Paulo Eduardo Oliveira

Estimation of a two-dimensional distribution function under association

Abstract By considering an associated and strictly stationary sequence of random variables, say X n , n⩾1 , we study the properties of an histogram estimator for the two-dimensional distribution function of (X1,Xk+1). We find conditions on the covariance structure of the original random variables for the almost sure convergence of the estimator and for the convergence in distribution of the finite dimensional distributions. We also characterize the mean square error (MSE) and find its convergence rate, under assumptions on the convergence rate of the covariances.

Serial position effects in Alzheimer's disease, mild cognitive impairment, and normal aging: Predictive value for conversion to dementia

Serial position effects in word list learning have been used to differentiate normal aging and dementia. Prominent recency and diminished primacy have consistently been observed in Alzheimers disease (AD). We examined serial position effects in patients with mild cognitive impairment (MCI), in patients with AD, and in normal healthy controls. Additionally, we classified MCI patients into those who progressed to AD (MCI-p) and those who did not (MCI-np). We compared two serial position measures: regional and standard scores. Regional scores, mainly the primacy effect, improved discrimination between MCI and controls and between MCI-np and MCI-p, proving to be more sensitive and specific than the recency effect.

A General Approach To Nonparametric Histogram Estimation

We note that some classical functional estimation problems may be reduced to a general unique framework and study an estimator within this general framework that reduces to the classical histogram type estimators in various examples presented. The convergence in probability and the almost complete convergence of this general estimator are studied obtaining convergence conditions which reduce to the classical conditions in each case. Finally, this general framework provides conditions for the convergence of the finite dimensional distributions of the associated empirical process.

Estimation of age at death from the pubic symphysis and the auricular surface of the ilium using a smoothing procedure

We discuss here the estimation of age at death from two indicators (pubic symphysis and the sacro-pelvic surface of the ilium) based on four different osteological series from Portugal, Great-Britain, South Africa or USA (European origin). These samples and the scoring system of the two indicators were used by Schmitt et al. (2002), applying the methodology proposed by Lucy et al. (1996). In the present work, the same data was processed using a modification of the empirical method proposed by Lucy et al. (2002). The various probability distributions are estimated from training data by using kernel density procedures and Jackknife methodology. Bayess theorem is then used to produce the posterior distribution from which point and interval estimates may be made. This statistical approach reduces the bias of the estimates to less than 70% of what was obtained by the initial method. This reduction going up to 52% if knowledge of sex of the individual is available, and produces an age for all the individuals that improves age at death assessment.

Histograms and Associated Point Processes

Nonparametric inference for point processes is discussed by way histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depends on a sequence of partitions, which we take to be nonembedded. This is quite natural in what regards applications, but presents some theoretical problems. In another direction, we drop the usual independence assumption on the sample, replacing it by an association assumption. Under this setting, we study the convergence of the histogram, in probability and almost surely which, under association, depends on conditions on the covariance structure. In the final section we prove that the finite dimensional distributions converge in distribution to a Gaussian centered vector with a specified covariance. The main tool of analysis is a decomposition of second order moment measures.

Relative smoothing of discrete distributions with sparse observations

Quite often we are faced with a sparse number of observations over a finite number of cells and are interested in estimating the cell probabilities. Some local polynomial smoothers or local likelihood estimators have been proposed to improve on the histogram, which would produce too many zero values. We propose a relativized local polynomial smoothing for this problem, weighting heavier the estimating errors in small probability cells. A simulation study about the estimators that are proposed show a good behaviour with respect to natural error criteria, especially when dealing with sparse observations.

Density estimation for associated sampling: a point process influenced approach

Let X_n , n\in {\open N} , be a sequence of associated variables with common density function. We study the kernel estimation of this density, based on the given sequence of variables. Sufficient conditions are given for the consistency and asymptotic normality of the kernel estimator. The assumptions made require that the distribution of pairs (X_i, X_j) decompose as the sum of an absolutely continuous measure with another measure concentrated on the diagonal of {\open R}\times {\open R} satisfying a further absolute continuity with respect to the Lebesgue measure on this diagonal. For the convergence in probability we find the usual convergence rate on the bandwidth, whereas for the almost sure convergence we need to require that the bandwidth does not decrease to fast and that the kernel is of bounded variation. This assumption on the kernel is also required for the asymptotic normality, together with a slightly strengthened version of the usual decrease rate on the bandwidth. The assumption of bounded variation on the kernel is needed as a consequence of the dependence structure we are dealing with, as association is only preserved by monotone transformations.

Recent Developments in Modeling and Applications in Statistics

Sampling and Modeling.- Estimation.- Extremes.- Testing Statistical Hypothesis.- Models with Stochastic Differential Equations.- Stochastic Processes.

Convergence rates for the estimation of two-dimensional distribution functions under association and estimation of the covariance of the limit empirical process

Let X n , n≥1, be an associated and strictly stationary sequence of random variables, having marginal distribution function F. The limit in distribution of the empirical process, when it exists, is a centred Gaussian process with covariance function depending on terms of the form ϕ k (s, t)=P(X 1 s, X k+1 t)−F(s)F(t). We prove the almost sure consistency for the histogram to estimate each ϕ k and also to estimate the covariance function of the limit empirical process, identifying, for both, uniform almost sure convergence rates. The convergence rates depend on a suitable version of an exponential inequality. The rates obtained, assuming the covariances to decrease geometrically, are of order n −1/3log2/3 n for the estimator of ϕ k and of order n −1/3log5/3 n for the estimator of the covariance function.

Histogram estimation of radon-nikodym derivatives for strong mixing data

Nonparametric inference for point processes is discussed by way of histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depends on a sequence of partitions, which we take tc be nonenibedded to allow partitions with sets of equal measure. This presents some theoretical problems, which are addressed with an assumption on the decomposition of second order moments. In another direction, we drop the usual independence assumption on the sample, replacing it by a strong mixing assumption. Under this setting, we study the convergence of the histogram in probability, which depends on approximation conditions between the distributions of random pairs and the product of their marginal distributions, and^almost completely, which is based on the decomposition of the second order moments. This last convergence is stated on two versions according to the assumption of Laplace transforms or the Cramer moment conditions. These are somewhat stronger, but enable us to recover the usual condition on the decrease rate of sets on each partition. In the final section we prove that the finite dimensional distributions converge in distribution to a Gaussian centered vector with a specified covariance.