Jaromír Antoch

Effect of dependence on statistics for determination of change

Abstract Quite a number of test statistics and estimators for detection of a change in the mean of a series of independent observations were proposed and studied. The purpose of this paper is to examine the behaviour of these statistics if the observations are dependent, particularly, if they form a linear process.

Change-point problem and bootstrap

We consider a class of simple estimators of the change-point m in a sequence of n independent random variables X1…,X n satisfying E(X i ) = θ0 for i = 1,…,m and E(X i ) = θ0+δ n for i = m +1,…n. (θ0 and δ n are unknown). We obtain rates of consistency for the estimator, derive its limiting distribution and show that the bootstrap approximation is asymptotically valid. The results are illustrated by some simulations.

Permutation tests in change point analysis

The critical values for various tests for changes in location model are obtained through the use of permutation tests principle. Theoretical results show that in the limit these new permutation tests behave in the same way as the classical tests stemming from both maximum likelihood and Bayes principles. However, the results of the simulation study show that the permutation tests behave considerably better than the corresponding classical tests if measured by the critical values attained.

On the possibilistic approach to linear regression models involving uncertain, indeterminate or interval data

Abstract We consider linear regression models where both input data (the observations of independent variables) and output data (the observations of the dependent variable) are affected by loss of information caused by uncertainty, indeterminacy, rounding or censoring. Instead of real-valued (crisp) data, only intervals are available. We study a possibilistic generalization of the least squares estimator, so called OLS-set for the interval model. Investigation of the OLS-set allows us to quantify whether the replacement of real-valued (crisp) data by interval values can have a significant impact on our knowledge of the value of the OLS estimator. We show that in the general case, very elementary questions about properties of the OLS-set are computationally intractable (assuming Pxa0≠xa0NP). We also focus on restricted versions of the general interval linear regression model to the crisp input case. Taking the advantage of the fact that in the crisp input – interval output model the OLS-set is a zonotope, we design both exact and approximate methods for its description. We also discuss special cases of the regression model, e.g. a model with repeated observations.

Functional Linear Regression with Functional Response: Application to Prediction of Electricity Consumption

Functional linear regression model linking observations of a functional response variable with measurements of an explanatory functional variable is considered. The slope function is estimated with a tensor product splines. Some computational issues are addressed by means of a simulation study. This model serves to analyze a real data set concerning electricity consumption in Sardinia. The interest lies in predicting either incoming weekend or incoming weekdays consumption curves if actual weekdays consumption is known.

Electricity consumption prediction with functional linear regression using spline estimators

A functional linear regression model linking observations of a functional response variable with measurements of an explanatory functional variable is considered. This model serves to analyse a real data set describing electricity consumption in Sardinia. The interest lies in predicting either oncoming weekends’ or oncoming weekdays’ consumption, provided actual weekdays’ consumption is known. A B-spline estimator of the functional parameter is used. Selected computational issues are addressed as well.

Environment for statistical computing

This paper is a short exposition on the current state of art as far as statistical software is concerned. The main aims are to take a look at current tendencies in information technologies for statistics and data analysis, especially for describing selected programs and systems. We start with statistical packages, i.e. a suite of computer programs that are specialized in statistical analysis, to enable people to obtain the results of standard statistical procedures without requiring low-level numerical programming, and to provide facilities of data management. A big surprise for many statisticians is that the most typical representative in this domain is Microsoft Excel. Aside from that, we touch upon a few commercial packages, a few general public license packages, and a few analysis packages with statistics add-ons. An integrated environment for statistical computing and graphics is essential for developing and understanding new techniques in statistics. Such an environment must essentially be a programming language. Therefore, we take a closer look at several typical representatives of these types of programmes, and on a few general purpose languages with statistics libraries. However, there exists quite a clear distinction between practical and theoretical approaches to most statistical work. The majority of software products for statistics are on the practical side, using numerical and graphical methods to provide the user access to existing methods. On the other hand, software packages specifically designed just for pure statistical-mathematical modelling do not exist. Nevertheless, all available computer algebra and/or mathematical systems offer tools for theoretical statistical work. Therefore, we take a look at some possibilities in this area. Finally, we summarize several major driving forces that will influence, according to our strong belief, the statistical software development process in the near future. Due to limited space, these discussions are cursory in nature for the most part. This paper is based on the personal experience of the author as described in [J. Antoch, Series of papers on statistical software and environments for statistical computing (in Czech for the Czech Statistical Society Newsletter and other publications). [1]] and on the information available on Internet. Very good and interesting source of information is especially Google search machine [Google search machine. [12]], Wikipedia [Wikipedia, a multilingual web-based, free content encyclopedia project. [25]] and the journal Scientific Computing World [Scientific Computing World Journal. [22]].

Off-Line Statistical Process Control

First part of this paper deals with tests on the stability of statistical models. The problem is formulated in terms of testing the null hypothesis H against the alternative hypothesis A. The null hypothesis H claims that the model remains the same during the whole observational period, usually it means that the parameters of the model do not change. The alternative hypothesis A claims that, at an unknown time point, the model changes, which means that some of the parameters of the model are subject to a change. In case we reject the null hypothesis H, i.e. when we decide that there is a change in the model, we concentrate on a number of questions that arise: n n nwhen has the model changed; n n nis there just one change or are there more changes; n n nwhat is the total number of changes etc.

Nonparametric regression M-quantiles

For the regression model Yi = m(xi)+[var epsilon]i, i = 1,...,n, robust nonparametric estimators are introduced and studied in Hardle and Gasser (1984). We show that these estimators can be viewed as regression M-quantiles. We then establish a probability inequality and a Bahadur representation for such quantiles and discuss some applications.

Statistical Analysis of Mitochondrial Pathologies in Childhood: Identification of Deficiencies using Principal Component Analysis

Mitochondrial pathologies are a heterogeneous group of metabolic disorders that are frequently characterized by anomalies of oxidative phosphorylation, especially in the respiratory chain. The identification of these anomalies may involve many investigations, and biochemistry is a main tool. However, considering the whole set of biochemical data, the interpretation of the results by the traditionally used statistical methods remains complex and does not always lead to an unequivocal conclusion about the presence or absence of a respiratory chain defect. This arises from three main problems: (a) the absence of an a priori-defined control population, because the determination of the control values are derived from the whole set of investigated patients, (b) the small size of the population studied, (c) the large number of variables collected, each of which creates a wide variability. To cope with these problems, the principal component analysis (PCA) has been applied to the biochemical data obtained from 35 muscle biopsies of children suspected of having a mitochondrial disease. This analysis makes it possible for each respiratory chain complex to distinguish between different subsets within the whole population (normal, deficient, and, in between, borderline subgroups of patients) and to detect the most discriminating variables. PCA of the data of all complexes together showed that mitochondrial diseases in this population were mainly caused by multiple deficits in respiratory chain complexes. This analysis allows the definition of a new subgroup of newborns, which have high respiratory chain complex activity values. Our results show that the PCA method, which simultaneously takes into account all of the concerned variables, allows the separation of patients into subgroups, which may help clinicians make their diagnoses.