Measuring association and dependence between random vectors

1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

Although the G allele variant of TERT-CLPTM1L rs4975616 has been confirmed to be negatively associated to the risk of lung cancer (LC), some other studies haven't found this negative association. The purpose of this study is to clarify the association of the rs4975616 with the risk of developing LC and the differences of this association among patients with different ethnicities (Caucasians and Asians), different subtypes of LC, and different smoking status. Relevant literatures published before July 20, 2023 in PubMed, EMbase, Web of Science, MEDLINE databases were searched through the Internet. Statistical analysis of data was performed in Revman5.3, including drawing forest plots, funnel plots and so on. Sensitivity and publication bias were performed in Stata 14.0. The stability of the results was assessed using Test Sequence Analysis (TSA) software. Registration number: CRD42024568348. The G allele variant of rs4975616 was negatively associated with the risk of LC ([OR] = 0.86, 95%CI [0.84, 0.88]), and that this negative association was present in both Caucasians ([OR] = 0.85, 95%CI [0.83, 0.87]) and Asians ([OR] = 0.91, 95%CI [0.86, 0.95]), and the strength of the negative association was higher in Caucasians than in Asians (subgroup differences: P = 0.02, I2 = 80.3%). Across LC subtypes, rs4975616[G] was negatively associated with the risk of NSCLC (LUAD, LUSC) in both Caucasians and Asians (P<0.05) and the strength of the association with NSCLC (LUAD) was higher in Caucasians than in Asians (Subgroup differences: I2>50%). In Caucasians, rs4975616[G] was negatively associated with the risk of LC in both smokers and non-smokers (P<0.05), and the strength of the association did not differ between smokers and non-smokers (Subgroup differences: P = 0.18, I2 = 45.0%). In Asians, rs4975616[G] was mainly negatively associated with the risk of LC in smokers (P<0.05) but not in non-smokers ([OR] = 0.97, 95%CI [0.78, 1.20]). Comparisons between the two populations showed that the strength of this negative association was higher in Caucasian non-smokers than in Asian non-smokers (Subgroup differences: P = 0.04, I2 = 75.3%), whereas the strength of this negative association was the same for Caucasian smokers as for Asian smokers (Subgroup differences: P = 0.42, I2 = 0%). Among the different LC subtypes, rs4975616[G] was negatively associated with the risk of NSCLC (LUAD) incidence in both Asian smokers and Caucasian non-smokers (P<0.05), whereas it was not associated with the risk of NSCLC development in Asian non-smokers (P>0.05). Comparisons between the two populations showed that the strength of the association was higher in Caucasian non-smokers than in Asian non-smokers (Subgroup differences: I2>50%). The G allele variant of rs4975616 is negatively associated with the risk of LC and NSCLC (LUAD, LUSC). Compared with Asians, Caucasians are more likely to have a higher risk of LC and NSCLC (LUAD) due to the rs4975616 variant. In Caucasians, smoking and other factors like non-smoking contribute to rs4975616 variations leading to LC, and other factors like non-smoking also induce rs4975616 variations leading to NSCLC (LUAD). In Asians, smoking is the major risk factor for the induction of rs4975616 variations leading to LC and NSCLC(LUAD).

Remarks on univariate elliptical distributions

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

Some simple methods for generating correlated categorical variates

Quantities governed by randomness correspond to functions on the probability space called random variables. The value taken by a random variable is subject to chance, and the associated likelihoods are described by a function called the distribution function. Two important classes of random variables are discussed, namely discrete variables and continuous variables. The law of averages, known also as the law of large numbers, states that the proportion of successes in a long run of independent trials converges to the probability of success in any one trial. This result provides a mathematical basis for a philosophical view of probability based on repeated experimentation. Worked examples involving random variables and their distributions are included, and the chapter terminates with sections on random vectors and on Monte Carlo simulation.

We extend the construction principle of multivariate phase-type distributions to establish an analytically tractable class of heavy-tailed multivariate random variables whose marginal distributions are of Mittag–Leffler type with arbitrary index of regular variation. The construction can essentially be seen as allowing a scalar parameter to become matrix-valued. The class of distributions is shown to be dense among all multivariate positive random variables and hence provides a versatile candidate for the modelling of heavy-tailed, but tail-independent, risks in various fields of application.

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed (i.i.d.), light-tailed, and nonlattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modeling. It has been extensively studied in the literature where state-independent, exponential-twisting-based importance sampling has been shown to be asymptotically efficient and a more nuanced state-dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point-based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Furthermore, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables.

Properties of the lattice or symmetric relations on a reference space Ω properties of the valued boolean lattices and an association between a random vector and an equivalence relation on tl are used to show how most or the formulae given by different authors concerning joint information of random vectors may easily be interpreted through introduction or a random object denoted A:B (A and B being random vectors). RA:Bis defined as the reflexive and symmetric relation associated with A:B. RA:B=RAwhere RA: equivalence relation associated with A RB: equivalence relation associated with B. A distance between random variables and a similarity relation having good independence properties are derived. These results are then extended to more complex structures of random objects.

Testing for mutual independence among several random vectors is a challenging problem, and in recent years, it has gained significant attention in statistics and machine learning literature. Most of the existing tests of independence deal with only two random vectors, and they do not have straightforward generalizations for testing mutual independence among more than two random vectors of arbitrary dimensions. On the other hand, there are various tests for mutual independence among several random variables, but these univariate tests do not have natural multivariate extensions. In this article, we propose two general recipes, one based on inter-point distances and the other based on linear projections, for multivariate extensions of these univariate tests. Under appropriate regularity conditions, these resulting tests turn out to be consistent whenever we have consistency for the corresponding univariate tests. We carry out extensive numerical studies to compare the empirical performance of these proposed methods with the state-of-the-art methods.

A random vector ${\bf X}$ is weakly stable if and only if for all $a,b \in {\mathbb R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where ${\bf X}'$ is an independent copy of ${\bf X}$ and $\Theta$ is independent of ${\bf X}$. This is equivalent (see [J. K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Studia Math., 167 (2005), pp. 195--213]) to the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that ${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta,$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define a weak generalized convolution of measures defined by the formula ${\cal L}(Q_1) \otimes_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$ if the former equation holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\bf X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this ...

SUMMARY A great deal of recent attention has focused on the estimation of survival distributions based on current status data, an extreme form of interval censored data. This particular data structure arises in a wide variety of applications where cross-sectional observation either naturally occurs or is preferred to more traditional forms of follow-up. Here we consider current status data in the context of competing risks. We briefly consider simple parametric models as a backdrop to nonparametric procedures. We make some brief comparisons and remarks regarding the nonparametric maximum likelihood estimator. The ideas are illustrated on the data of Krailo & Pike (1983) which considers estimation of the age distribution at both natural and operative menopause. We also consider the case where there is exact observation of failure times due to one of the competing risks when failure occurs prior to the monitoring time. Basic techniques of survival analysis focus on estimation of the survival distribution based on various forms of censored data. An extreme form of censoring arises where the only information on studied individuals is their survival status at a single monitoring time. This particular data structure is known as current status data. Nonparametric estimation of the survival function and semiparametric techniques for related regression models, based on current status data, have been much studied of late. Here, we consider estimation of sub-survival functions based on current status observations in the presence of competing risks. For simplicity, we describe only two competing risks, but the ideas are easily extended when there are more than two possible causes of failure. Let )j denote the cause-specific hazard function for cause j = 1, 2 (Kalbfleisch & Prentice, 2002, p. 252). If T and J are the random variables that measure time to failure and cause of failure, respectively, the two sub-distribution functions of interest are

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

Deforestation and child diet diversity: a geospatial analysis of 15 sub-Saharan African countries

Input modeling is the selection of a probability distribution to capture the uncertainty in the input environment of a stochastic system. Example applications of input modeling include the representation of the randomness in the time to failure for a machining process, the time between arrivals of calls to a call center, and the demand received for a product of an inventory system. Building simulations of stochastic systems requires the development of input models that adequately represent the uncertainty in such random variables. Since there are an abundance of probability distributions that can be used for this purpose, a natural question to ask is how to identify the probability distribution that best represents the particular situation under study. For example, is the exponential distribution a reasonable choice to represent the time to failure for a machining process, or is it better to use an empirical distribution function obtained from the historical time-to-failure data? Recognizing the fact that there is no true input model waiting to be found, the goal of stochastic input modeling is to obtain an approximation that captures the key characteristics of the system inputs. The development of a good input model requires the collection of as much information as possible about the relevant randomness in the system as well as the historical data consisting of the past realizations of the random variables of interest. In the presence of a data set, the input model can be identified by fitting a probability distribution to the historical data. However, it may be difficult and/or costly to collect data for the stochastic system under study; it can also be impossible to properly collect any data at all such as when the proposed system does not exist. In the absence of historical data, any relevant information (e.g., expert opinion and the conventional bounds suggested by the underlying physical situation) can be used for input modeling. This article addresses the key issues that arise in stochastic input modeling both in the presence and in the absence of historical data. The first step in input modeling is to identify the sources of randomness in the input environment of the system under study. Many stochastic systems contain multiple sources of uncertainty, e.g., the completion time of an item on a particular machine, the potential breakdown of the machine, and the percentage of defective items produced by the machine might be among the sources of uncertainty in a manufacturing setting. Throughout, the random vector X = (X1, X2, …, X K )′ is used to represent the collection of K different inputs of a stochastic system, where X k is the random variable denoting the kth system input. The K components of this random vector might also be correlated with each other. Therefore, the stochastic properties of the random inputs X k , k = 1, 2, …, K, are captured in the joint probability