Search results for "probability"
showing 10 items of 3417 documents
Adaptive Modifications of Hypotheses After an Interim Analysis
2001
It is investigated how one can modify hypotheses in a trial after an interim analysis such that the type I error rate is controlled. If only a global statement is desired, a solution was given by Bauer (1989). For a general multiple testing problem, Kieser, Bauer and Lehmacher (1999) and Bauer and Kieser (1999) gave solutions, by means of which the initial set of hypotheses can be reduced after the interim analysis. The same techniques can be applied to obtain more flexible strategies, as changing weights of hypotheses, changing an a priori order, or even including new hypotheses. It is emphasized that the application of these methods requires very careful planning of a trial as well as a c…
Dynamics of a financial market index after a crash
2002
We discuss the statistical properties of index returns in a financial market just after a major market crash. The observed non-stationary behavior of index returns is characterized in terms of the exceedances over a given threshold. This characterization is analogous to the Omori law originally observed in geophysics. By performing numerical simulations and theoretical modelling, we show that the nonlinear behavior observed in real market crashes cannot be described by a GARCH(1,1) model. We also show that the time evolution of the Value at Risk observed just after a major crash is described by a power-law function lacking a typical scale.
Volatility in Financial Markets: Stochastic Models and Empirical Results
2002
We investigate the historical volatility of the 100 most capitalized stocks traded in US equity markets. An empirical probability density function (pdf) of volatility is obtained and compared with the theoretical predictions of a lognormal model and of the Hull and White model. The lognormal model well describes the pdf in the region of low values of volatility whereas the Hull and White model better approximates the empirical pdf for large values of volatility. Both models fails in describing the empirical pdf over a moderately large volatility range.
The stabilizing effect of volatility in financial markets
2017
In financial markets, greater volatility is usually considered synonym of greater risk and instability. However, large market downturns and upturns are often preceded by long periods where price returns exhibit only small fluctuations. To investigate this surprising feature, here we propose using the mean first hitting time, i.e. the average time a stock return takes to undergo for the first time a large negative or positive variation, as an indicator of price stability, and relate this to a standard measure of volatility. In an empirical analysis of daily returns for $1071$ stocks traded in the New York Stock Exchange, we find that this measure of stability displays nonmonotonic behavior, …
Inhomogeneity and complexity measures for spatial patterns
2002
In this work, we examine two different measures for inhomogeneity and complexity that are derived from non-extensive considerations à la Tsallis. Their performance is then tested on theoretically generated patterns. All measures are found to exhibit a most sensitive behaviour for Sierpinski carpets. The procedures here introduced provide us with new, powerful Tsallis’ tools for analysing the inhomogeneity and complexity of spatial patterns.
Escape Times in Fluctuating Metastable Potential and Acceleration of Diffusion in Periodic Fluctuating Potentials
2004
The problems of escape from metastable state in randomly flipping potential and of diffusion in fast fluctuating periodic potentials are considered. For the overdamped Brownian particle moving in a piecewise linear dichotomously fluctuating metastable potential we obtain the mean first-passage time (MFPT) as a function of the potential parameters, the noise intensity and the mean rate of switchings of the dichotomous noise. We find noise enhanced stability (NES) phenomenon in the system investigated and the parameter region of the fluctuating potential where the effect can be observed. For the diffusion of the overdamped Brownian particle in a fast fluctuating symmetric periodic potential w…
Geometric Entropies of Mixing (EOM)
2005
Trigonometric and trigonometric-algebraic entropies are introduced. Regularity increases the entropy and the maximal entropy is shown to result when a regular $n$-gon is inscribed in a circle. A regular $n$-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy which occurs in the asymptotic infinite $n$ limit. EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex sets. These expressions are shown to be related to the phase functions obtained from the WKB approximation for Bessel and Hermite functions.
Identification of clusters of companies in stock indices via Potts super-paramagnetic transitions
2000
The clustering of companies within a specific stock market index is studied by means of super-paramagnetic transitions of an appropriate q-state Potts model where the spins correspond to companies and the interactions are functions of the correlation coefficients determined from the time dependence of the companies' individual stock prices. The method is a generalization of the clustering algorithm by Domany et. al. to the case of anti-ferromagnetic interactions corresponding to anti-correlations. For the Dow Jones Industrial Average where no anti-correlations were observed in the investigated time period, the previous results obtained by different tools were well reproduced. For the Standa…
Heavy-tailed targets and (ab)normal asymptotics in diffusive motion
2010
We investigate temporal behavior of probability density functions (pdfs) of paradigmatic jump-type and continuous processes that, under confining regimes, share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general L\'evy stable distribution) in its long time asymptotics. For diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when initially infinite number of the pdf moments drops down to a few or none at all. The time-dependence of the variance (if in existence…
In-Fiber All-Optical Fractional Differentiator Using an Asymmetrical Moiré Fiber Grating
2023
In this work, it is demonstrated numerically that an asymmetric Moiré fiber grating operated in reflection can provide the required spectral response to implement an all-optical fractional differentiator. In our case, the accumulated phase shift is not associated with a point phase shift, as when working with fiber Bragg gratings and long-period gratings with punctual defects, but is distributed all over the grating. The proposed device is supported by numerical simulations, and a dimensionless deviation factor is calculated to make quantitative analysis feasible. The performance of the proposed device is analyzed using numerical simulations by computing the fractional time derivatives of t…