Absolute Regularity Research Articles

How Do Individuals View Euthanasia In Different Cultural Backgrounds Based on Different Ideas of Moral Psychology

To this day, euthanasia remains a topic of lively discussion. The debate over whether it should be practiced or not never stops. This paper presents basic information about euthanasia and analyzes different people's views on euthanasia in the context of religious beliefs and philosophical ideas based on moral psychology. People who grow up in different cultures will have different views on euthanasia. This paper argues that Catholicism, Christianity, and traditional Confucianism are associated with the central idea of deontology (absolute regularity) and tend to reject euthanasia. In contrast, Taoism is associated with the central idea of utilitarianism (the pursuit of the greatest happiness of mankind) and tends to accept the practice of euthanasia. Various ideas have their reasons for rejecting or accepting the practice of euthanasia, each of which thoroughly explains their respective claims. Each idea also has its staunch followers. But regardless of the view one supports, one should be cautious regarding issues related to their own life.

Coregularity of Fano varieties

The absolute regularity of a Fano variety, denoted by reg^(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{\ extrm{reg}}(X)$$\\end{document}, is the largest dimension of the dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be coreg^(X):=dimX-reg^(X)-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\hat{\ extrm{coreg}}(X):= \\dim X - \\hat{\ extrm{reg}}(X)-1. \\end{aligned}$$\\end{document}The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of X. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.

The β-Delaunay tessellation IV: Mixing properties and central limit theorems

Various mixing properties of [Formula: see text]-, [Formula: see text]- and Gaussian-Delaunay tessellations in [Formula: see text] are studied. It is shown that these tessellation models are absolutely regular, or [Formula: see text]-mixing. In the [Formula: see text]- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the [Formula: see text]-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of [Formula: see text]- and Gaussian-Delaunay tessellations are established. This includes the number of [Formula: see text]-dimensional faces and the [Formula: see text]-volume of the [Formula: see text]-skeleton for [Formula: see text].

Significance testing of rank cross-correlations between autocorrelated time series with short-range dependence

Statistical dependency measures such as Kendall’s Tau or Spearman’s Rho are frequently used to analyse the coherence between time series in environmental data analyses. Autocorrelation of the data can, however, result in spurious cross correlations if not accounted for. Here, we present the asymptotic distribution of the estimators of Spearman’s Rho and Kendall’s Tau, which can be used for statistical hypothesis testing of cross-correlations between autocorrelated observations. The results are derived using U-statistics under the assumption of absolutely regular (or β-mixing) processes. These comprise many short-range dependent processes, such as ARMA-, GARCH- and some copula-based models relevant in the environmental sciences. We show that while the assumption of absolute regularity is required, the specific type of model does not have to be specified for the hypothesis test. Simulations show the improved performance of the modified hypothesis test for some common stochastic models and small to moderate sample sizes under autocorrelation. The methodology is applied to observed climatological time series of flood discharges and temperatures in Europe. While the standard test results in spurious correlations between floods and temperatures, this is not the case for the proposed test, which is more consistent with the literature on flood regime changes in Europe.

Mixing properties of non-stationary INGARCH(1,1) processes

We derive mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we prove absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. We provide easily verifiable sufficient conditions for absolute regularity for a variety of models including classical (log-)linear models. Finally, we illustrate the practical use of our results for hypothesis testing.

Bootstrap for integer‐valued GARCH(p, q) processes

AbstractWe consider integer‐valued processes with a linear or nonlinear generalized autoregressive conditional heteroscedastic models structure, where the count variables given the past follow a Poisson distribution. We show that a contraction condition imposed on the intensity function yields a contraction property of the Markov kernel of the process. This allows almost effortless proofs of the existence and uniqueness of a stationary distribution as well as of absolute regularity of the count process. As our main result, we construct a coupling of the original process and a model‐based bootstrap counterpart. Using a contraction property of the Markov kernel of the coupled process we obtain bootstrap consistency for different types of statistics.

Asymptotic Behaviour of the Empirical Distance Covariance for Dependent Data

We give two asymptotic results for the empirical distance covariance on separable metric spaces without any iid assumption on the samples. In particular, we show the almost sure convergence of the empirical distance covariance for any measure with finite first moments, provided that the samples form a strictly stationary and ergodic process. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity of the samples and extend these results to certain types of pseudometric spaces. In the process, we derive a general theorem concerning the asymptotic distribution of degenerate V-statistics of order 2 under a strong mixing condition.

Absolute regularity of semi-contractive GARCH-type processes

AbstractWe prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.

A nonparametric model check for time series when the random vectors are nonstationary and absolutely regular

In this Note, we study some general methods for testing the goodness-of-fit of a parametric model for a real-valued Markovian time series under nonstationarity and absolute regularity. For that, we define a marked empirical process based on residuals, which converges in distribution to a Gaussian process with respect to the Skorohod topology. This method was first introduced by Koul and Stute [1], and then widely developed by Ngatchou-Wandji [2,3] under more general conditions. Applications to general AR-ARCH models are given.

Dependent Wild Bootstrap for the Empirical Process

In this paper, we propose a model‐free bootstrap method for the empirical process under absolute regularity. More precisely, consistency of an adapted version of the so‐called dependent wild bootstrap, which was introduced by Shao () and is very easy to implement, is proved under minimal conditions on the tuning parameter of the procedure. We show how our results can be applied to construct confidence intervals for unknown parameters and to approximate critical values for statistical tests. In a simulation study, we investigate the size properties of a bootstrap‐aided Kolmogorov‐Smirnov test and show that our method is competitive to standard block bootstrap methods in finite samples.

On a ‘Replicating Character String’ Model

In Chaudhuri and Dasgupta's 2006 paper a certain stochastic model for ‘replicating character strings’ (such as in DNA sequences) was studied. In their model, a random ‘input’ sequence was subjected to random mutations, insertions, and deletions, resulting in a random ‘output’ sequence. In this paper their model will be set up in a slightly different way, in an effort to facilitate further development of the theory for their model. In their 2006 paper, Chaudhuri and Dasgupta showed that, under certain conditions, strict stationarity of the ‘input’ sequence would be preserved by the ‘output’ sequence, and they proved a similar ‘preservation’ result for the property of strong mixing with exponential mixing rate. In our setup, we will in spirit slightly extend their ‘preservation of stationarity’ result, and also prove a ‘preservation’ result for the property of absolute regularity with summable mixing rate.

On a ‘Replicating Character String’ Model

In Chaudhuri and Dasgupta's 2006 paper a certain stochastic model for ‘replicating character strings’ (such as in DNA sequences) was studied. In their model, a random ‘input’ sequence was subjected to random mutations, insertions, and deletions, resulting in a random ‘output’ sequence. In this paper their model will be set up in a slightly different way, in an effort to facilitate further development of the theory for their model. In their 2006 paper, Chaudhuri and Dasgupta showed that, under certain conditions, strict stationarity of the ‘input’ sequence would be preserved by the ‘output’ sequence, and they proved a similar ‘preservation’ result for the property of strong mixing with exponential mixing rate. In our setup, we will in spirit slightly extend their ‘preservation of stationarity’ result, and also prove a ‘preservation’ result for the property of absolute regularity with summable mixing rate.

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pn-basis which generalizes the pn-basis defined in [1], and the concept of an pn-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pn-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1, 3]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9]).

Absolute regularity and Brillinger-mixing of stationary point processes

We study the following problem: How to verify Brillinger-mixing of stationary point processes in \( {{\mathbb{R}}^d} \) by imposing conditions on a suitable mixing coefficient? For this, we define an absolute regularity (or β-mixing) coefficient for point processes and derive, in terms of this coefficient, an explicit condition that implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed \( k\geqslant 2 \). To prove this, we introduce higher-order covariance measures and use Statulevicius’ representation formula for mixed cumulants in case of random (counting) measures. To illustrate our results, we consider some Brillinger-mixing point processes occurring in stochastic geometry.

Self-crossing Points of a Line Segment Process

This paper is devoted to planar stationary line segment processes. The segments are assumed to be independent, identically distributed, and independent of the locations (reference points). We consider a point process formed by self-crossing points between the line segments. Its asymptotic variance is explicitly expressed for Poisson segment processes. The main result of the paper is the central limit theorem for the number of intersection points in expanding rectangular sampling window. It holds not only for Poisson processes of reference points but also for stationary point processes satisfying certain conditions on absolute regularity (β-mixing) coefficients. The proof is based on the central limit theorem for β-mixing random fields. Approximate confidence intervals for the intensity of intersections can be constructed.

Strong mixing properties of max-infinitely divisible random fields

Let η=(η(t))t∈T be a sample continuous max-infinitely random field on a locally compact metric space T. For a closed subset S⊂T, we denote by ηS the restriction of η to S. We consider β(S1,S2), the absolute regularity coefficient between ηS1 and ηS2, where S1,S2 are two disjoint closed subsets of T. Our main result is a simple upper bound for β(S1,S2) involving the exponent measure μ of η: we prove that β(S1,S2)≤2∫P[η≮S1f,η≮S2f]μ(df), where f≮Sg means that there exists s∈S such that f(s)≥g(s).If η is a simple max-stable random field, the upper bound is related to the so-called extremal coefficients: for countable disjoint sets S1 and S2, we obtain β(S1,S2)≤4∑(s1,s2)∈S1×S2(2−θ(s1,s2)), where θ(s1,s2) is the pair extremal coefficient.As an application, we show that these new estimates entail a central limit theorem for stationary max-infinitely divisible random fields on Zd. In the stationary max-stable case, we derive the asymptotic normality of three simple estimators of the pair extremal coefficient.

Absolute regularity and ergodicity of Poisson count processes

We consider a class of observation-driven Poisson count processes where the current value of the accompanying intensity process depends on previous values of both processes. We show under a contractive condition that the bivariate process has a unique stationary distribution and that a stationary version of the count process is absolutely regular. Moreover, since the intensities can be written as measurable functionals of the count variables, we conclude that the bivariate process is ergodic. As an important application of these results, we show how a test method previously used in the case of independent Poisson data can be used in the case of Poisson count processes.

Testing in nonparametric varying coefficient additive models

We consider the problem of testing for a general parametric form against a nonparametric alternative for a coefficient function in a varying coefficient mul- tivariate regression model. We propose a test statistic and derive its asymptotic null and alternative distributions. We analyze the asymptotic power of the test in shrinking neighborhoods of the null hypothesis and show that the test is asymp- totically optimal. These results are derived under the fairly general condition of absolute regularity (β-mixing) for the predictor variables. We give numerical re- sults that support the theory. We also illustrate usefulness of the method through an application to a body fat dataset where we build a simple, yet accurate, model that predicts individual body fat values well.

Uniform Laws of Averages for Sequences of Dependent Random Elements

This paper studies classes of sequences with dependence properties generalizing the absolute regularity and mixing by Ibragimov. The coupling method permits us to obtain the uniform laws of averages for the same classes of functions, as in the case when the sequence under consideration consists of independent random elements.

Central limit theorem and the bootstrap for [formula omitted]-statistics of strongly mixing data

The asymptotic normality of U -statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley [R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983),69–81] to prove a new generalized covariance inequality similar to Yoshihara’s [K. Yoshihara, Limiting behavior of U -statistics for stationary, absolutely regular processes, Z. Wahrsch. Verw. Gebiete 35 (1976), 237–252]. It follows from the Hoeffding-decomposition and this inequality that U -statistics of strongly mixing observations converge to a normal limit if the kernel of the U -statistic fulfills some moment and continuity conditions. The validity of the bootstrap for U -statistics has until now only been established in the case of iid data (see [P.J. Bickel, D.A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196–1217]. For mixing data, Politis and Romano [D.N. Politis, J.P. Romano, A circular block resampling procedure for stationary data, in: R. Lepage, L. Billard (Eds.), Exploring the Limits of Bootstrap, Wiley, New York, 1992, pp. 263–270] proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean’s distribution. We extend these results to U -statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U -statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U -statistic and give some simulation results.