In a nonparametric regression setting with multiple random predictor variables, we give the asymptotic distributions of estimators of global integral functionals of the regression surface. We apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination. This coefficient, which is also called Pearson's correlation ratio, gives the fraction of the total variability of a response that can be explained by a given set of covariates. It can be used to construct measures of nonlinearity of regression and relative importance of subsets of regressors, and to assess the validity of other model restrictions. In addition to providing asymptotic results, we propose several data-based bandwidth selection rules and carry out a Monte Carlo simulation study of finite sample properties of these rules and associated estimators of explanatory power. We also provide two real data examples.

Based on a random sample of size $n$ from an unknown density $f$ on the real line, the nonparametric estimation of $\theta_k = \int\{f^{(k)}(x)\}^2 dx, k = 0, 1,\ldots$, is considered. These functionals are important in a number of contexts. The proposed estimates of $\theta_k$ is constructed in the frequency domain by using the sample characteristic function. It is known that the sample characteristic function at high frequency is dominated by sample variation and does not contain much information about $f$. Hence, the variation of the estimate can be reduced by modifying the sample characteristic function beyond some cutoff frequency. It is proposed to select adaptively the cutoff frequency by a generalization of the (smoothed) cross-validation. The exact convergence rate of the proposed estimate to $\theta_k$ is established. It depends solely on the smoothness of $f$. For sufficiently smooth $f$, it is shown that the proposed estimate is asymptotically normal, attains the optimal $O_p(n^{-1/2})$ rate and achieves the information bound. Finally, to improve the performance of the proposed estimate at small to moderately large $n$, two modifications are proposed. One modification is for estimating $\theta_0$; it reduces bias of the estimate. The other modification is for estimating $\theta_k, k \geq 1$; it reduces sample variation of the estimate. In simulation studies the superior performance of the proposed procedures is clearly demonstrated.

Usual nonparametric regression estimators often show many little wiggles which do not appear to be necessary for a good description of the data. The new "Wp" smoother is a maximum penalized likelihood (MPL) estimate with a novel roughness penalty. It penalizes a relative change of curvature. This leads to disjoint classes of functions, each with given number, $n_w$, of inflection points. For a "Wp" estimate, $f"(x) = \pm (x - w_1)\cdots (x - w_{n_w}) \cdot \exp h_f(x)$, which is semiparametric, with parameters $w_j$ and nonparametric part $h_f(\cdot)$. The main mathematical result is a convenient form of the characterizing differential equation for a very general class of MPL estimators.

It is shown that, in the nonparametric setting for the so-called Wicksell problem, the distribution function of the squared radii of the balls cannot be estimated at a rate faster than $n^{-1/2}\sqrt{\log n}$. We present an isotonic estimator of the distribution function which attains this rate and derive its asymptotic (normal) distribution. It is shown that the variance of this limiting distribution is exactly half the asymptotic variance of the naive plug-in estimator.

Let $\mathscr{M}_\mu$ be the set of all probability densities equivalent to a given reference probability measure $\mu$. This set is thought of as the maximal regular (i.e., with strictly positive densities) $\mu$-dominated statistical model. For each $f \in \mathscr{M}_\mu$ we define (1) a Banach space $L_f$ with unit ball $\mathscr{V}_f$ and (2) a mapping $s_f$ from a subset $\mathscr{U}_f$ of $\mathscr{M}_\mu$ onto $\mathscr{V}_f$, in such a way that the system $(s_f, \mathscr{U}_f, f \in \mathscr{M}_\mu)$ is an affine atlas on $\mathscr{M}_\mu$. Moreover each parametric exponential model dominated by $\mu$ is a finite-dimensional affine submanifold and each parametric statistical model dominated by $\mu$ with a suitable regularity is a submanifold. The global geometric framework given by the manifold structure adds some insight to the so-called geometric theory of statistical models. In particular, the present paper gives some of the developments connected with the Fisher information metrics (Rao) and the Hilbert bundle introduced by Amari.

The parametric statistical models with suitable regularity conditions have a natural Riemannian manifold structure, given by the information metric. Since the parameters are merely labels for the probability measures, an inferential statement should be formulated through intrinsic objects, invariant under reparametrizations. In this context the estimators will be random objects valued on the manifold corresponding to the statistical model. In spite of these considerations, classical measures of an estimator's performance, like the bias and the mean square error, are clearly dependent on the statistical model parametrizations. In this paper the authors work with extended notions of mean value and moments of random objects which take values on a Hausdorff and connected manifold, equipped with an affine connection. In particular, the Riemannian manifold case is considered. This extension is applied to the bias and the mean square error study in statistical point estimation theory. Under this approach an intrinsic version of the Cramer-Rao lower bound is obtained: a lower bound, which depends on the intrinsic bias and the curvature of the statistical model, for the mean square of the Rao distance, the invariant measure analogous to the mean square error. Further, the behavior of the mean square of the Rao distance of an estimator when conditioning with respect to a sufficient statistic is considered, obtaining intrinsic versions of the Rao-Blackwell and Lehmann-Scheffe theorems. Asymptotic properties complete the study.

Let $M$ be a compact, smooth, orientable manifold without boundary, and let $f: M \rightarrow \mathbb{R}$ be a smooth function. Let $dm$ be a volume form on $M$ with total volume 1, and denote by $X$ the corresponding random variable. Using a theorem of Kirwan, we obtain necessary conditions under which the method of stationary phase returns an exact evaluation of the characteristic function of $f(X)$. As an application to the Langevin distribution on the sphere $S^{d-1}$, we deduce that the method of stationary phase provides an exact evaluation of the normalizing constant for that distribution when, and only when, $d$ is odd.

Chen and Loh showed that the Box-Cox transformed two-sample $t$-test is more powerful than the ordinary $t$-test under Pitman alternatives where the location shifts appear in the untransformed scale. In this article, we prove that Chen and Loh's result also holds for a general family of transformations. An upper bound on the asymptotic relative efficiency (ARE) is obtained. In addition, we investigate bounds on the ARE under Pitman location shift alternatives in the transformed scale. We find that when the estimate for $\lambda$ is consistent, a lower bound on the ARE is the reciprocal of Fisher information of the standard transformed distribution. This lower bound is close to 1 for commonly used symmetric distributions.

The data driven method of selecting the number of components in Neyman's smooth test for uniformity, introduced by Ledwina, is extended. The resulting tests consist of a combination of Schwarz's Bayesian information criterion (BIC) procedure and smooth tests. The upper bound of the dimension of the exponential families in applying Schwarz's rule is allowed to grow with the number of observations to infinity. Simulation results show that the data driven version of Neyman's test performs very well for a wide range of alternatives and is competitive with other recently introduced (data driven) procedures. It is shown that the data driven smooth tests are consistent against essentially all alternatives. In proving consistency, new results on Schwarz's selection rule are derived, which may be of independent interest.

This paper establishes the asymptotic admissibility of the likelihood ratio (LR) test for a general class of testing problems in which a nuisance parameter is present only under the alternative hypothesis. The paper also establishes the finite sample admissibility of the LR test for testing problems of this sort that arise in Gaussian linear regression models with known variance.

Assuming the model $f(\lambda) \sim G\lambda^{1-2H}$, as $\lambda \rightarrow 0 +$, for the spectral density of a covariance stationary process, we consider an estimate of $H \in (0, 1)$ which maximizes an approximate form of frequency domain Gaussian likelihood, where discrete averaging is carried out over a neighbourhood of zero frequency which degenerates slowly to zero as sample size tends to infinity. The estimate has several advantages. It is shown to be consistent under mild conditions. Under conditions which are not greatly stronger, it is shown to be asymptotically normal and more efficient than previous estimates. Gaussianity is nowhere assumed in the asymptotic theory, the limiting normal distribution is of very simple form, involving a variance which is not dependent on unknown parameters, and the theory covers simultaneously the cases $f(\lambda) \rightarrow \infty, f(\lambda) \rightarrow 0$ and $f(\lambda) \rightarrow C \in (0, \infty)$, as $\lambda \rightarrow 0$. Monte Carlo evidence on finite-sample performance is reported, along with an application to a historical series of minimum levels of the River Nile.

An autoregressive time series is said to be unstable if all of its characteristic roots lie on or outside the unit circle, with at least one on the unit circle. This paper aims at developing asymptotic inferential schemes for an unstable autoregressive model generated by long-memory innovations. This setting allows both nonstationarity and long-memory behavior in the modeling of low-frequency phenomena. In developing these procedures, a novel weak convergence result for a sequence of long-memory random variables to a stochastic integral of fractional Brownian motions is established. Results of this paper can be used to test for unit roots in a fractional AR model.

We suggest an estimator, based on the periodogram, of the fractal index and fractal dimension of a continuous, stationary Gaussian process. We argue that the cosine part of the periodogram is more appropriate than the full periodogram for this application. The term "semiperiodogram" is used to describe the cosine component, and our estimator is based on simple linear regression of the logarithm of the semiperiodogram on the algorithm of frequency. Theoretical properties of the estimator, including its bias, variance and asymptotic distribution, are derived. Consistency is possible using only a small trace of the process, recorded over a fixed interval. We do not need to model the covariance function parametrically, and assume only mild conditions on the behaviour of the covariance in the neighbourhood of the origin. The issue of aliasing is discussed in both theoretical and numerical terms, and the numerical properties of the estimator are assessed in a simulation study.

The additive-multiplicative hazard model specifies that the hazard function for the counting process associated with a multidimensional covariate process $Z = (W^T, X^T)^T$ takes the form of $\lambda(t\mid Z) = g\{\beta^T_0 W(t)\} + \lambda_0(t)h\{\gamma^T_0X(t)\}$, where $\theta_0 = (\beta^T_0, \gamma^T_0)^T$ is a vector of unknown regression parameters, $g$ and $h$ are known link functions and $\lambda_0$ is an unspecified "baseline hazard function." In this paper, we develop a class of simple estimating functions for $\theta_0$, which contains the partial likelihood score function in the special case of proportional hazards models. The resulting estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions. Weak convergence of the Aalen-Breslow type estimators for the cumulative baseline hazard function $\Lambda_0(t) = \int^t_0\lambda_0(u) du$ is also established. Furthermore, we construct adaptive estimators for $\theta_0$ and $\Lambda_0$ that achieve the (semiparametric) information bounds. Finally, a real example is provided along with some simulation results.

We introduce a new kernel hazard estimator in a nonparametric model where the stochastic hazard depends on the current value of time and on the current value of a time dependent covariate or marker. We establish the pointwise and global convergence of our estimator.

Methods are provided for regression parameter and cumulative baseline hazard estimation in the Cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law. The key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes, leading to estimation methods parallel to those for full cohort data. These techniques are illustrated for a number of sampling designs, including three novel designs: counter-matching with additional randomly sampled controls; quota sampling of controls; and nested case-control sampling with number of controls dependent on the failure's exposure status. General asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs.

We obtain an exponential probability inequality for martingales and a uniform probability inequality for the process $\int g dN$, where $N$ is a counting process and where $g$ varies within a class of predictable functions $\mathscr{G}$. For the latter, we use techniques from empirical process theory. The uniform inequality is shown to hold under certain entropy conditions on $\mathscr{G}$. As an application, we consider rates of convergence for (nonparametric) maximum likelihood estimators for counting processes. A similar result for discrete time observations is also presented.

The order of the Kolmogorov-Smirnov distance for the bootstrap of $U$-quantiles is considered. We observe that the order of the bootstrap of $U$-quantiles depends on the order of the local variance of the first term of the Hoeffding decomposition at the $U$-quantile. This order can be smaller than the order of the bootstrap of quantiles: $U$-quantiles can be smoother than quantiles.

This paper calculates the influence functions and asymptotic distributions of $M$-estimators of the rotation $A$ in a spherical regression model on the unit sphere in $p$ dimensions with isotropic errors. The problem arises in the reconstruction of the motion of a rigid body on the surface of the sphere. The comparable model for $p$-dimensional Euclidean space data is that $(\nu_1, \ldots, \nu_n)$ are independent with $\nu_i$ symmetrically distributed around $\gamma A \cdot u_i + b, u_i$ known, where the real constant $\gamma > 0, p \times p$ rotation matrix $A$ and $p$-vector $b$ are the parameters to be estimated. This paper also calculates the influence functions and asymptotic distributions of $M$-estimators for $\gamma, A$ and $b$. Besides rigid body motion, this problem arises in image registration from landmark data. Particular attention is paid to how the geometry of the rigid body or landmarks affects the statistical properties of the estimators.

We consider independent random elements $X_1, \ldots, X_n, n \in \mathbb{N}$, with values in a measurable space $(\mathscr{X}, \mathscr{B})$ so that $X_1, \ldots, X_{\lbrack n\theta\rbrack}$ have a common distribution $\nu_1$ and the remaining $X_{\lbrack n\theta\rbrack + 1}, \ldots, X_n$ have a common distribution $\nu_2 \neq \nu_1$, for some $\theta \in (0, 1)$. The change point $\theta$ as well as the distributions are unknown. A family of tests is introduced for the nonstandard change-point problem $H_0: \theta \in \Theta_0$ versus $H_1: \theta \not\in \Theta_0$, where $\Theta_0$ is an arbitrary subset of (0, 1). The tests are shown to be asymptotic level-$\alpha$ tests and to be consistent on a large class of alternatives. The same holds for the corresponding bootstrap versions of the tests. Moreover, we present a detailed investigation of the local power.