## Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos (2000)

Citations: | 8 - 2 self |

### BibTeX

@MISC{Shintani00nonparametricneural,

author = {Mototsugu Shintani and Oliver Linton},

title = {Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos},

year = {2000}

}

### OpenURL

### Abstract

This paper derives the asymptotic distribution of the nonparametric neural network estimator of the Lyapunov exponent in a noisy system. Positivity of the Lyapunov exponent is an operational definition of chaos. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated Lyapunov exponents and a consistent variance estimator. A simulation study to evaluate small sample performance is reported. We also apply our procedures to daily stock return data. In most cases, the hypothesis of chaos in the stock return series is rejected at the 1 % level with an exception in some higher power transformed absolute returns.

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Citation Context ...ult to the multidimensional case. For d = 1, we denote Z = Â and our goal is to obtain the convergence rate for sup x2Â ¯ ¯Db µ(x) ¡ Dµ0(x) Note that interpolation inequality (See Gabushin, 1967, and =-=Shen and Wong, 1994-=-) implies ¯ : kg(x) ¡ g0(x)k 1 · K kg(x) ¡ g0(x)k 2(m¡1)=2m kD m g(x) ¡ D m g0(x)k 1=2m : where K is a …xed constant. Substituting g(x) = D b µ(x), g0(x) = Dµ0(x), m = 1 yields ° °Db ° µ(x) ¡ Dµ0(x) °... |

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Citation Context ... then extend the result to the multidimensional case. For d = 1, we denote Z = Â and our goal is to obtain the convergence rate for sup x2Â ¯ ¯Db µ(x) ¡ Dµ0(x) Note that interpolation inequality (See =-=Gabushin, 1967-=-, and Shen and Wong, 1994) implies ¯ : kg(x) ¡ g0(x)k 1 · K kg(x) ¡ g0(x)k 2(m¡1)=2m kD m g(x) ¡ D m g0(x)k 1=2m : where K is a …xed constant. Substituting g(x) = D b µ(x), g0(x) = Dµ0(x), m = 1 yield... |

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