Empirical Process

This paper introduces a new stochastic process, a collection of U-statis-tics indexed by a family of symmetric kemels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation ...

Consider a random vector $(X,Y)$ and let $m(x)=E(Y|X=x)$. We are interested in testing $H_0:m\in {\cal M}_{\Theta,{\cal G}}=\{\gamma(\cdot,\theta,g):\theta \in \Theta,g\in {\cal G}\}$ for some known function $\gamma$, some compact set $\Theta \subset $IR$^p$ and some function set ${\cal G}$ of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis. To test this null hypothesis, we make use of the so-called marked empirical process introduced by \citeD and studied by \citeSt for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study.

We present some extensions of the distributions of the maximum of the Brownian bridge in [0,t] when the conditioning event is placed at a future timeu>t or at an intermediate timeu<t. The standard distributions of Brownian motion and Brownian bridge are obtained as limiting cases. These results permit us to derive also the distribution of the first-passage time of the Brownian bridge. Similar generalizations are carried out for the Brownian bridge with drift μ; in this case, it is shown that the maximal distribution is independent of μ (whenu≥t). Finally, the case of the two-sided maximal distribution of Brownian motion in [0,t], conditioned onB(u)=η (for bothu>t andu<t), is considered.

Recent methodological disputes amongst econometricians are shown to depend critically upon a sequence of differing conceptions of the empirical process, although they are nowhere spelled out in detail. The third conception of critical postmodernism is further described, comparing practices across physics, psychology, and economics using both narratives from recent philosophy of science and the technique of Birge ratios.

For projects that rely on empirical process control and deliver frequently working versions of software, developers and project managers regularly need to examine the status of their software quality. This study illustrates that simple goal-oriented changes or extensions to the existing data of projects&amp;amp;amp;amp;#x27; respective defect tracking systems could provide valuable and prompt information to improve their software quality assessment

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{\mu_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $\mu_n=(1/n)\sum_{i=1}^n\delta_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

Wang & Wells [&quot;J. Amer. Statist. Assoc.&quot; 95 (2000) 62] describe a non-parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [&quot;J. Amer. Statist. Assoc.&quot; 88 (1993) 1034] and later studied by Barbe &quot;et al&quot;. [&quot;J. Multivariate Anal.&quot; 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its &quot;L&quot;-super-2-norm. This paper shows how to compute asymptotic &quot;p&quot;-values for various goodness-of-fit test statistics based on a non-truncated version of Kendall&#39;s process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical beh...

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_n≥ 1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_n≥ 0, if it is adapted to (G_n)_n≥ 0 and, for each n≥ 0, (X_k)_k&gt;n is identically distributed given the past G_n. In case G_0=∅,Ω and G_n=σ(X_1,...,X_n), a result of Kallenberg implies that (X_n)_n≥ 1 is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X_n)_n≥ 1 is exchangeable if and only if (X_τ(n))_n≥ 1 is c.i.d. for any finite permutation τ of 1,2,..., and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-σ-field. Moreover, (1/n)∑_k=1^nX_k converges a.s. and in L^1 whenever (X_n)_n≥ 1 is (real-valued) c.i.d. and E[| X_1| ]&lt;∞. As to the CLT, three types of random centering are considered. One such centering, significant in B...

We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results.

Consider an ensemble of N ×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4 + ε) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of 1 / √ N and letting N → ∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f) = ∑N k=1 f(λk) where λ1,λ2,...,λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2...