计量经济分析 = Econometric analysis

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner.

In this paper a new nonparametric estimate of conditional quantiles is proposed, that avoids the problem of crossing quantile curves [calculated for various p 2 (0;1)]: The method uses an initial estimate of the conditional distribution function in a flrst step and solves the problem of inversion and monotonization with respect to p 2 (0;1) simultaneously. It is demonstrated that the new estimates are asymptotically normal distributed and asymptotically flrst order equivalent to quantile estimates obtained by local constant or local linear smoothing of the conditional distribution function. The performance of the new procedure is illustrated by means of a simulation study and some comparisons with the currently available procedures which are similar in spirit with the proposed method are presented.

/pdf/non-crossing-non-parametric-estimates-of-quantile-curves-2vy4b4djj9.pdf

Non-crossing non-parametric estimates of quantile curves

Non-crossing nonparametric estimates of quantile curves

The increased availability of massive data sets provides a unique opportunity to discover subtle patterns in their distributions, but also imposes overwhelming computational challenges. To fully utilize the information contained in big data, we propose a two-step procedure: (i) estimate conditional quantile functions at different levels in a parallel computing environment; (ii) construct a conditional quantile regression process through projection based on these estimated quantile curves. Our general quantile regression framework covers both linear models with fixed or growing dimension and series approximation models. We prove that the proposed procedure does not sacrifice any statistical inferential accuracy provided that the number of distributed computing units and quantile levels are chosen properly. In particular, a sharp upper bound for the former and a sharp lower bound for the latter are derived to capture the minimal computational cost from a statistical perspective. As an important application, the statistical inference on conditional distribution functions is considered. Moreover, we propose computationally efficient approaches to conducting inference in the distributed estimation setting described above. Those approaches directly utilize the availability of estimators from subsamples and can be carried out at almost no additional computational cost. Simulations confirm our statistical inferential theory.

Distributed inference for quantile regression processes

https://www.ruhr-uni-bochum.de/imperia/md/content/mathematik3/publications/empcop_website_111111.pdf

Empirical and sequential empirical copula processes under serial dependence

In this paper, we present an alternative method for the spectral analysis of a univariate, strictly stationary time series $\{Y_{t}\}_{t\in\mathbb{Z} }$ We define a “new” spectrum as the Fourier transform of the differences between copulas of the pairs $(Y_{t},Y_{t-k})$ and the independence copula This object is called a copula spectral density kernel and allows to separate the marginal and serial aspects of a time series We show that this spectrum is closely related to the concept of quantile regression Like quantile regression, which provides much more information about conditional distributions than classical location-scale regression models, copula spectral density kernels are more informative than traditional spectral densities obtained from classical autocovariances In particular, copula spectral density kernels, in their population versions, provide (asymptotically provide, in their sample versions) a complete description of the copulas of all pairs $(Y_{t},Y_{t-k})$ Moreover, they inherit the robustness properties of classical quantile regression, and do not require any distributional assumptions such as the existence of finite moments In order to estimate the copula spectral density kernel, we introduce rank-based Laplace periodograms which are calculated as bilinear forms of weighted $L_{1}$-projections of the ranks of the observed time series onto a harmonic regression model We establish the asymptotic distribution of those periodograms, and the consistency of adequately smoothed versions The finite-sample properties of the new methodology, and its potential for applications are briefly investigated by simulations and a short empirical example

/pdf/of-copulas-quantiles-ranks-and-spectra-an-l1-approach-to-3moaxzreps.pdf

Stanislav Volgushev

Papers

Non-crossing non-parametric estimates of quantile curves

Non-crossing nonparametric estimates of quantile curves

Distributed inference for quantile regression processes

Empirical and sequential empirical copula processes under serial dependence

Of Copulas, Quantiles, Ranks and Spectra - An L1-Approach to Spectral Analysis