Multivariate Cumulative Distribution Function Research Articles

A New Kernel Estimator of Copulas Based on Beta Quantile Transformations

A copula is a multivariate cumulative distribution function with marginal distributions Uniform(0,1). For this reason, a classical kernel estimator does not work and this estimator needs to be corrected at boundaries, which increases the difficulty of the estimation and, in practice, the bias boundary correction might not provide the desired improvement. A quantile transformation of marginals is a way to improve the classical kernel approach. This paper shows a Beta quantile transformation to be optimal and analyses a kernel estimator based on this transformation. Furthermore, the basic properties that allow the new estimator to be used for inference on extreme value copulas are tested. The results of a simulation study show how the new nonparametric estimator improves alternative kernel estimators of copulas. We illustrate our proposal with a financial risk data analysis.

Determining the optimal window length of the time-varying copula parameter: a simulation study

A time-varying copula is a multivariate cumulative distribution function that can accommodate the temporal dependence of random variables. To capture the temporal dependence of random variables using a copula, we can form a time variation on copula parameters. Some studies reported that a function of a latent variable following an autoregressive process defines time-varying copula parameters. The process consists of an autoregressive coefficient describing the influence of the parameters in previous times and a forcing variable representing the relationship between the marginal variables which utilize previous information over a certain number of window lengths of the observations. Previous studies reported that ARMA(1,10) defines time-varying copula parameters. However, the optimal parameters of time-varying copulas do not always require ten last observations. Therefore, we determine the optimal window length of the time-varying copula parameter by conducting a numerical simulation on four types of copula with different strengths and directions of dependence. The window length is defined as a new parameter and the optimal parameters can be obtained by finding the values producing the smallest AIC. We also provide a numerical simulation on how to improve the accuracy of the predicted series by changing the mean and standard deviation of the residual pairs.

2‐dimensional human‐like driver model for autonomous vehicles in mixed traffic

Classical artificial potential approach of motion planning is extended for emulating human driving behaviour in two dimensions. Different stimulus parameters including type of ego-vehicle, type of obstacles, relative velocity, relative acceleration, and lane offset are used. All the surrounding vehicles are considered to influence drivers' decisions. No emphasis is laid on vehicle control; instead, an ego vehicle is assumed to reach the desired state. The study is on human-like driving behaviour modelling. The developed motion planning algorithm formulates repulsive and attractive potentials in a data-driven way in contrast to the classical arbitrary formulation. Interaction between the stimulus parameters is explicitly considered by using multivariate cumulative distribution functions. Comparison of two-dimensional (lateral and longitudinal) performance indicators with a baseline model and generative adversarial networks indicate the effectiveness and suitability of the developed motion planning algorithm in the mixed traffic environment.

Constructing gene regulatory networks from microarray data using non-Gaussian pair-copula Bayesian networks.

Many biological and biomedical research areas such as drug design require analyzing the Gene Regulatory Networks (GRNs) to provide clear insight and understanding of the cellular processes in live cells. Under normality assumption for the genes, GRNs can be constructed by assessing the nonzero elements of the inverse covariance matrix. Nevertheless, such techniques are unable to deal with non-normality, multi-modality and heavy tailedness that are commonly seen in current massive genetic data. To relax this limitative constraint, one can apply copula function which is a multivariate cumulative distribution function with uniform marginal distribution. However, since the dependency structures of different pairs of genes in a multivariate problem are very different, the regular multivariate copula will not allow for the construction of an appropriate model. The solution to this problem is using Pair-Copula Constructions (PCCs) which are decompositions of a multivariate density into a cascade of bivariate copula, and therefore, assign different bivariate copula function for each local term. In fact, in this paper, we have constructed inverse covariance matrix based on the use of PCCs when the normality assumption can be moderately or severely violated for capturing a wide range of distributional features and complex dependency structure. To learn the non-Gaussian model for the considered GRN with non-Gaussian genomic data, we apply modified version of copula-based PC algorithm in which normality assumption of marginal densities is dropped. This paper also considers the Dynamic Time Warping (DTW) algorithm to determine the existence of a time delay relation between two genes. Breast cancer is one of the most common diseases in the world where GRN analysis of its subtypes is considerably important; Since by revealing the differences in the GRNs of these subtypes, new therapies and drugs can be found. The findings of our research are used to construct GRNs with high performance, for various subtypes of breast cancer rather than simply using previous models.

Relaxations for probabilistically constrained stochastic programming problems: review and extensions

We consider probabilistically constrained stochastic programming problems, in which the random variables are in the right-hand sides of the stochastic inequalities defining the joint chance constraints. Problems of that kind arise in a variety of contexts, and are particularly difficult to solve for random variables with continuous joint distributions, because the calculation of the cumulative distribution function and its gradient values involves numerical integration and/or simulation in large dimensional spaces. We revisit known and provide new relaxations extensions to various probability bounding schemes that permit to approximate the feasible set of joint probabilistic constraints. The derived mathematical formulations relax the requirement to handle large multivariate cumulative distribution functions and involve instead the computation of marginal and bivariate cumulative distribution functions. We analyze the convexity of and computational challenges posed by the inferred relaxations

On Selection Combiner Output Multivariate Distributions in Correlated Generalized-Rician Fading

Theoretical developments on selection combiner output (SCO) multivariate distributions of popular correlated fading environments, such as Rayleigh, Rician, and Nakagami- $m$ , have been studied in the literature, but those of generalized-Rician fading have not been rigorously studied. The generalized-Rician fading environment, comprising of $n$ underlying non-zero mean independent and identically distributed Gaussian random variables, can be generalized to Rayleigh, Rician, and Nakagami- $m$ fading. In addition, available findings on this topic in the open literature are scattered with mathematical mistakes. This letter thus derives SCO correlated generalized-Rician multivariate cumulative distribution function, then identifies mathematical errors in the open literature, and corrects them. Cross verification of well-known results is mathematically employed. Simulation details of average bit error rate for binary frequency-shift keying are also given for verification purposes.

Hybrid copula estimators

An extension of the empirical copula is considered by combining an estimator of a multivariate cumulative distribution function with estimators of the marginal cumulative distribution functions for marginal estimators that are not necessarily equal to the margins of the joint estimator. Such a hybrid estimator may be reasonable when there is additional information available for some margins in the form of additional data or stronger modelling assumptions. A functional central limit theorem is established and some examples are developed.

Computing bounds for the probability of the union of events by different methods

Let A1,…,An be arbitrary events. The underlying problem is to give lower and upper bounds on the probability P(A1∪⋯∪An) based on \(P(A_{i_{1}}\cap\cdots\cap A_{i_{k}})\), 1≤i1<⋯<ik≤n, where k=1,…,d, and d≤n (usually d≪n) is a certain integer, called the order of the problem or the bound. Most bounding techniques fall in one of the following two main categories: those that use (hyper)graph structures and the ones based on binomial moment problems. In this paper we compare bounds from the two categories with each other, in particular the bounds yielded by univariate and multivariate moment problems are compared with Bukszar’s hypermultitree bounds. In the comparison we considered several numerical examples, most of which have important practical applications, e.g., the approximation of the values of multivariate cumulative distribution functions or the calculation of network reliability. We compare the bounds based on how close they are to the real value and the time required to compute them, however, the problems arising in the implementations of the methods as well as the limitations of the usability of the bounds are also illustrated.

Kernel estimation of multivariate cumulative distribution function

A smooth kernel estimator is proposed for multivariate cumulative distribution functions (cdf), extending the work of Yamato [H. Yamato, Uniform convergence of an estimator of a distribution function, Bull. Math. Statist. 15 (1973), pp. 69–78.] on univariate distribution function estimation. Under assumptions of strict stationarity and geometrically strong mixing, we establish that the proposed estimator follows the same pointwise asymptotically normal distribution of the empirical cdf, while the new estimator is a smooth instead of a step function as the empirical cdf. We also show that under stronger assumptions the smooth kernel estimator has asymptotically smaller mean integrated squared error than the empirical cdf, and converges to the true cdf uniformly almost surely at a rate of (n −1/2log n). Simulated examples are provided to illustrate the theoretical properties. Using the smooth estimator, survival curves for US gross domestic product (GDP) growth are estimated conditional on the unemployment growth rate to examine how GDP growth rate depends on the unemployment policy. Another example of gold and silver price returns is given.

Saddlepoint approximation for moment generating functions of truncated random variables

We consider the problem of approximating the moment generating function (MGF) of a truncated random variable in terms of the MGF of the underlying (i.e., untruncated) random variable. The purpose of approximating the MGF is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables. Two important statistical applications are the following: the approximation of certain multivariate cumulative distribution functions; and the approximation of passage time distributions in ion channel models which incorporate time interval omission. We derive two types of representation for the MGF of a truncated random variable. One of these representations is obtained by exponential tilting. The second type of representation, which has two versions, is referred to as an exponential convolution representation. Each representation motivates a different approximation. It turns out that each of the three approximations is extremely accurate in those cases “to which it is suited.” Moreover, there is a simple rule of thumb for deciding which approximation to use in a given case, and if this rule is followed, then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate.

Microdata disclosure by resampling ? Empirical findings for business survey data

One specific problem statistical offices and research institutes are faced with when releasing microdata is the preservation of confidentiality. Traditional methods to avoid disclosure often destroy the structure of the data, and information loss is potentially high. In this paper an alternative technique of creating scientific–use files is discussed, which reproduces the characteristics of the original data quite well. It is based on Fienberg (1997, 1994) who estimates and resamples from the empirical multivariate cumulative distribution function of the data in order to get synthetic data. The procedure creates data sets – the resample – which have the same characteristics as the original survey data. The paper includes some applications of this method with (a) simulated data and (b) innovation survey data, the Mannheim Innovation Panel (MIP), and a comparison between resampling and a common method of disclosure control (disturbance with multiplicative error) with regard to confidentiality on the one hand and the appropriateness of the disturbed data for different kinds of analyses on the other. The results show that univariate distributions can be better reproduced by unweighted resampling. Parameter estimates can be reproduced quite well if the resampling procedure implements the correlation structure of the original data as a scale or if the data is multiplicatively perturbed and a correction term is used. On average, anonymization of data with multiplicatively perturbed values protects better against re–identification than the various resampling methods used.

A link between the multivariate cumulative distribution function and the hitting function for random closed sets

The hitting function for random closed sets in R d is shown to be a generalization of the multivariate cumulative distribution function by finding a suitable reduction of the compact hitting function test sets in the simple random vector case.

Multivariate local polynomial fitting for a probability distribution function and its partial derivatives*

We present a general framework for simultaneous estimation of a smooth multivariate cumulative distribution function and its partial derivatives. This achieved by means of a multi-variate version of the popular local polynomial fitting technique. We investigate some asymptotic properties of the obtained estimators.

Saddlepoint Approximation for Multivariate Cumulative Distribution Functions and Probability Computations in Sampling Theory and Outlier Testing

Four multivariate distributions commonly arise in sampling theory: the multinomial, multivariate hypergeometric, Dirichlet, and multivariate Pólya distributions. Second-order saddlepoint approximations are given for approximating these multivariate cumulative distribution functions (cdf's) in their most general settings. Probabilities of rectangular regions associated with these cdf's are also approximated directly using second-order saddlepoint methods. All the approximations follow from characterizations of the multivariate distributions as conditional distributions. Applications to outlier discordancy tests and slippage tests are discussed.

Saddlepoint Approximation for Multivariate Cumulative Distribution Functions and Probability Computations in Sampling Theory and Outlier Testing

Saddlepoint Approximation for Multivariate Cumulative Distribution Functions and Probability Computations in Sampling Theory and Outlier Testing