Ulla Holst

Local polynomial variance-function estimation

The conditional variance function in a heteroscedastic, nonparametric regression model is estimated by linear smoothing of squared residuals. Attention is focused on local polynomial smoothers. Both the mean and variance functions are assumed to be smooth, but neither is assumed to be in a parametric family. The biasing effect of preliminary estimation of the mean is studied, and a degrees-of-freedom correction of bias is proposed. The corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known. A proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions. The proposal is illustrated with data from the LIDAR method of measuring atmospheric pollutants and from turbulence-model computations.

Locally weighted least squares kernel regression and statistical evaluation of LIDAR measurements

The LIDAR technique is an efficient tool in monitoring the distribution of atmospheric species of importance. We study the concentration of atmospheric atomic mercury in an Italian geothermal field and discuss the possibility of using recent results from local polynomial kernel regression theory for the evaluation of the derivative of the DIAL curve. A MISE-optimal bandwidth selector, which takes account of the heteroscedasticity in the regression is suggested. Further, we estimate the integrated amount of mercury in a certain area.

Recursive estimation in mixture models with Markov regime

A recursive algorithm is proposed for estimation of parameters in mixture models, where the observations are governed by a hidden Markov chain. The often badly conditioned information matrix is estimated, and its inverse is incorporated into the algorithm. The performance of the algorithm is studied by simulations of a symmetric normal mixture. The algorithm seems to be stable and produce approximately normally distributed estimates, provided the adaptive matrix is kept well conditioned. Some numerical examples are included. >

VALIDATION OF GREY BOX MODELS

Many methods for validation of grey box models are closely related to validation techniques for black box models, employing essentially up to second order statistics of the data and residual distributions. The available information used for validation, i.e. data, structure, parameters, residuals etc. is also the same. However, the advantage of grey box models is the prior information about the structure and the parameters. This implies a special demand on the statistical tests, which have to be evaluated in the domain of the claimed information. Furthermore due to the usually nonlinear character of the basic knowledge, the inference basis for the validation must include not only the first and second order moments of the data and residual distributions but higher order moments and eventually the total distributions as well. Statistical validation methods can be performed either in a Bayesian or non Bayesian framework. Both of these validation environments are considered.

Recursive estimators for stationary, strong mixing processes—a representation theorem and asymptotic distributions

Many generalizations of the Robbins-Monro process have been proposed for the purpose of recursive estimation. In this paper it is shown that the recursive estimates can be represented as sums of possibly dependent random variables and can therefore be studied using limit theorems for sums. One application which is particularly studied is recursive M-estimators of location and scale for dependent strong mixing sequences.

Methods for recursive robust estimation of AR parameters

Abstract In many technical applications, such as automatic control or supervision of systems, on-line predictions are required. Since the system of interest might change as time is passing, the model used for predictions must follow these changes. The estimation method therefore has to be adaptive implying the use of a recursive estimation algorithm. Furthermore, because of the possibility of outliers among the observations, it must be required that the applied estimation algorithm minimizes the influence of any sort of outliers. In this paper two recursive robust estimation algorithms for estimation of AR models are derived. One of them implements a recursive minimization of a criterion function, in which prediction errors enter through the weight function proposed by Huber (1964). The other algorithm is a recursive version of the bounded-influence estimator proposed by Krasker and Welsch (1982). This estimator is an extension of the Huber estimator where a measure of the amount of aberrant information in each observation is used to down-weight the influence of observations that stand out among the rest. By these derivations a general procedure for obtaining recursive algorithms is demonstrated. In a simulation study the proposed methods are compared with ordinary recursive least squares, as well as a modification of this, in which classified outliers imply the corresponding observations to be left out of the estimation.

Recursive estimation of quantiles using recursive density estimators

Recursive estimation of quantiles may be obained via adaptive stochastic approximation approximation theorms can be used to obtained the asympotic properties when the obervation are independent. for dependent sequences matingale theory cannot be applied straight forwardly as the tool for asympototic analysis.In this paper we consider both the case when the observation are i.i.d. and when they form a stationary and strongly regular process.the main result is sufficient condition for almost sure convergence in the strongly regular case.

Estimating the Total Rate of DNA Replication Using Branching Processes

Increasing the knowledge of various cell cycle kinetic parameters, such as the length of the cell cycle and its different phases, is of considerable importance for several purposes including tumor diagnostics and treatment in clinical health care and a deepened understanding of tumor growth mechanisms. Of particular interest as a prognostic factor in different cancer forms is the S phase, during which DNA is replicated. In the present paper, we estimate the DNA replication rate and the S phase length from bromodeoxyuridine-DNA flow cytometry data. The mathematical analysis is based on a branching process model, paired with an assumed gamma distribution for the S phase duration, with which the DNA distribution of S phase cells can be expressed in terms of the DNA replication rate. Flow cytometry data typically contains rather large measurement variations, however, and we employ nonparametric deconvolution to estimate the underlying DNA distribution of S phase cells; an estimate of the DNA replication rate is then provided by this distribution and the mathematical model.

Influence of solar zenith angles on observed trends in the NOAA/NASA 8‐km Pathfinder normalized difference vegetation index over the African Sahel

The strong systematic change in solar zenith angles (SZA) due to annual orbital drift of the NOAA satellites has raised the suspicion of the influence of residual illumination on the calibrated normalized difference vegetation index (NDVI) derived from the Pathfinder AVHRR Land (PAL) database. The aim of this work is to analyse if trends in AVHRR NDVI from 1982 to 2000 over the Sahel region in Africa depend on variations in SZA. The analysis uses both ordinary least squares regression and cointegration to analyse possible linear dependencies between NDVI and SZA on a per satellite basis. Tests for integration and cointegration fail to find any significant evidence for either. This, together with the ability of simple deterministic models to explain primarily SZA constitutes evidence against integration and cointegration, indicating that linear relationships can be examined using ordinary linear regression. Regression gives no consistent relationship between NDVI and SZA and the explanatory power (R 2) of the regression is low (on average 0.08). However there is some evidence for downward bias in NDVI due to nonlinear interactions between NDVI and SZA when SZA is large (⩾80°) leading to the conclusion that PAL data from the year 2000 should not be used for analyses in these environments.

A Markov model approach shows a large variation in the length of S phase in MCF-7 breast cancer cells.

The potential doubling time of a tumor has been suggested to be a measurement of tumor aggressiveness; therefore, it is of interest to find reliable methods to estimate this time. Because of variability in length of the various cell cycle phases, stochastic modeling of the cell cycle might be a suitable approach.