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## Model-based clustering of probability density functions

### Advances in Data Analysis and Classification (2013-06-27): 1-19 , June 27, 2013

Complex data such as those where each statistical unit under study is described not by a single observation (or vector variable), but by a unit-specific sample of several or even many observations, are becoming more and more popular. Reducing these sample data by summary statistics, like the average or the median, implies that most inherent information (about variability, skewness or multi-modality) gets lost. Full information is preserved only if each unit is described by a whole distribution. This new kind of data, a.k.a. “distribution-valued data”, require the development of adequate statistical methods. This paper presents a method to group a set of probability density functions (pdfs) into homogeneous clusters, provided that the pdfs have to be estimated nonparametrically from the unit-specific data. Since elements belonging to the same cluster are naturally thought of as samples from the same probability model, the idea is to tackle the clustering problem by defining and estimating a proper mixture model on the space of pdfs. The issue of model building is challenging here because of the infinite-dimensionality and the non-Euclidean geometry of the domain space. By adopting a wavelet-based representation for the elements in the space, the task is accomplished by using mixture models for hyper-spherical data. The proposed solution is illustrated through a simulation experiment and on two real data sets.

## Time and Spatial Series

### Large Sample Techniques for Statistics (2010-01-01) 0: 283-315 , January 01, 2010

Time series occur naturally in a wide range of practices. For example, the opening price of a certain stock at the New York Stock Exchange, the monthly rainfall total of a certain region, and the CD4+ cell count over time of an individual infected with the HIV virus may all be viewed as time series.

## Guides

### The Grammar of Graphics (1999-01-01): 329-338 , January 01, 1999

Guides show us the way. They give us absolute directions in a relative world. We need guides in graphics to help us associate categories or quantities with aesthetics. Position, for example, allows us to compare magnitudes among graphics in a frame, but we need a positional guide to help us to relate magnitudes to non-graphical information. Color can help us discriminate categories, but we cannot associate colors with specific categories without a color guide.

## Expectation

### All of Statistics (2004-01-01): 48-61 , January 01, 2004

The mean, or expectation, of a random variable *X* is the average value of *X*.

## An asymptotic test for a geometric process against a lattice distribution with monotone hazard

### Journal of the Italian Statistical Society (1997-12-01) 6: 213-231 , December 01, 1997

### Summary

In this paper a simple characterization of the geometric distribution, in the class of discrete distributions with monotone hazard ratio, is provided. This result is used to construct a test for the hypothesis that the anival process of a discrete queueing model is a geometric process. The properties of the test, as well as those of its «bootstrapped version », are studied both theoretically and by Monte Carlo simulation.

## Economic and Financial Modeling with Mathematica®

### Economic and Financial Modeling with Mathematica® (1993-01-01) , January 01, 1993

## Projection pursuit approach to robust canonical correlation analysis

### COMPSTAT (2000-01-01): 415-420 , January 01, 2000

Projection pursuit techniques are used to build new robust estimators for the parameters of the canonical correlation model. A simulation study shows that for non-ideal data these estimators can perform as well as other robust estimators. However, they can have much higher breakdown points. This advantage makes these estimators the right choice for use with real data, where potential outlying observations are very frequent.

## Social Choice

### Mathematics and Politics (2008-01-01): 1-48 , January 01, 2008

In the present chapter we consider the situation wherein a group of voters is collectively trying to choose among several alternatives. When people speak of the area of “social choice,” it is typically this context that they have in mind.

In the case where there are only two alternatives, the standard democratic process is to let each person vote for his or her preferred alternative, with the social choice (the “winner”) being the alternative receiving the most votes. The situation, however, becomes complicated if there are more than two alternatives. In particular, if we proceed exactly as we did above where we had two alternatives, then we are not taking advantage of some individual comparisons among the several alternatives that could be made.

## Within-Schätzung bei anonymisierten Paneldaten

### AStA Wirtschafts- und Sozialstatistisches Archiv (2008-10-02) 2: 277-297 , October 02, 2008

### Zusammenfassung

Dieser Beitrag untersucht die Auswirkungen der beiden datenverändernden Anonymisierungsverfahren ‚variablenspezifische abstandsorientierte Mikroaggregation‘ (vaabMA, im Englischen: Individual Ranking) und ‚multiplikative stochastische Überlagerung‘ auf die „Within“-Schätzung eines linearen Panelmodells mit Individualeffekten. Es wird gezeigt, dass der „Within“-Schätzer auf der Grundlage der mittels vaabMA anonymisierten Daten konsistent bleibt. Bei der multiplikativen stochastischen Überlagerung wird neben der allgemeinen Form der Überlagerung eine spezielle Variante analysiert, bei der die Variablen zuerst mit einem konstantem Grundüberlagerungsfaktor und danach zusätzlich additiv überlagert werden. Es wird weiterhin gezeigt, dass beide Varianten der Überlagerung zu Inkonsistenz der „Within“-Schätzer führen. Anschließend werden korrigierte Schätzer hergeleitet.