In this paper, we investigate the theoretical and empirical properties of $L_2$ boosting with kernel regression estimates as weak learners. We show that each step of $L_2$ boosting reduces the bias of the estimate by two orders of magnitude, while it does not deteriorate the order of the variance. We illustrate the theoretical findings by some simulated examples. Also, we demonstrate that $L_2$ boosting is superior to the use of higher-order kernels, which is a well-known method of reducing the bias of the kernel estimate.

Increasing practical interest has been shown in regression problems where the errors, or disturbances, are centred in a way that reflects particular characteristics of the mechanism that generated the data. In economics this occurs in problems involving data on markets, productivity and auctions, where it can be natural to centre at an end-point of the error distribution rather than at the distribution’s mean. Often these cases have an extreme-value character, and in that broader context, examples involving meteorological, record-value and production-frontier data have been discussed in the literature. We shall discuss nonparametric methods for estimating regression curves in these settings, showing that they have features that contrast so starkly with those in better understood problems that they lead to apparent contradictions. For example, merely by centring errors at their end-points rather than their means the problem can change from one with a familiar nonparametric character, where the optimal convergence rate is slower than $n^{−1/2}$, to one in the super-efficient class, where the optimal rate is faster than $n^{−1/2}$. Moreover, when the errors are centred in a non-standard way there is greater intrinsic interest in estimating characteristics of the error distribution, as well as of the regression mean itself. The paper will also address this aspect of the problem.

We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate $n^{−1/4}$. Moreover, we construct estimates which are robust to finite activity jumps.

Let $\{(Y_{\mathbf i}, \mathbf X_{\mathbf i }), \mathbf i ∈ℤ^N\}$ be a stationary real-valued $(d+1)$-dimensional spatial processes. Denote by $\mathbf x ↦q_p(\mathbf x ), p∈(0, 1), \mathbf x ∈ℝ^d$, the spatial quantile regression function of order $p$, characterized by $\mathrm{P}\{Y_{\mathbf i} ≤q_p(\mathbf x )|\mathbf X _{\mathbf i }=\mathbf x \}=p$. Assume that the process has been observed over an $N$-dimensional rectangular domain of the form $\mathcal{I}_{\mathbf{n}}:=\{\mathbf{i}=(i_{1},\ldots,i_{N})\in\mathbb{Z}^{N}\vert1\leq i_{k}\leq n_{k},k=1,\ldots,N\}$, with $\mathbf n =(n_1, …, n_N)∈ℤ^N$. We propose a local linear estimator of $q_p$. That estimator extends to random fields with unspecified and possibly highly complex spatial dependence structure, the quantile regression methods considered in the context of independent samples or time series. Under mild regularity assumptions, we obtain a Bahadur representation for the estimators of $q_p$ and its first-order derivatives, from which we establish consistency and asymptotic normality. The spatial process is assumed to satisfy general mixing conditions, generalizing classical time series mixing concepts. The size of the rectangular domain $\mathcal{I}_{\mathbf{n}}$ is allowed to tend to infinity at different rates depending on the direction in $ℤ^N$ (non-isotropic asymptotics). The method provides much richer information than the mean regression approach considered in most spatial modelling techniques.

We consider a microstructure model for a financial asset, allowing for price discreteness and for a diffusive behavior at large sampling scale. This model, introduced by Delattre and Jacod, consists in the observation at the high frequency $n$, with round-off error $α_n$, of a diffusion on a finite interval. We give from this sample estimators for different forms of the integrated volatility of the asset. Our method is based on variational properties of the process associated with wavelet techniques. We prove that the accuracy of our estimation procedures is $α_n∨n^{−1/2}$. Using compensated estimators, limit theorems are obtained.

We extend de Finetti’s [Ann. Inst. H. Poincaré 7 (1937) 1–68] notion of exchangeability to finite and countable sequences of variables, when a subject’s beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We derive representation theorems in both the finite and countable cases, in terms of sampling without and with replacement, respectively.

The statistical literature discusses different types of Markov properties for chain graphs that lead to four possible classes of chain graph Markov models. The different models are rather well understood when the observations are continuous and multivariate normal, and it is also known that one model class, referred to as models of LWF (Lauritzen–Wermuth–Frydenberg) or block concentration type, yields discrete models for categorical data that are smooth. This paper considers the structural properties of the discrete models based on the three alternative Markov properties. It is shown by example that two of the alternative Markov properties can lead to non-smooth models. The remaining model class, which can be viewed as a discrete version of multivariate regressions, is proven to comprise only smooth models. The proof employs a simple change of coordinates that also reveals that the model’s likelihood function is unimodal if the chain components of the graph are complete sets.

Several important properties of positive semidefinite processes of Ornstein–Uhlenbeck type are analysed. It is shown that linear operators of the form $X↦AX+XA^T$ with $A∈M_d(ℝ)$ are the only ones that can be used in the definition provided one demands a natural non-degeneracy condition. Furthermore, we analyse the absolute continuity properties of the stationary distribution (especially when the driving matrix subordinator is the quadratic variation of a $d$-dimensional Lévy process) and study the question of how to choose the driving matrix subordinator in order to obtain a given stationary distribution. Finally, we present results on the first and second order moment structure of matrix subordinators, which is closely related to the moment structure of positive semidefinite Ornstein–Uhlenbeck type processes. The latter results are important for method of moments based estimation.

Scaling of proposals for Metropolis algorithms is an important practical problem in MCMC implementation. Criteria for scaling based on empirical acceptance rates of algorithms have been found to work consistently well across a broad range of problems. Essentially, proposal jump sizes are increased when acceptance rates are high and decreased when rates are low. In recent years, considerable theoretical support has been given for rules of this type which work on the basis that acceptance rates around 0.234 should be preferred. This has been based on asymptotic results that approximate high dimensional algorithm trajectories by diffusions. In this paper, we develop a novel approach to understanding 0.234 which avoids the need for diffusion limits. We derive explicit formulae for algorithm efficiency and acceptance rates as functions of the scaling parameter. We apply these to the family of elliptically symmetric target densities, where further illuminating explicit results are possible. Under suitable conditions, we verify the 0.234 rule for a new class of target densities. Moreover, we can characterise cases where 0.234 fails to hold, either because the target density is too diffuse in a sense we make precise, or because the eccentricity of the target density is too severe, again in a sense we make precise. We provide numerical verifications of our results.

Let $$ Y_j=f_{\ast}(X_j)+\xi_j,\qquad j=1,\dots, n, $$ where $X, X_1,\dots, X_n$ are i.i.d. random variables in a measurable space $(S,\mathcal{A})$ with distribution $\Pi$ and $\xi, \xi_1,\dots ,\xi_n$ are i.i.d. random variables with ${\mathbb E}\xi=0$ independent of $(X_1,\dots, X_n).$ Given a dictionary $h_1,\dots, h_{N}: S\mapsto{\mathbb R},$ let $ f_{\lambda}:=\sum_{j=1}^N \lambda_j h_j$, $ \lambda=(\lambda_1,\dots, \lambda_N)\in{\mathbb R}^N. $ Given $\varepsilon>0,$ define $$ \hat\Lambda_{\varepsilon}:=\Biggl\{\lambda\in{\mathbb R}^N: \max_{1\leq k\leq N} \Biggl|n^{-1}\sum_{j=1}^n \bigl(f_{\lambda}(X_j)-Y_j\bigr)h_k(X_j)\Biggr| \leq\varepsilon \Biggr\} $$ and $$\hat\lambda:=\hat\lambda^{\varepsilon}\in \operatorname{Argmin}_{\lambda\in\hat\Lambda_{\varepsilon}}\|\lambda\| _{\ell_1}. $$ In the case where $f_{\ast}:=f_{\lambda^{\ast}}, \lambda^{\ast}\in {\mathbb R}^N,$ Candes and Tao Ann. Statist. 35 (2007) 2313-2351] suggested using $\hat\lambda$ as an estimator of $\lambda^{\ast}.$ They called this estimator “the Dantzig selector”. We study the properties of $f_{\hat\lambda}$ as an estimator of $f_{\ast}$ for regression models with random design, extending some of the results of Candes and Tao (and providing alternative proofs of these results).

Expansions are provided for the moments of the number of collisions $X_n$ in the $β(2, b)$-coalescent restricted to the set $\{1, …, n\}$. We verify that $X_{n}/\mathbb{E}X_{n}$ converges almost surely to one and that $X_n$, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in $β(a, b)$-coalescents with $a∈(0, 2)$ and $b=1$, and $a>2$ and $b>0$. The case $a=2$ is a kind of ‘border situation’ which seems not to be amenable to approaches used for $a≠2$.

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H∈(0, 1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to be random and even anticipating. The main technique used in this work is an adaptation of the anticipating Girsanov transformation of Buckdahn [Mem. Amer. Math. Soc. 111 (1994)] for the Brownian motion case. By extending a fundamental theorem of Kusuoka [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 567–597] using fractional calculus, we are able to prove that the anticipating Girsanov transformation holds for the fractional Brownian motion case as well. We then use this result to prove the well-posedness of the SDE.

Let $M, N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities:

(i) If $M\geq0$ and $0<p\leq1$, then \[ \Vert N\Vert_{p,\infty} \leq2\Vert M\Vert_p \] and the constant is the best possible.

(ii) If $M\geq0$ and $p\geq2$, then \[ \Vert N\Vert_{p,\infty} \leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p \] and the constant is the best possible.

(iii) If $1\leq p\leq2$ and $M$ and $N$ are orthogonal, then \[ \Vert N\Vert_{p,\infty} \leq K_p \Vert M\Vert_p, \] where \[ K_p^p=\frac{1}{\Gamma(p+1)}\cdot\biggl(\frac{\pi}{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+ 1/5^2+1/7^2+\cdots}{ 1-1/3^{p+1}+1/5^{p+1}-1/7^{p+1}+\cdots}. \] The constant is the best possible.

We also provide related estimates for harmonic functions on Euclidean domains.

We propose a bivariate model for a pair of dependent unit vectors which is generated by Brownian motion. Both marginals have uniform distributions on the sphere, while the conditionals follow so-called “exit” distributions. Some properties of the proposed model, including parameter estimation and a pivotal statistic, are investigated. Further study is undertaken for the bivariate circular case by transforming variables and parameters into the form of complex numbers. Some desirable properties, such as a multiplicative property and infinite divisibility, hold for this submodel. Two estimators for the parameter of the submodel are studied and a simulation study is carried out to investigate the finite sample performance of the estimators. In an attempt to produce more flexible models, some methods to generalize the proposed model are discussed. One of the generalized models is applied to wind direction data. Finally, we show how it is possible to construct distributions in the plane and on the cylinder by applying bilinear fractional transformations to the proposed bivariate circular model.

We correct a condition in a result of Johnson and Samworth (Bernoulli 11 (2005) 829–845) concerning convergence to stable laws in Mallows distance. We also give an improved version of this result, setting it in the more familiar context of a Lindeberg-like condition.