Continuous Random Variables Research Articles

On the asymptotic behavior of randomly weighted averages

In this paper we assume that X1,X2,… is a sequence of independent continuous centered random variables with finite variances σ12,σ22,…. Then we present a central limit theorem for the randomly weighted averages Sn=R1X1+⋯+RnXn, where the random weights R1,…,Rn are the cuts of (0,1) by the order statistics of a random sample of size n−1 from a uniform distribution on (0,1). Indeed we prove that under certain assumptions on the variances, n+1Sn converges in distribution to the normal distribution with mean zero and variance 2c, c=limn→∞(1/n)∑i=1nσi2.

Optimal capacity design under $$k$$ k -out-of- $$n$$ n and consecutive $$k$$ k -out-of- $$n$$ n type probabilistic constraints

We formulate and solve probabilistic constrained stochastic programming problems, where we prescribe lower and upper bounds for $$k$$ k -out-of- $$n$$ n and consecutive- $$k$$ k -out-of- $$n$$ n reliabilities in the form of probabilistic constraints. Problems of this kind arise in many practical situations where stockout in a number of consecutive or a certain fraction of all periods causes irreparable damage. Examples are determination of reservoir capacities used for irrigation or capital reserves of banks. The optimal solution is typically obtained by the solution of a“reliability equation” involving discrete or continuous random variables. The reliability here is approximated by the use of binomial Boolean probability bounds. For $$k$$ k -out-of- $$n$$ n reliability, the properties of Gamma distribution are used to approximate the reliability equation. For the consecutive $$k$$ k -out-of- $$n$$ n case, Binomial probability bounds ( $$S_1, S_2, S_3$$ S 1 , S 2 , S 3 sharp lower bounds, Hunter’s upper bound and Cherry tree upper bound) are used to create lower and upper bounds for the reliability constraint and therefore to solve the reliability equation. A bi-section algorithm is finally applied to determine the optimal capacity level. Copyright Springer Science+Business Media New York 2015

A Geometrical Framework for Covariance Matrices of Continuous and Categorical Variables

It is well known that a categorical random variable can be represented geometrically by a simplex. Accordingly, several measures of association between categorical variables have been proposed and discussed in the literature. Moreover, the standard definitions of covariance and correlation coefficient for continuous random variables have been extended to categorical variables. In this article, we present a geometrical framework where both continuous and categorical data are represented by simplices and lines in a high-dimensional space, respectively. We introduce a function whose direct minimization leads to a single definition of covariance between categorical–categorical, categorical–continuous, and continuous–continuous data. The novelty of this general approach is that a single space and a single distance function can be used for describing both continuous and categorical data. It thus provides a unified geometrical description of the measure of association, in particular between categorical and continuous data. We discuss virtues and limitations of such a geometrical framework and provide examples with possible applications to sociological surveys.

Almost opposite regression dependence in bivariate distributions

Let \(X\),\(Y\) be two continuous random variables. Investigating the regression dependence of \(Y\) on \(X\), respectively, of \(X\) on \(Y\), we show that the two of them can have almost opposite behavior. Indeed, given any \(\epsilon >0\), we construct a bivariate random vector \((X,Y)\) such that the respective regression dependence measures \(r_{2|1}(X,Y), r_{1|2}(X,Y) \in [0,1]\) introduced in Dette et al. (Scand. J. Stat. 40(1):21–41, 2013) satisfy \(r_{2|1}(X,Y) = 1\) as well as \(r_{1|2}(X,Y) < \epsilon \).

D-optimal Factorial Designs under Generalized Linear Models

Generalized linear models (GLMs) have been used widely for modeling the mean response both for discrete and continuous random variables with an emphasis on categorical response. Recently Yang, Mandal and Majumdar (2013) considered full factorial and fractional factorial locally D-optimal designs for binary response and two-level experimental factors. In this article, we extend their results to a general setup with response belonging to a single-parameter exponential family and for multilevel predictors.

Sumset and Inverse Sumset Inequalities for Differential Entropy and Mutual Information

The sumset and inverse sumset theories of Freiman, Pl\"{u}nnecke and Ruzsa, give bounds connecting the cardinality of the sumset $A+B=\{a+b\;;\;a\in A,\,b\in B\}$ of two discrete sets $A,B$, to the cardinalities (or the finer structure) of the original sets $A,B$. For example, the sum-difference bound of Ruzsa states that, $|A+B|\,|A|\,|B|\leq|A-B|^3$, where the difference set $A-B= \{a-b\;;\;a\in A,\,b\in B\}$. Interpreting the differential entropy $h(X)$ of a continuous random variable $X$ as (the logarithm of) the size of the effective support of $X$, the main contribution of this paper is a series of natural information-theoretic analogs for these results. For example, the Ruzsa sum-difference bound becomes the new inequality, $h(X+Y)+h(X)+h(Y)\leq 3h(X-Y)$, for any pair of independent continuous random variables $X$ and $Y$. Our results include differential-entropy versions of Ruzsa's triangle inequality, the Pl\"{u}nnecke-Ruzsa inequality, and the Balog-Szemer\'{e}di-Gowers lemma. Also we give a differential entropy version of the Freiman-Green-Ruzsa inverse-sumset theorem, which can be seen as a quantitative converse to the entropy power inequality. Versions of most of these results for the discrete entropy $H(X)$ were recently proved by Tao, relying heavily on a strong, functional form of the submodularity property of $H(X)$. Since differential entropy is {\em not} functionally submodular, in the continuous case many of the corresponding discrete proofs fail, in many cases requiring substantially new proof strategies. We find that the basic property that naturally replaces the discrete functional submodularity, is the data processing property of mutual information.

The Lossy Common Information of Correlated Sources

The two most prevalent notions of common information (CI) are due to Wyner and Gács-Körner and both the notions can be stated as two different characteristic points in the lossless Gray-Wyner region. Although the information theoretic characterizations for these two CI quantities can be easily evaluated for random variables with infinite entropy (e.g., continuous random variables), their operational significance is applicable only to the lossless framework. The primary objective of this paper is to generalize these two CI notions to the lossy Gray-Wyner network, which hence extends the theoretical foundation to general sources and distortion measures. We begin by deriving a single letter characterization for the lossy generalization of Wyner's CI, defined as the minimum rate on the shared branch of the Gray-Wyner network, maintaining minimum sum transmit rate when the two decoders reconstruct the sources subject to individual distortion constraints. To demonstrate its use, we compute the CI of bivariate Gaussian random variables for the entire regime of distortions. We then similarly generalize Gács and Körner's definition to the lossy framework. The latter half of this paper focuses on studying the tradeoff between the total transmit rate and receive rate in the Gray-Wyner network. We show that this tradeoff yields a contour of points on the surface of the Gray-Wyner region, which passes through both the Wyner and Gács-Körner operating points, and thereby provides a unified framework to understand the different notions of CI. We further show that this tradeoff generalizes the two notions of CI to the excess sum transmit rate and receive rate regimes, respectively.

Sensitivity analysis of ranked data: from order statistics to quantiles

In this paper we provide the mathematical theory for sensitivity analysis of order statistics of continuous random variables, where the sensitivity is with respect to a distributional parameter. Sensitivity analysis of order statistics over a finite number of observations is discussed before addressing sensitivity analysis of quantiles. In particular, we provide a central limit theorem and related confidence intervals for the proposed sensitivity estimators, and we improve the known convergence results for IPA. Our analysis provides guidelines for sensitivity analysis of order statistic related performance measures such as basic order-statistics or quantiles. Our findings are corroborated by a series of numerical examples from different areas of operations research: stochastic activity networks, queues, reliability, and financial engineering.

A dual-inexact fuzzy stochastic model for water resources management and non-point source pollution mitigation under multiple uncertainties

Abstract. In this research, a dual-inexact fuzzy stochastic programming (DIFSP) method was developed for supporting the planning of water and farmland use management system considering the non-point source pollution mitigation under uncertainty. The random boundary interval (RBI) was incorporated into DIFSP through integrating fuzzy linear programming (FLP) and chance-constrained programming (CCP) approaches within an interval linear programming (ILP) framework. This developed method could effectively tackle the uncertainties expressed as intervals and fuzzy sets. Moreover, the lower and upper bounds of RBI are continuous random variables, and the correlation existing between the lower and upper bounds can be tackled in RBI through the joint probability distribution function. And thus the subjectivity of decision making is greatly reduced, enhancing the stability and robustness of obtained solutions. The proposed method was then applied to solve a water and farmland use planning model (WFUPM) with non-point source pollution mitigation. The generated results could provide decision makers with detailed water supply–demand schemes involving diversified water-related activities under preferred satisfaction degrees. These useful solutions could allow more in-depth analyses of the trade-offs between humans and environment, as well as those between system optimality and reliability. In addition, comparative analyses on the solutions obtained from ICCP (Interval chance-constraints programming) and DIFSP demonstrated the higher application of this developed approach for supporting the water and farmland use system planning.

A note on mean absolute deviation

We introduce the idea of Mean Absolute Deviation (MAD). We provide a simple method for calculating the MAD that circumvents the use of the absolute value. We develop this both for the case of continuous and finite valued random variables. We introduce the idea of sub-mean and show how the MAD is related to the difference between the mean and sub-mean. We look at the MAD for random variables with symmetric probability density functions. We consider problem of determining the best estimate of a random variable under various measures of error.

Control-loop diagnosis using continuous evidence through kernel density estimation

While most previous work in the subject of Bayesian Fault diagnosis and control loop diagnosis use discretized evidence for performing diagnosis (an example of evidence being a monitor reading), discretizing continuous evidence can result in information loss. This paper proposes the use of kernel density estimation, a non-parametric technique for estimating the density functions of continuous random variables. Kernel density estimation requires the selection of a bandwidth parameter, used to specify the degree of smoothing, and a number of bandwidth selection techniques (optimal Gaussian, sample-point adaptive, and smoothed cross-validation) are discussed and compared. Because kernel density estimation is known to have reduced performance in high dimensions, this paper also discusses a number of existing preprocessing methods that can be used to reduce the dimensionality (grouping according to dependence, and independent component analysis). Bandwidth selection and dimensionality reduction techniques are tested on a simulation and an industrial process.

Fractional integral inequalities for continuous random variables

By introducing new concepts on the probability theory, new integral inequalities are established for the fractional expectation and the fractional variance for continuous random variables. These inequalities generalize some interested results in [N.S. Barnett, P. Cerone, S.S. Dragomir and J. Roumeliotis: Some inequalities for the dispersion of a random variable whose p.d.f. is defined on a finite interval, J. Inequal. Pure Appl. Math., Vol. 2 Iss. 1 Art. 1 (2001), 1-18].

Fast and Accurate Multivariate Gaussian Modeling of Protein Families: Predicting Residue Contacts and Protein-Interaction Partners

In the course of evolution, proteins show a remarkable conservation of their three-dimensional structure and their biological function, leading to strong evolutionary constraints on the sequence variability between homologous proteins. Our method aims at extracting such constraints from rapidly accumulating sequence data, and thereby at inferring protein structure and function from sequence information alone. Recently, global statistical inference methods (e.g. direct-coupling analysis, sparse inverse covariance estimation) have achieved a breakthrough towards this aim, and their predictions have been successfully implemented into tertiary and quaternary protein structure prediction methods. However, due to the discrete nature of the underlying variable (amino-acids), exact inference requires exponential time in the protein length, and efficient approximations are needed for practical applicability. Here we propose a very efficient multivariate Gaussian modeling approach as a variant of direct-coupling analysis: the discrete amino-acid variables are replaced by continuous Gaussian random variables. The resulting statistical inference problem is efficiently and exactly solvable. We show that the quality of inference is comparable or superior to the one achieved by mean-field approximations to inference with discrete variables, as done by direct-coupling analysis. This is true for (i) the prediction of residue-residue contacts in proteins, and (ii) the identification of protein-protein interaction partner in bacterial signal transduction. An implementation of our multivariate Gaussian approach is available at the website http://areeweb.polito.it/ricerca/cmp/code.

Anatomy of a domain of continuous random variables I

In this paper we study the family of thin probability measures on the domain A∞ of finite and infinite words over a finite alphabet A. This structure is inspired by work of Jean Goubault-Larrecq and Daniele Varacca, who recently proposed a model of continuous random variables over bounded complete domains. Their presentation leaves out many details, and also misses some motivations. In this and a related paper we attempt to fill in some of these details, and in the process, we reveal some features of their model. Our approach to constructing the thin probability measures uses domain theory, and we show the family forms a bounded complete algebraic domain over A∞. In the second paper in this series, we explore using the thin probability measures to reconstruct the bounded complete domain of continuous random variables over any bounded complete domain due originally to Goubault-Larrecq and Varacca.

Diffusion Processes Satisfying a Conservation Law Constraint

We investigate coupled stochastic differential equations governingNnonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

Independence tests for continuous random variables based on the longest increasing subsequence

Abstract We propose a new class of nonparametric tests for the supposition of independence between two continuous random variables X and Y . Given a size n sample, let π be the permutation which maps the ranks of the X observations on the ranks of the Y observations. We identify the independence assumption of the null hypothesis with the uniform distribution on the permutation space. A test based on the size of the longest increasing subsequence of π ( L n ) is defined. The exact distribution of L n is computed from Schensted’s theorem (Schensted, 1961). The asymptotic distribution of L n was obtained by Baik et al. (1999). As the statistic L n is discrete, there is a small set of possible significance levels. To solve this problem we define the J L n statistic which is a jackknife version of L n , as well as the corresponding hypothesis test. A third test is defined based on the J L M n statistic which is a jackknife version of the longest monotonic subsequence of π . On a simulation study we apply our tests to diverse dependence situations with null or very small correlations where the independence hypothesis is difficult to reject. We show that L n , J L n and J L M n tests have very good performance on that kind of situations. We illustrate the use of those tests on two real data examples with small sample size.

Best-possible bounds on the set of copulas with given degree of non-exchangeability

We establish best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability. These bounds are shown to be best-possible bounds as well for the set of copulas with given degree of non-exchangeability, and, consequently, also on the set of bivariate distribution functions of continuous random variables with given margins and given degree of non-exchangeability. Non-exchangeability of a (quasi-)copula is measured in the sense of Nelsen, i.e. proportional to the maximal absolute difference between this (quasi-)copula and its transpose.

Some applications for the difference of two CDFs

It is known that the dierence in the length between two location parameters of two random variables is equivalent to the difference in the area between two cumulative distribution functions. In this paper, we suggest two applications by using the difference of distribution functions. The first is that the difference of expectations of a certain function of two continuous random variables such as the differences of two kth moments and two moment generating functions could be defined by using the difference between two univariate distribution functions. The other is that the difference in the volume between two empirical bivariate distribution functions is derived. If their covariance is estimated to be zero, the difference in the volume between two empirical bivariate distribution functions could be defined as the difference in two certain areas.

Teaching probability theory by using a web based assessment system together with computer algebra

In the course of Maths Basics 2, the Faculty of Economic Science students of Kaposvár University learn the classical chapters of Probability Theory, namely random variables and the well-known probability distributions. Our teaching experiences show that students' achievement is weaker in case of problems concerning continuous random variables. From school year 2012/13 we have had an opportunity to take Maple TA, the web-based test- and assessment system, into the course of education. It is sufficient for the users of Maple TA to have a browser. Maple computer algebra system, which runs on the server, assesses students' answers in an intelligent way, and compares them with the answers that are considered correct by the teacher. In our presentation we introduce some elements of Maple TA system, the didactic considerations the test sheets were made by, as well as our research results concerning the use of Maple TA.

Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling

A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.