For an absolutely continuous (integer-valued) r.v. $X$ of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order $k$ holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237–260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. $X$, expressions that seem to be known only in particular cases (for the Normal, see [Houdré and Kagan, J. Theoret. Probab. 8 (1995) 23–30]; see also [Houdré and Pérez-Abreu, Ann. Probab. 23 (1995) 400–419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819–1837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0–1 point processes and then applied Stein’s method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein’s method, thereby expressing the error bounds in terms of the mean measures of the individual point processes, which is not possible with Schuhmacher’s approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia [Ann. Probab. 32 (2004) 2545–2569] and apply the main theorem to the superposition of thinned point processes and of renewal processes.

We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.

We propose a new algorithm to compute numerically the distribution function of the sum of $d$ dependent, non-negative random variables with given joint distribution.

We compare estimators of the (essential) supremum and the integral of a function $f$ defined on a measurable space when $f$ may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of $f$ by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum, the criterion is based on the stochastic order of estimators.

This paper deals with convergence of the maximum a posterior probability path estimator in hidden Markov models. We show that when the state space of the hidden process is continuous, the optimal path may stabilize in a way which is essentially different from the previously considered finite-state setting.

A closed-form expression for Wigner–Yanase skew information in mixed-state quantum systems is derived. It is shown that limit values of the mixing coefficients exist such that Wigner–Yanase information is equal to Helstrom information. The latter constitutes an upper bound for the classical expected Fisher information, hence the inverse Wigner–Yanase information provides an approximate lower bound to the variance of an unbiased estimator of the parameter of interest. The advantage of approximating Helstrom’s sharp bound lies in the fact that Wigner–Yanase information is straightforward to compute, while it is often very difficult to obtain a feasible expression for Helstrom information. In fact, the latter requires the solution of an implicit second order matrix differential equation, while the former requires just scalar differentiation.

The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of a Lévy process, is considered here under a discrete-sampling scheme. In this setting, the jumps are latent variables, the statistical properties of which can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the Lévy density based on Grenander’s method of sieves was proposed in Figueroa-López [IMS Lecture Notes 57 (2009) 117–146]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the Lévy density. In the pointwise case, our estimators converge to the Lévy density at a rate that is arbitrarily close to the rate of the minimax risk of estimation on smooth Lévy densities. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to $\log^{−1/2}(n) ⋅ n^{−1/3}$, where $n$ is the number of observations. The convergence rates are valid, provided that $s$ is smooth enough and that the time horizon $T_n$ and the dimension of the sieve are appropriately chosen in terms of $n$.

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur’s representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that $$\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^{n}\phi\bigl(X_{n:i}^{(1)},\ldots,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}Z_{n,i}+\mathrm{o}_{P}(1)$$ as $n → ∞$, where $γ̄>$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.

A classical condition for fast learning rates is the margin condition, first introduced by Mammen and Tsybakov. We tackle in this paper the problem of adaptivity to this condition in the context of model selection, in a general learning framework. Actually, we consider a weaker version of this condition that allows one to take into account that learning within a small model can be much easier than within a large one. Requiring this “strong margin adaptivity” makes the model selection problem more challenging. We first prove, in a general framework, that some penalization procedures (including local Rademacher complexities) exhibit this adaptivity when the models are nested. Contrary to previous results, this holds with penalties that only depend on the data. Our second main result is that strong margin adaptivity is not always possible when the models are not nested: for every model selection procedure (even a randomized one), there is a problem for which it does not demonstrate strong margin adaptivity.

In this paper, we will consider the estimation of a monotone regression (or density) function in a fixed point by the least-squares (Grenander) estimator. We will show that this estimator is locally asymptotic minimax, in the sense that, for each $f_0$, the attained rate of the probabilistic error is uniform over a shrinking $L^2$-neighborhood of $f_0$ and there is no estimator that attains a significantly better uniform rate over these shrinking neighborhoods. Therefore, it adapts to the individual underlying function, not to a smoothness class of functions. We also give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.

It is widely admitted that structured nonparametric modeling that circumvents the curse of dimensionality is important in nonparametric estimation. In this paper we show that the same holds for semi-parametric estimation. We argue that estimation of the parametric component of a semi-parametric model can be improved essentially when more structure is put into the nonparametric part of the model. We illustrate this for the partially linear model, and investigate efficiency gains when the nonparametric part of the model has an additive structure. We present the semi-parametric Fisher information bound for estimating the parametric part of the partially linear additive model and provide semi-parametric efficient estimators for which we use a smooth backfitting technique to deal with the additive nonparametric part. We also present the finite sample performances of the proposed estimators and analyze Boston housing data as an illustration.

A statistic based on increment ratios (IR’s) and related to zero crossings of an increment sequence is defined and studied for the purposes of measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (which we shall call the IR-roughness) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. First, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Second, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Third, the IR-roughness of a Lévy process with an $α$-stable tangent process is established and can be used to estimate the fractional parameter $α ∈ (0, 2)$ following a central limit theorem.

We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a Lévy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation, we design a test for the presence of a continuous martingale component in the process and a test for establishing whether the jumps have finite or infinite variation, based on observations on a discrete-time grid. We evaluate the performance of our tests using simulations of various stochastic models and use the tests to investigate the fine structure of the DM/USD exchange rate fluctuations and SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian component and a finite variation jump component.

Let $ξ$ be a (possibly killed) subordinator with Laplace exponent $ϕ$ and denote by $I_ϕ = ∫_0^∞e^{−ξ_s} ds$, the so-called exponential functional. Consider the positive random variable $I_{ψ1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95–106], is determined by its negative entire moments as follows: $$\mathbb {E}[I_{\psi_{1}}^{-n}]=\prod_{k=1}^{n}\phi(k),\qquad n=1,2,\ldots.$$ In this note, we show that $I_{ψ1}$ is a positive self-decomposable random variable whenever the Lévy measure of $ξ$ is absolutely continuous with a monotone decreasing density. In fact, $I_{ψ1}$ is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95–106] the following factorization of the exponential law e: $$I_{\phi}/I_{\psi_{1}}\stackrel {\mathrm {(d)}}{=}{\mathbf {e}},$$ where $I_{ψ1}$ is taken to be independent of $I_ϕ$. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that $S(α)^α$ is a self-decomposable random variable, where $S(α)$ is a positive stable random variable of index $α ∈ (0, 1)$.