15 research outputs found
Resampling Procedures with Empirical Beta Copulas
The empirical beta copula is a simple but effective smoother of the empirical
copula. Because it is a genuine copula, from which, moreover, it is
particularly easy to sample, it is reasonable to expect that resampling
procedures based on the empirical beta copula are expedient and accurate. In
this paper, after reviewing the literature on some bootstrap approximations for
the empirical copula process, we first show the asymptotic equivalence of
several bootstrapped processes related to the empirical copula and empirical
beta copula. Then we investigate the finite-sample properties of resampling
schemes based on the empirical (beta) copula by Monte Carlo simulation. More
specifically, we consider interval estimation for some functionals such as rank
correlation coefficients and dependence parameters of several well-known
families of copulas, constructing confidence intervals by several methods and
comparing their accuracy and efficiency. We also compute the actual size and
power of symmetry tests based on several resampling schemes for the empirical
copula and empirical beta copula.Comment: 22 pages, 8 table
The Empirical Beta Copula
Given a sample from a multivariate distribution , the uniform random
variates generated independently and rearranged in the order specified by the
componentwise ranks of the original sample look like a sample from the copula
of . This idea can be regarded as a variant on Baker's [J. Multivariate
Anal. 99 (2008) 2312--2327] copula construction and leads to the definition of
the empirical beta copula. The latter turns out to be a particular case of the
empirical Bernstein copula, the degrees of all Bernstein polynomials being
equal to the sample size.
Necessary and sufficient conditions are given for a Bernstein polynomial to
be a copula. These imply that the empirical beta copula is a genuine copula.
Furthermore, the empirical process based on the empirical Bernstein copula is
shown to be asymptotically the same as the ordinary empirical copula process
under assumptions which are significantly weaker than those given in Janssen,
Swanepoel and Veraverbeke [J. Stat. Plan. Infer. 142 (2012) 1189--1197].
A Monte Carlo simulation study shows that the empirical beta copula
outperforms the empirical copula and the empirical checkerboard copula in terms
of both bias and variance. Compared with the empirical Bernstein copula with
the smoothing rate suggested by Janssen et al., its finite-sample performance
is still significantly better in several cases, especially in terms of bias.Comment: 23 pages, 3 figure
Weak convergence and the prediction process
We first consider convergence in law of measurable processes with a general parameter set and a state space. To this end, we need to investigate topological properties of the space of measurable functions which is the paths space of measurable processes. Also a characterization of compact sets in the space is derived and some functionals on the space are discussed.We then proceed to prove some properties of probability measures on the space of measurable functions. After investigating conditions on function spaces which guarantee that weak convergence may be proved by establishing finite-dimensional convergence and tightness, we prove necessary and sufficient conditions for convergence in law of measurable processes. These results are then applied to convergence of the prediction process; a special case where the given process is Markov is studied in detail.We also derive two results on the prediction process which is not in the field of weak convergence. One is on the asymptotic behavior of the prediction process under absolutely continuity, and the other is an example which shows that the prediction process may lost Markov property if one uses a larger filtration than the natural one.U of I OnlyETDs are only available to UIUC Users without author permissio
A Limit Theorem for the Prediction Process under Absolute Continuity
Consider a stochastic process with two probability laws, one of which is absolutely continuous with respect to the other. Under each law, we look at a process consisting of the conditional distributions of the future given the past. Blackwell and Dubins (1962) showed in discrete case that those conditional distributions merge as we observe more and more; more precisely, the total variation distance between them converges to 0 a.s. In this paper we prove its extension to continuous time case using the prediction process of F. B. Knight.
A Note on the Prediction Process
In this note, we give an example to show that the prediction process may lost Markov property if the future of the process which generates the known past is not included in the future to be predicted.
Convergence in Law of Measurable Processes with Applications to the Prediction Process
We study convergence in law of measurable processes with a general state space and a parameter set. The space of measurable functions are first investigated and we examine properties of probability measure on the space. A necessary and sufficient condition for convergence in law of measurable processes is obtained. These general results are applied to the prediction process, and we show that convergence of the prediction processes implies that of given processes. We also find a simple condition for convergence of the prediction processes when given processes are Markovian.
Two Transformation Models and Rank Estimation
In the first half of the paper, we shall investigate the relation of two models both of which have been called a transformation model. One model is in terms of distribution functions and the other is in terms of random variables. We shall show that the former class is larger than the latter and we give an explicit relation between these models. The second half deals with estimation procedures for the regression parameters in the transformation model in terms of distribution functions. After reviewing and extending previously proposed estimators for the model, we derive a new estimator based on ranks. Monte Carlo simulation is performed to compare the empirical properties of several estimators for the Cox model, which is a particular case of our transformation model.