypothesis testings and confidence intervals of locations of any unknown distributions.

Multivariate count autoregression

We are studying linear and log-linear models for multivariate count time series data with Poisson marginals. For studying the properties of such processes we develop a novel conceptual framework which is based on copulas. Earlier contributions impose the copula on the joint distribution of the vector of counts by employing a continuous extension methodology. Instead we introduce a copula function on a vector of associated continuous random variables. This construction avoids conceptual difficulties related to the joint distribution of counts yet it keeps the properties of the Poisson process marginally. Furthermore, this construction can be employed for modeling multivariate count time series with other marginal count distributions. We employ Markov chain theory and the notion of weak dependence to study ergodicity and stationarity of the models we consider. Suitable estimating equations are suggested for estimating unknown model parameters. The large sample properties of the resulting estimators are studied in detail. The work concludes with some simulations and a real data example.

A stochastic programming model of vaccine preparation and administration for seasonal influenza interventions.

This study considers the integration of vaccine preparation and administration decisions for seasonal influenza interventions. We examine actual vaccination activities of sharing multiple vaccine products and supplementary vaccinations. A two-stage stochastic program is formulated to determine the optimal ordering and allocation of vaccines under uncertain attack rates, vaccine efficacies, and demands. We present an algorithm based on the sample average approximation and warm-start solution to solve the stochastic integer program with continuous random variables. Furthermore, the optimal solution for the deterministic model using the expected value is analyzed and obtained directly. Our analysis compares the deterministic and stochastic solutions to assess the impact of uncertainties on the immunization outcomes and costs. The result shows that the stochastic programming model provides a more robust solution than the deterministic model, and uncertain characteristics should consider when making public health decisions.

Binary Trees for Dependence Structure

In a data set with many categorical variables and several continuous valuables, the relationship between continuous random variables may differ from category to category for a given categorical variable. To study how categorical variables may affect the dependent structure of continuous variables, we proposed two splitting criteria constructed based on copula entropy to build decision trees serving for different purposes. One type of tree can be used to identify the attributes or combinations of them under which the continuous variables have a strong relationship. The other type of tree is used to classify regions with different strength of relationship. Applying these methods to the survey data on the status of poor families of Sichuan province, it is found that the method successfully evaluated the effectiveness of the poverty alleviation policies.

A unified approach to constructing correlation coefficients between random variables

Measuring the correlation between two random variables is an important goal in various statistical applications. The standardized covariance is a widely used criterion for measuring the linear association. In this paper, first, we propose a covariance-based unified measure of variability for a continuous random variable X and show that several measures of variability and uncertainty, such as variance, Gini mean difference and cumulative residual entropy arise as special cases. Then, we propose a unified measure of correlation between two continuous random variables X and Y, with distribution functions (DFs) F and G. Assuming that H is a continuous DF, the proposed measure is defined based on the covariance between X and the transformed random variable $$H^{-1}G(Y)$$ (known as the Q-transformation of H on G). We show that our proposed measure of association subsumes some of the existing measures of correlation. Under some mild condition on H, it is shown that the suggested index ranges in $$[-1,1]$$ where the extremes of the range, i.e., $$-1$$ and 1, are attainable by the Frechet bivariate minimal and maximal DFs, respectively. A special case of the proposed correlation measure leads to a variant of the Pearson correlation coefficient which has absolute values greater than or equal to Pearson correlation. The results are examined numerically for some well known bivariate DFs.

Estimates of bounds on the weighted Simpson type inequality and their applications

Based on the established integral identity, some bounds involving the weighted Simpson type inequality are obtained where the first derivative of considered mappings is (<i>m</i>, <i>h</i>)-preinvex, boundedness or Lipschitzian. As applications, certain generalized inequalities in connection with weighted Simpson type quadrature formula, continuous random variables and <i>F</i>-divergence measures are investigated, respectively.