Statistical Paradigms

Recent Advances and Reconciliations

Ashis SenGupta, Tapas Samanta, Ayanendranath Basu

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814644112
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This volume consists of a collection of research articles on classical and emerging Statistical Paradigms — parametric, non-parametric and semi-parametric, frequentist and Bayesian — encompassing both theoretical advances and emerging applications in a variety of scientific disciplines. For advances in theory, the topics include: Bayesian Inference, Directional Data Analysis, Distribution Theory, Econometrics and Multiple Testing Procedures. The areas in emerging applications include: Bioinformatics, Factorial Experiments and Linear Models, Hotspot Geoinformatics and Reliability.

Contents:

• Reviews:
  • Weak Paradoxes and Paradigms (Jayanta K Ghosh)
  • Nonparametrics in Modern Interdisciplinary Research: Some Perspectives and Prospectives (Pranab K Sen)
• Parametric:
  • Bounds on Distributions Involving Partial, Marginal and Conditional Information: The Consequences of Incomplete Prior Specification (Barry C Arnold)
  • Stepdown Procedures Controlling a Generalized False Discovery Rate (Wenge Guo and Sanat K Sarkar)
  • On Confidence Intervals for Expected Response in 2 n Factorial Experiments with Exponentially Distributed Response Variables (H V Kulkarni and S C Patil)
  • Predictive Influence of Variables in a Linear Regression Model when the Moment Matrix is Singular (Md Nurul Haque Mollah and S K Bhattacharjee)
  • New Wrapped Distributions — Goodness of Fit (A V Dattatreya Rao, I Ramabhadra Sarma and S V S Girija)
• Semi-Parametric:
  • Non-Stationary Samples and Meta-Distribution (Dominique Guégan)
  • MDL Model Selection Criterion for Mixed Models with an Application to Spline Smoothing (Antti Liski and Erkki P Liski)
  • Digital Governance and Hotspot Geoinformatics with Continuous Fractional Response (G P Patil, S W Joshi and R E Koli)
  • Bayesian Curve Registration of Functional Data (Z Zhong, A Majumdar and R L Eubank)
• Non-Parametric & Probability:
  • Nonparametric Estimation in a One-Way Error Component Model: A Monte Carlo Analysis (Daniel J Henderson and Aman Ullah)
  • GERT Analysis of Consecutive- k Systems: An Overview (Kanwar Sen, Manju Agarwal and Pooja Mohan)
  • Moment Bounds for Strong-Mixing Processes with Applications (Ratan Dasgupta)

Readership: Researchers, professionals and advanced students working on Bayesian and frequentist approaches to statistical modeling and on interfaces for both theory and applications.
Key Features:

• A scholarly and motivating review of non-parametric methods by P K Sen, winner of the Wilks Medal in 2010
• Discussion of paradoxes of the frequentist and Bayesian paradigms, related counterexamples, and their implications
• Stands out in terms of the width and depth

PART 3

SEMI-PARAMETRIC

Chapter 8

Non-Stationary Samples and Meta-Distribution

Dominique Guégan

University Paris 1 Panthéon-Sorbonne, France

In this chapter, we focus on the building of an invariant distribution function associated with a non-stationary sample. After discussing some specific problems encountered by non-stationarity inside samples like the “spurious” long memory effect, we build a sequence of stationary processes permitting us to define the concept of meta-distribution for a given non-stationary sample. We use this new approach to discuss some interesting econometric issues in a non-stationary setting, namely forecasting and risk management strategy.

Keywords: Copula, Non-stationarity, Risk management, SETAR processes, Switching processes.

Contents

8.1Introduction

8.2Empirical Evidence

8.2.1Non-stationary stylized facts

8.2.2Asymptotic behavior of the ACF of the (Y_t^δ)t process

8.3A Meta-Distribution Function

8.4Conclusion

8.5Appendix: Proof of Proposition 8.1

References

8.1.Introduction

During decades time series modelling has focused on stationary linear models, and later also on stationary non-linear models. Recently the question of non-stationarity has arised, and a sudden interest focuses on the modelling of non-stationary and/or non-linear time series.

An interesting type of questions has emerged when it has been observed that stationary non-linear processes exhibited empirical autocorrelation function with a hyperbolic decreasing rate, although they are characterized by a theoretical short memory behavior (exponential decrease of the autocorrelation function). This observation has highlighted the fact that the behavior of some statistical tools under non-stationarity must be questioned. Indeed, the statistical tools we are using are meaningful only under certain assumptions, the most crucial one being the stationarity. Hence, the question arises what the statistical tools are telling us when used on non-stationary data.

Thus, defining a correct framework in which we can analyse data sets is fundamental before choosing the class of models we will use. Following this idea, we develop here a framework permitting us to work in a stationary setting including the structural non-stationarities. We assume that a process (Y_t)_t is characterized by changes in the k-th order moments all along the information set (corresponding in practice to the observed trajectory). This corresponds to structural changes in financial time series causing the time series over long intervals to deviate significantly from stationarity. This means that we assume that non-global stationarity is verified for the sample. Then it is plausible that by relaxing the assumptions of stationarity in an adequate way, we may obtain better fit followed by robust forecasts and management theory. Doing that, we will see that we can get new insight to approximate the unknown distribution function for complex non-stationary processes.

Without using the whole sample, which is the source of non-stationarity, we define a new way to analyse and model this information set dividing it in subsamples on which stationarity is achieved. In this paper, our objectives are twofold. First, we show that the non-stationarities observed on the empirical moments pollute the theoretical properties of the statistics defined inside a “global” stationary framework, and thus a new framework needs to be developed. Second we propose a new way to study finite sample data sets in presence of k-order non-stationarity.

As the non-stationarity affects nearly all the moments of a time series when it is present, we first study the impact of the non-stationarity on a non-linear transformation of the observed data set (Y_t)_t considering (Y_t^δ)t, for any δ ∈ R⁺, looking at its sample autocovariance function (ACF). We exhibit the strange behavior of this ACF in presence of non stationarity and illustrate it throu...

Table of contents

• Cover Page
• Title Page
• Copyright
• Foreword
• Preface
• List of Contributors
• Contents
• Reviews
• Parametric
• Semi-Parametric
• Non-Parametric & Probability