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Infinite Variance (infinite + variance)

Distribution by Scientific Domains
Mathematics and Statistics80%

Selected Abstracts

Modelling Long-memory Time Series with Finite or Infinite Variance: a General Approach

JOURNAL OF TIME SERIES ANALYSIS, Issue 1 2000
Remigijus Leipus
We present a class of generalized fractional filters which is stable with respect to series and parallel connection. This class extends the so-called fractional ARUMA and fractional ARMA filters previously introduced by e.g. Goncalves (1987) and Robinson (1994) and recently studied by Giraitis and Leipus (1995) and Viano et al. (1995). Conditions for the existence of the induced stationary S,S and L2 processes are given. We describe the asymptotic dependence structure of these processes via the codifference and the covariance sequences respectively. In the L2 case, we prove the weak convergence of the normalized partial sums. [source]

Nonparametric smoothing using state space techniques

THE CANADIAN JOURNAL OF STATISTICS, Issue 1 2001
Patrick E. Brown
Abstract The authors examine the equivalence between penalized least squares and state space smoothing using random vectors with infinite variance. They show that despite infinite variance, many time series techniques for estimation, significance testing, and diagnostics can be used. The Kalman filter can be used to fit penalized least squares models, computing the smoothed quantities and related values. Infinite variance is equivalent to differencing to stationarity, and to adding explanatory variables. The authors examine constructs called "smoothations" which they show to be fundamental in smoothing. Applications illustrate concepts and methods. Les auteurs examinent l'équivalence entre les moindres carrés pénalisés et le lissage de l'espace d'états au moyen de vecteurs aléatoires à variance infinie. Ils montrent que malgré le problème de variance infinie, plusieurs techniques de diagnostic, d'estimation et de test de signification propres aux chroniques restent valables. Le filtre de Kalman permet d'évaluer les modèles des moindres carrés pénalisés en fournissant entre autres des valeurs lissées. La variance infinie est équivalente à la différenciation à des fins de stationnarité et à l'ajout de variables explicatives. Les auteurs étudient en outre des quantités appelées "lissations," dont ils montrent l'importance pour le lissage. Des applications illustrent les méthodes et procédures décrites. [source]

The lognormal distribution is not an appropriate null hypothesis for the species,abundance distribution

JOURNAL OF ANIMAL ECOLOGY, Issue 3 2005
MARK WILLIAMSON
Summary 1Of the many models for species,abundance distributions (SADs), the lognormal has been the most popular and has been put forward as an appropriate null model for testing against theoretical SADs. In this paper we explore a number of reasons why the lognormal is not an appropriate null model, or indeed an appropriate model of any sort, for a SAD. 2We use three empirical examples, based on published data sets, to illustrate features of SADs in general and of the lognormal in particular: the abundance of British breeding birds, the number of trees > 1 cm diameter at breast height (d.b.h.) on a 50 ha Panamanian plot, and the abundance of certain butterflies trapped at Jatun Sacha, Ecuador. The first two are complete enumerations and show left skew under logarithmic transformation, the third is an incomplete enumeration and shows right skew. 3Fitting SADs by ,2 test is less efficient and less informative than fitting probability plots. The left skewness of complete enumerations seems to arise from a lack of extremely abundant species rather than from a surplus of rare ones. One consequence is that the logit-normal, which stretches the right-hand end of the distribution, consistently gives a slightly better fit. 4The central limit theorem predicts lognormality of abundances within species but not between them, and so is not a basis for the lognormal SAD. Niche breakage and population dynamical models can predict a lognormal SAD but equally can predict many other SADs. 5The lognormal sits uncomfortably between distributions with infinite variance and the log-binomial. The latter removes the absurdity of the invisible highly abundant half of the individuals abundance curve predicted by the lognormal SAD. The veil line is a misunderstanding of the sampling properties of the SAD and fitting the Poisson lognormal is not satisfactory. A satisfactory SAD should have a thinner right-hand tail than the lognormal, as is observed empirically. 6The SAD for logarithmic abundance cannot be Gaussian. [source]

Nonparametric smoothing using state space techniques

THE CANADIAN JOURNAL OF STATISTICS, Issue 1 2001
Patrick E. Brown
Abstract The authors examine the equivalence between penalized least squares and state space smoothing using random vectors with infinite variance. They show that despite infinite variance, many time series techniques for estimation, significance testing, and diagnostics can be used. The Kalman filter can be used to fit penalized least squares models, computing the smoothed quantities and related values. Infinite variance is equivalent to differencing to stationarity, and to adding explanatory variables. The authors examine constructs called "smoothations" which they show to be fundamental in smoothing. Applications illustrate concepts and methods. Les auteurs examinent l'équivalence entre les moindres carrés pénalisés et le lissage de l'espace d'états au moyen de vecteurs aléatoires à variance infinie. Ils montrent que malgré le problème de variance infinie, plusieurs techniques de diagnostic, d'estimation et de test de signification propres aux chroniques restent valables. Le filtre de Kalman permet d'évaluer les modèles des moindres carrés pénalisés en fournissant entre autres des valeurs lissées. La variance infinie est équivalente à la différenciation à des fins de stationnarité et à l'ajout de variables explicatives. Les auteurs étudient en outre des quantités appelées "lissations," dont ils montrent l'importance pour le lissage. Des applications illustrent les méthodes et procédures décrites. [source]

EXPONENTIAL SMOOTHING AND NON-NEGATIVE DATA

AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 4 2009
Muhammad Akram
Summary The most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non-negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non-negative data. We explore exponential smoothing state space models for non-negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models that do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non-Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short-term forecasting performance of the various models using data on the weekly sales of over 300 items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one-step-ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed. [source]