Suppose that $k$ series, all having the same autocorrelation function, are observed in parallel at $n$ points in time or space. From a single series of moderate length, the autocorrelation parameter $β$ can be estimated with limited accuracy, so we aim to increase the information by formulating a suitable model for the joint distribution of all series. Three Gaussian models of increasing complexity are considered, two of which assume that the series are independent. This paper studies the rate at which the information for $β$ accumulates as $k$ increases, possibly even beyond $n$. The profile log likelihood for the model with $k(k+1)/2$ covariance parameters behaves anomalously in two respects. On the one hand, it is a log likelihood, so the derivatives satisfy the Bartlett identities. On the other hand, the Fisher information for $β$ increases to a maximum at $k=n/2$, decreasing to zero for $k≥n$. In any parametric statistical model, one expects the Fisher information to increase with additional data; decreasing Fisher information is an anomaly demanding an explanation.

Under left truncation, data $(X_i, Y_i)$ are observed only when $Y_i≤X_i$. Usually, the distribution function $F$ of the $X_i$ is the target of interest. In this paper, we study linear functionals $∫\varphi \mathrm{d}F_n$ of the nonparametric maximum likelihood estimator (MLE) of $F$, the Lynden-Bell estimator $F_n$. A useful representation of $∫\varphi \mathrm{d}F_n$ is derived which yields asymptotic normality under optimal moment conditions on the score function $\varphi$. No continuity assumption on $F$ is required. As a by-product, we obtain the distributional convergence of the Lynden-Bell empirical process on the whole real line.

Let $\mathbf{Y}=\mathbf{X \Theta Z}'+\mathbf{\mathcal {E}}$ be the growth curve model with $\mathbf{\mathcal{E}}$ distributed with mean $\mathbf{0}$ and covariance $\mathbf{I}_n⊗\mathbf{Σ}$, where $\mathbf{Θ}$, $\mathbf{Σ}$ are unknown matrices of parameters and $\mathbf{X}$, $\mathbf{Z}$ are known matrices. For the estimable parametric transformation of the form $\mathbf{γ}=\mathbf{CΘD}'$ with given $\mathbf{C}$ and $\mathbf{D}$, the two-stage generalized least-squares estimator $\mathbf{γ̂}(\mathbf{Y})$ defined in (7) converges in probability to $\mathbf{γ}$ as the sample size $n$ tends to infinity and, further, $\sqrt{n}[\hat{\mathbf{\gamma}}(\mathbf{Y})-{\mathbf{\gamma}}]$ converges in distribution to the multivariate normal distribution $\mathcal{N}(\mathbf{0},(\mathbf{C}\mathbf{R}^{-1}\mathbf{C}')\otimes(\mathbf{D}(\mathbf{Z}'\mathbf{\Sigma }^{-1}\mathbf{Z})^{-1}\mathbf{D}'))$ under the condition that $\lim_{n→∞} \mathbf{X}'\mathbf{X}/n=\mathbf{R}$ for some positive definite matrix $\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator $\mathbf{\hat Σ}(\mathbf{Y})$ defined in (6) is also proved to be consistent with the second-order parameter matrix $\mathbf{Σ}$.

We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value distribution of a maximum-type test statistic which is asymptotically equivalent to the maximally selected likelihood ratio. The resulting test is easy to apply and has good size and power, even in small samples.

In regression with random design, we study the problem of selecting a model that performs well for out-of-sample prediction. We do not assume that any of the candidate models under consideration are correct. Our analysis is based on explicit finite-sample results. Our main findings differ from those of other analyses that are based on traditional large-sample limit approximations because we consider a situation where the sample size is small relative to the complexity of the data-generating process, in the sense that the number of parameters in a ‘good’ model is of the same order as sample size. Also, we allow for the case where the number of candidate models is (much) larger than sample size.

This work is intended as a contribution to the theory of a wavelet-based adaptive estimator of the memory parameter in the classical semi-parametric framework for Gaussian stationary processes. In particular, we introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish a central limit theorem for the estimator of the memory parameter with the minimax rate of convergence (up to a logarithm factor). The quality of the estimators is demonstrated via simulations.

Thus far, likelihood-based interval estimates for quantiles have not been studied in the literature on interval censored case 2 data and partly interval censored data, and, in this context, the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval censored data. The fourth order expansion of the weighted empirical log-likelihood ratio is derived and the theoretical coverage accuracy equation for the proposed WELRCI is established, which generally guarantees at least ‘first order’ accuracy. In particular, for right censored data, we show that the coverage accuracy is at least $O(n^{−1/2})$ and our simulation studies show that in comparison with empirical likelihood-based methods, the smoothing used in WELRCI generally provides a shorter confidence interval with comparable coverage accuracy. For interval censored data, it is interesting to find that with an adjusted rate $n^{−1/3}$, the weighted empirical log-likelihood ratio has an asymptotic distribution completely different from that obtained by the empirical likelihood approach and the resulting WELRCI perform favorably in the available comparison simulation studies.

A new independence property of univariate beta distributions, related to the results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a characterization of univariate beta laws through this independence property is proved. A related characterization of a family of 2×2 random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation.

We investigate the relation of the semigroup probability density of an infinite activity Lévy process to the corresponding Lévy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the Lévy measure and the third method uses the analytic continuation of the Lévy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.

Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques by an application involving linear and quadratic functionals of generalized Ornstein–Uhlenbeck processes, as well as examples concerning random hazard rates.

We study the asymptotic expansions with respect to $h$ of $$ \mathrm{E}[Δ_hf(X_t)], \mathrm{E}[Δ_hf(X_t)|ℱ_t^X] \mathrm{and} \mathrm{E}[Δ_hf(X_t)|X_t], $$ where $Δ_hf(X_t)=f(X_{t+h})−f(X_t)$, when $f:ℝ→ℝ$ is a smooth real function, $t≥0$ is a fixed time, $X$ is the solution of a one-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst index $H>1/2$ and $ℱ^X$ is its natural filtration.

A random vector $X$ with representation $X=∑_{j≥0}A_jZ_j$ is considered. Here, $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, ‘predictable’ with respect to the sequence $(Z_j)$. The distribution of $Z_1$ is assumed to be multivariate regular varying. Moment conditions on the matrices $(A_j)$ are determined under which the distribution of $X$ is regularly varying and, in fact, ‘inherits’ its regular variation from that of the $(Z_j)$’s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.

Denote by $H(t)=(H_1(t), …, H_N(t))$ a function in $t∈ℝ_+^N$ with values in $(0, 1)^N$. Let $\{B^{H(t)}(t)\}=\{B^{H(t)}(t), t∈ℝ_+^N\}$ be an $(N, d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$. Under some regularity conditions on the function $H(t)$, we prove the existence, joint continuity and the Hölder regularity of the local times of $\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the level sets of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.