Single-index modeling is widely applied in,for example,econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models. By such modeling the statistical analysis usually focuses on estimating the index coefficients. The average derivative estimator (ADE) of the index vector is based on the fact that the average gradient of a single index function $f(x^{\top}\beta)$ is proportional to the index vector $\beta$. Unfortunately,a straightforward application of this idea meets the so-called “curse of dimensionality” problem if the dimensionality $d$ of the model is larger than 2. However, prior information about the vector $\beta$ can be used for improving the quality of gradient estimation by extending the weighting kernel in a direction of small directional derivative. The method proposed in this paper consists of such iterative improvements of the original ADE. The whole procedure requires at most 2 $\log n$ iterations and the resulting estimator is $\sqrt{n}$-consistent under relatively mild assumptions on the model independently of the dimensionality $d$.

We suggest a method for monotonizing general kernel-type estimators, for example local linear estimators and Nadaraya .Watson estimators. Attributes of our approach include the fact that it produces smooth estimates, indeed with the same smoothness as the unconstrained estimate. The method is applicable to a particularly wide range of estimator types, it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented. Therefore,an experimenter may use his or her favorite kernel estimator, and their favorite bandwidth selector, to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way. Implementation involves only an off-the-shelf programming routine. The method is based on maximizing fidelity to the conventional empirical approach, subject to monotonicity.We adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change, in the sense of a distance measure, subject to imposing the constraint of monotonicity.

In this paper, we consider nonparametric least squares estimators of the mode of an unknown unimodal regression function. We establish almost sure convergence of these estimators with nearly optimal convergence rates, under the assumption of the exponential tail for the error distributions.

Ghosh and Ramamoorthi studied posterior consistency for survival models and showed that the posterior was consistent when the prior on the distribution of survival times was the Dirichlet process prior. In this paper,we study posterior consistency of survival models with neutral to the right process priors which include Dirichlet process priors. A set of sufficient conditions for posterior consistency with neutral to the right process priors are given. Interestingly, not all the neutral to the right process priors have consistent posteriors, but most of the popular priors such as Dirichlet processes, beta processes and gamma processes have consistent posteriors. With a class of priors which includes beta processes, a necessary and sufficient condition for the consistency is also established. An interesting counter-intuitive phenomenon is found. Suppose there are two priors centered at the true parameter value with finite variances. Surprisingly, the posterior with smaller prior variance can be inconsistent, while that with larger prior variance is consistent.

We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include in finite dimensional cases. The rates are driven by two quantities: the size of the space, as measured by bracketing entropy, and the degree to which the prior concentrates in a small ball around the true parameter. We consider two examples.

Log-linear modelling plays an important role in many statistical applications, particularly in the analysis of contingency table data.With the advent of powerful new computational techniques such as reversible jump MCMC, Bayesian analyses of these models, and in particular model selection and averaging, have become feasible. Coupled with this is the desire to construct and use suitably flexible prior structures which allow efficient computation while facilitating prior elicitation. The latter is greatly improved in the case where priors can be specified on interpretable parameters about which relevant experts can express their beliefs.

In this paper, we show how the specification of a general multivariate normal prior on the log-linear parameters induces a multivariate log- normal prior on the corresponding cell counts of a contingency table. We derive the parameters of this distribution in an explicit practical form and state the corresponding mean and covariances of the cell counts. We discuss the importance of these results in terms of applying both uninformative and informative priors to the model parameters and provide an illustration in the context of the analysis of a 2³ contingency table.

When fitting, by least squares, a linear model (with an intercept term) with $p$ parameters to $n$ data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided $p^3 \log^2 (p) /n \to 0$ for any error density having finite variance and a bounded first derivative. No further conditions are imposed on the sequence of design matrices. The result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares.

E-optimal and standardized-E-optimal designs for various types of rational regression models are determined. In most cases, optimal designs are found for every parameter subsystem. The design points and weights are given explicitlyin terms of Bernstein-Szegő polynomials.The analysis is based on a general theorem on E-optimal designs for Chebyshev systems.

We discuss here a general approach to the calculation of the asymptotic covariance of $M$-estimates for location parameters in statistical group models when an invariant objective function is used. The calculation reduces to standard tools in group representation theory and the calculation of some constants. Only the constants depend upon the precise forms of the density or of the objective function. If the group is sufficiently large this represents a major simplification in the computation of the asymptotic covariance.

Following the approach of Chang and Tsai we define a regression model for group models and derive the asymptotic distribution of estimates in the regression model from the corresponding distribution theory for the location model. The location model is not, in general, a subcase of the regression model.

We illustrate these techniques using Stiefel manifolds.The Stiefel manifold $\mathcal{V}_{p,m}$ is the collection of $p\times m$ matrices $\mathbf{X}$ which satisfy the condition $\mathbf{X}^T \mathbf{X} = \mathbf{I}_m$ where $m \le p$. Under the assumption that $\mathbf{X}$ has a distribution proportional to $\exp(Tr(\mathbf{F}^T \mathbf{X}))$ for some $p\times m$ matrix $\mathbf{F}$ Downs (1972) gives approximations to maximum likelihood estimation of $\\mathbf{F}$. In this paper, we consider a somewhat differenct location problem: under the assumption that $\mathbf{X}$ has a distribution of the form $f(Tr(\mathbf{\theta}^T_0 \mathbf{X})$ for some $\mathbf{\theta}_0 \in \mathcal{V}_{p,m}$, we calculate the asymptotic distribution of $M$-estimates which minimize an objective function of the form $\sum_i \rho(Tr(\mathbf{\theta}^T \mathbf{X}_i))$. i i The assumptions on the form of the density and the objective function can be relaxed to a more general invariant form. In this case, the calculation of the asymptotic distribution of $\hat{\mathbf{\theta}}$ reduces to the calculation of four constants and we present consistent estimators for these constants.

Prentice (1989)introduced a regression model for Stiefel manifolds. In the Prentice model, $\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_ n \in \mathcal{V}_{p,m}$ are fixed, $\mathbf{V}_1, \mathbf{V}_2, \ldots, \mathbf{V}_ n \in \mathcal{V}_{p,m}$ are independent random so that the distribution of $\mathbf{V}_i$ depends only upon $Tr(\mathbf{V}^T_i \mathbf{A}_2 \mathbf{u}_i \mathbf{A}^T_1$ for unknown $(\mathbf{A}_1, \mathbf{A}_2) \in SO(m) \times SO(p)$ .We discuss here M-estimation of $\mathcal{A}_1$ and $\mathcal{A}_2$ under general invariance conditions for both the density and the objective function.

Using a well-studied example on vector cardiograms we discuss the physical interpretation of the invariance assumption as well as of the parameters (A₁ A₂) in the Prentice regression model. In particular, A₁ represents a rotation of the u’s to the V’s in a coordinate systemrelative to the u’s and A₂ represents a rotation of the u’s to the V’s in a coordinate system fixed to the external world.

We consider abstractly defined time series arrays y _t(T), 1 \le t\le T, requiring only that their sample lagged second moments converge and that their end values y_1+j(T) and y_T-j(T) be of order less than T^½ for each j \ge 0. We show that,under quite general assumptions, various types of arrays that arise naturally in time series analysis have these properties,including regression residuals from a time series regression, seasonal adjustments and infinite variance processes rescaled by their sample standard deviation. We establish a useful uniform convergence result,namely that these properties are preserved in a uniform way when relatively compact sets of absolutely summable filters are applied to the arrays. This result serves as the foundation for the proof, in a companion paper by Findley, Pötscher and Wei, of the consistency of parameter estimates specified to minimize the sample mean squared multistep-ahead forecast error when invertible short-memory models are fit to (short- or long-memory)time series or time series arrays.

We study the problem of estimatingsome unknown regression function in a $\beta$-mixing dependent framework. To this end, we consider some collection of models which are finite dimensional spaces. A penalized least-squares estimator (PLSE) is built on a data driven selected model among this collection. We state non asymptotic risk bounds for this PLSE and give several examples where the procedure can be applied (autoregression, regression with arithmetically $\beta$-mixing design points, regression with mixing errors, estimation in additive frameworks, estimation of the order of the autoregression). In addition we show that under a weak moment condition on the errors, our estimator is adaptive in the minimax sense simultaneously over some family of Besov balls.

Work due to Junker and more recently due to Junker and Ellis characterized desired latent properties of an educational testing procedure in terms of a collection of other manifest properties. This is important because one can only propose tests for manifest quantities, not latent ones. Here, we complete the conversion of a pair of latent properties to equivalent conditions in terms of four manifest quantities and identify a general method for producing tests for manifest properties.

We study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations. Several applications to finite population statistics are given:we establish one-term Edgeworth expansions for general asymptotically normal symmetric statistics, prove an Efron-Stein inequality and the consistency of the jackknife esti- mator of variance. Our expansions provide second order a.s. approximations to Wu’s jackknife histogram.