Positron emission tomography (PET) is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo. A number of biological and physical factors limit the resolution of PET so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study. To a first approximation, reconstruction involves the solution of a linear inverse problem with a line-integral Radon-type transform. Standard…
If an experimenter wants to determine the degree of a polynomial regression on the basis of a sample of observations, Anderson showed that the following method is optimal. Starting with the highest (specified) degree the procedure is to test in sequence whether the coefficients are 0. In this paper optimal designs for Anderson's procedure are determined explicitly. The optimal design maximizes the minimum power of a given set of alternatives.
An asymptotic expansion is provided for the transformation kernel density estimator introduced by Ruppert and Cline. Let $h_k$ be the band-width used in the $k$th iteration, $k = 1,2,\ldots, t$. If all bandwidths are of the same order, the leading bias term of the $l$th derivative of the $t$th iterate of the density estimator has the form $\bar{b}^{(l)}_t(x) \pi^t_{k=1} h^2_k$, where the bias factor $\bar{b}_t(x)$ depends on the second moment of the kernel $K$, as well as on…
A smoothing spline is a nonparametric curve estimate that is defined as the solution to a minimization problem. One problem with this representation is that it obscures the fact that a spline, like most other nonparametric estimates, is a local, weighted average of the observed data. This property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates. Although equivalent kernels…
A geometric combination estimator is proposed for $d$-dimensional ordinal contingency tables. The proposed estimator is nonnegative. It is shown that, assuming sufficient smoothness and boundary conditions for the underlying probabilities, the rate of convergence of mean summed squared error (MSSE) of this estimator is $O(K^{-1}N^{-8/(d+8)})$ for $d$-dimensional tables $(d \leq 4)$ with $K$ cells and sample size $N$. This rate is optimal under the smoothness assumptions, and…
Prediction of categorical responses in panel studies is considered. Prediction functions based on general conditional $\log$-linear models are investigated for statistical properties both from a population perspective and a sampling perspective. Problems such as existence and uniqueness of optimal prediction functions are addressed, and basic properties of measures of prediction quality are examined. Estimation, consistency and asymptotic normality are studied for the…
We use the method of maximum likelihood and regression splines to derive estimates of the parametric and nonparametric components of semiparametric generalized linear models. The resulting estimators of both components are shown to be consistent. Also, the asymptotic theory for the estimator of the parametric component is derived, indicating that the parametric component can be estimated efficiently without under-smoothing the nonparametric component.
Finite-state Markov chains with either a discrete or continuous time parameter, Markov renewal processes and Markov-additive processes are considered. We prove that their likelihood functions, in the nonsequential as well as in various sequential cases, belong to special $(n + k, n)$-curved exponential families in general, for which limit results are easily established. Subsequently, asymptotic normality of the corresponding nonsequential and sequential maximum likelihood…
This paper discusses the estimation of multiple time series models which allow elements of the spectral density matrix to tend to infinity or zero at zero frequency and be unrestricted elsewhere. A form of log-periodogram regression estimate of differencing and scale parameters is proposed, which can provide modest efficiency improvements over a previously proposed method (for which no satisfactory theoretical justification seems previously available) and further…
In this paper we construct an efficient weighted least squares estimator for the mean and more generally for the regression parameters in certain Gaussian long range dependent regression models, including polynomial regression. The form of the estimator does not depend on the whole dependence structure of the residuals, but only on the local behaviour of the spectral density at zero. By using an estimator of the self-similarity parameter, we give a fully efficient estimator.…
This paper addresses the problem of testing the hypothesis that an observed series is difference stationary. The alternative hypothesis is that the series is another nonstationary process; in particular, an autoregressive model with a random parameter is used. A locally best invariant test is developed assuming Gaussianity, and a representation of its asymptotic distribution as a mixture of Brownian motions is found. The performance of the test in finite samples is…
We consider the fixed-design regression model with long-range dependent normal errors and show that the finite-dimensional distributions of the properly normalized Gasser-Muller and Priestley-Chao estimators of the regression function converge to those of a white noise process. Furthermore, the distributions of the suitably renormalized maximal deviations over an increasingly finer grid converge to the Gumbel distribution. These results contrast with our previous findings…
Dehling and Taqqu established the weak convergence of the empirical process for a long-range dependent stationary sequence under Gaussian subordination. We show that the corresponding density process, based on kernel estimators of the marginal density, converges weakly with the same normalization to the derivative of their limiting process. The phenomenon, which carries on for higher derivatives and for functional laws of the iterated logarithm, is in contrast with…
Bounded influence functions are used for robust estimation in semiparametric models. In this paper, we generalize Hampel's variational problem to semiparametric models and define the optimal B-robust influence function as the one solving the variational problem. We identify the lowest bounds for influence functions and establish the existence and uniqueness of the optimal influence functions in general semiparametric models. Explicit optimal influence functions are given for…
We propose a general semiparametric variance function model in a fixed design regression setting. In this model, the regression function is assumed to be smooth and is modelled nonparametrically, whereas the relation between the variance and the mean regression function is assumed to follow a generalized linear model. Almost all variance function models that were considered in the literature emerge as special cases. Least-squares-types estimates for the parameters of this…
This paper concerns estimating a mixing density function $g$ and its derivatives based on iid observations from $f(x) = \int f(x \mid \theta)g(\theta)d\theta$, where $f(x \mid \theta)$ is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If $f(x \mid \theta_0) \geq \varepsilon^{x + 1} \forall x$…
The traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example, the normal, while not losing much in precision when the true density is…
Consider a statistical procedure (Method A) which is based on $n$ observations and a less effective procedure (Method B) which requires a larger number $k_n$ of observations to give equal performance under a certain criterion. To compare two different procedures, Hodges and Lehmann suggested that the difference $k_n - n$, called the deficiency of Method B with respect to Method A, is the most natural quantity to examine. In this article, the performance of two kernel…
It is shown in infinite-dimensional Gaussian problems that affine estimators minimax the variance among all estimators of a linear functional subject to a constraint on the bias. Likewise, affine estimators also minimax the square of the bias among all estimates of a linear functional subject to a constraint on the variance.
Most hazard regression models in survival analysis specify a given functional form to describe the influence of the covariates on the hazard rate. For instance, Cox's model assumes that the covariates act multiplicatively on the hazard rate, and Aalen's additive risk model stipulates that the covariates have a linear additive effect on the hazard rate. In this paper we study a fully nonparametric model which makes no assumption on the association between the hazard rate and…
The common unknown probability law $P$ of a random sample $Y_1,\ldots, Y_n$ is assigned a Dirichlet process prior with index $\alpha$. It is shown that the posterior joint density of several moments of $P$ converges, as $\alpha(\mathbb{R})\rightarrow 0$, to a multivariate B-spline, which is, therefore, the Bayesian bootstrap joint density of the moments. The result provides the basis for possible default nonparametric Bayesian inference on unknown moments.
This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for $K$ location parameters, with $J$ observations each, the number of iterations required for convergence (for large $K$ and $J$) is a constant times $(1 + \log K/\log J)$. This is one of the first rigorous, a priori results about time to convergence for the Gibbs sampler. A quantitative…
In this paper, the problem of computing the exact value of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise is considered. A method based on constructing difference equations for the appropriate moments is presented and used to show that the exact variance of the Pitman estimator is $16\zeta(3)$, where $\zeta$ is the Riemann zeta function.
The paper develops a one-step triangular array Edgeworth expansion for multivariate martingales that are, essentially, asymptotically ergodic. Both discrete and continuous time are covered. The expansion is in a test function topology. We investigate when the expansion has the usual Edgeworth form, looking in particular at likelihood inference, including Cox regression, and at inference for stationary time series. The triangular array nature of the results make them useful…
An asymptotic expansion is provided for the transformation kernel density estimator introduced by Ruppert and Cline. Let hk be the band-width used in the kth iteration, k=1,2,…,t. If all bandwidths are of the same order, the leading bias term of the lth derivative of the tth iterate of the density estimator has the form bˉt(l)(x)πk=1thk2, where the bias factor bˉt(x) depends on the second moment of the kernel K, as well as on…
A Geometric Combination Estimator for d-Dimensional Ordinal Sparse Contingency Tables
A geometric combination estimator is proposed for d-dimensional ordinal contingency tables. The proposed estimator is nonnegative. It is shown that, assuming sufficient smoothness and boundary conditions for the underlying probabilities, the rate of convergence of mean summed squared error (MSSE) of this estimator is O(K−1N−8/(d+8)) for d-dimensional tables (d≤4) with K cells and sample size N. This rate is optimal under the smoothness assumptions, and…
Prediction of categorical responses in panel studies is considered. Prediction functions based on general conditional log-linear models are investigated for statistical properties both from a population perspective and a sampling perspective. Problems such as existence and uniqueness of optimal prediction functions are addressed, and basic properties of measures of prediction quality are examined. Estimation, consistency and asymptotic normality are studied for the…
Finite-state Markov chains with either a discrete or continuous time parameter, Markov renewal processes and Markov-additive processes are considered. We prove that their likelihood functions, in the nonsequential as well as in various sequential cases, belong to special (n+k,n)-curved exponential families in general, for which limit results are easily established. Subsequently, asymptotic normality of the corresponding nonsequential and sequential maximum likelihood…
This paper concerns estimating a mixing density function g and its derivatives based on iid observations from f(x)=∫f(x∣θ)g(θ)dθ, where f(x∣θ) is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If f(x∣θ0)≥εx+1∀x…
Consider a statistical procedure (Method A) which is based on n observations and a less effective procedure (Method B) which requires a larger number kn of observations to give equal performance under a certain criterion. To compare two different procedures, Hodges and Lehmann suggested that the difference kn−n, called the deficiency of Method B with respect to Method A, is the most natural quantity to examine. In this article, the performance of two kernel…
The common unknown probability law P of a random sample Y1,…,Yn is assigned a Dirichlet process prior with index α. It is shown that the posterior joint density of several moments of P converges, as α(R)→0, to a multivariate B-spline, which is, therefore, the Bayesian bootstrap joint density of the moments. The result provides the basis for possible default nonparametric Bayesian inference on unknown moments.
This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for K location parameters, with J observations each, the number of iterations required for convergence (for large K and J) is a constant times (1+logK/logJ). This is one of the first rigorous, a priori results about time to convergence for the Gibbs sampler. A quantitative…
In this paper, the problem of computing the exact value of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise is considered. A method based on constructing difference equations for the appropriate moments is presented and used to show that the exact variance of the Pitman estimator is 16ζ(3), where ζ is the Riemann zeta function.