arXiv subjects

Uwe Saint-Mont

Publications and source records attributed to Uwe Saint-Mont.

5 recordsLinked to original sources

arXiv2021

Toward a deeper understanding of a basic cascade

Towards the end of the last century, B. Mandelbrot saw the importance, revealed the beauty, and robustly promoted (multi-)fractals. Multiplicative cascades are closely related and provide simple models for the study of turbulence and chaos. For pedagogical reasons, but also due to technical difficulties, continuous stochastic models have been favoured over discrete cascades. Particularly important are the $α$ and the $p$ model. It is the aim of this contribution to introduce original concepts that shed new light on a variant of the latter paradigmatic cascade and allow key features to be derived in a rather elementary fashion. To this end, we introduce and study a discrete version of the $p$ model which is based on a new kind of sampling. Technical machinery can be kept simple, therefore proofs are straightforward and formulas are explicit. It is hoped that the proposed line of investigation may enhance understanding and simplify received multifractal analyses.

math.ST↗

arXiv2018

On the Logic (plus some history and philosophy) of Statistical Tests and Scientific Investigation

Every scientific endeavour consists of (at least) two components: A hypothesis on the one hand and data on the other. There is always a more or less abstract level - some theory, a set of concepts, certain relations of ideas - and a concrete level, i.e., empirical evidence, experiments or some observations which constitute matters of fact. The focus of this contribution is on elementary models connecting both levels that have been very popular in psychological research - statistical tests. Going from simple to complex we will examine four paradigms of statistical testing (Fisher, Likelihood, Bayes, Neyman & Pearson) and an elegant contemporary treatment. In a nutshell, testing is an easy problem that has a straightforward mathematical solution. However, it is rather surprising that the statistical mainstream has pursued a different line of argument. The application of the latter theory in psychology and other fields has brought some progress but has also impaired scientific thinking.

stat.AP↗

arXiv2017

Beyond the law of large numbers: Introducing progressive sampling, weaving, the geometric triangle, and corresponding distributions

In probability theory and statistics, the IID model represents a single population, and a large, potentially infinite sample from this population. Main theorems, in particular the central limit theorem and laws of large number (LLN) assure convergence, making asymptotic statistics possible. To avoid convergence, it is thus straightforward to consider two populations and a sample that ceaselessly fluctuates between them. It is the aim of this contribution to study the effects that thus occur. To this end, we introduce "progressive sampling," leading to a straightforward model that is analytically tractable. With a minimum of technical overhead, a number of interesting results thus ensue: In particular, one encounters a multiplicate structure (similar to Pascal's triangle) that is associated with a new class of distributions (related to the binomial). Although the argument is completely probabilistic, it entails a well-known fractal structure. It also turns out that the new (global) operation of "weaving" is equivalent to a certain (local) cascade process.

math.ST↗

arXiv2014

Randomization does not help much, comparability does

Following Fisher, it is widely believed that randomization "relieves the experimenter from the anxiety of considering innumerable causes by which the data may be disturbed." In particular, it is said to control for known and unknown nuisance factors that may considerably challenge the validity of a result. Looking for quantitative advice, we study a number of straightforward, mathematically simple models. However, they all demonstrate that the optimism with respect to randomization is wishful thinking rather than based on fact. In small to medium-sized samples, random allocation of units to treatments typically yields a considerable imbalance between the groups, i.e., confounding due to randomization is the rule rather than the exception. In the second part of this contribution, we extend the reasoning to a number of traditional arguments for and against randomization. This discussion is rather non-technical, and at times even "foundational" (Frequentist vs. Bayesian). However, its result turns out to be quite similar. While randomization's contribution remains questionable, comparability contributes much to a compelling conclusion. Summing up, classical experimentation based on sound background theory and the systematic construction of exchangeable groups seems to be advisable.

stat.ME↗

arXiv2014

Comparing Different Information Levels

Given a sequence of random variables ${\bf X}=X_1,X_2,\ldots$ suppose the aim is to maximize one's return by picking a `favorable' $X_i$. Obviously, the expected payoff crucially depends on the information at hand. An optimally informed person knows all the values $X_i=x_i$ and thus receives $E (\sup X_i)$. We will compare this return to the expected payoffs of a number of observers having less information, in particular $\sup_i (EX_i)$, the value of the sequence to a person who only knows the first moments of the random variables. In general, there is a stochastic environment (i.e. a class of random variables $\cal C$), and several levels of information. Given some ${\bf X} \in {\cal C}$, an observer possessing information $j$ obtains $r_j({\bf X})$. We are going to study `information sets' of the form $$ R_{\cal C}^{j,k} = \{ (x,y) | x = r_j({\bf X}), y=r_k({\bf X}), {\bf X} \in {\cal C} \}, $$ characterizing the advantage of $k$ relative to $j$. Since such a set measures the additional payoff by virtue of increased information, its analysis yields a number of interesting results, in particular `prophet-type' inequalities.

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