Comparison Theorems for Small Deviations of Random Series

Gao, Fuchang; Hannig, Jan; Torcaso, Fred

doi:10.1214/ejp.v8-147

Cited by 18 publications

(16 citation statements)

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“…Gao et al (2003)) similar to the polynomial decay. In this paper however, we shall con…ne our attention to the case of the polynomial law for expositional simplicity and consistency of presentation, since the asymptotic behaviour of the small ball is not yet known in the general case for choices other than the polynomial decay as in D3.In practice, we would require some ordering for the marginal regressors in the static regressions case A1, since the in ‡uence of marginals is set to decrease via the bandwidth adjustments as discussed just above.…”

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confidence: 72%

Nonparametric estimation of infinite order regression and its application to the risk-return tradeoff

Hong

Linton

2020

Journal of Econometrics

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This paper studies nonparametric estimation of the infinite order regression E (|Ft-1), k ∈ ℤ with stationary and weakly dependent data. We propose a Nadaraya-Watson type estimator that operates with an infinite number of conditioning variables. The established theories are applied to examine the intertemporal risk-return relation for the aggregate stock market, and some new empirical evidence is reported. With a bandwidth sequence that shrinks the effects of long lags, the influence of all conditioning information is modelled in a natural and flexible way, and the issues of omitted information bias and specification error are effectively handled. Asymptotic properties of the estimator are shown under a wide range of static and dynamic regressions frameworks, thereby allowing various kinds of conditioning variables to be used. We establish pointwise/uniform consistency and CLTs. It is shown that the convergence rates are at best logarithmic, and depend on the smoothness of the regression, the distribution of the marginal regressors and their dependence structure in a non-trivial way via the Lambert W function. The empirical studies on S&P 500 daily data from 1950-2016 using our estimator report an overall positive risk-return relation. We also .find evidence of strong time variation and counter-cyclical behaviour in risk aversion. These conclusions are attributable to the inclusion of otherwise neglected information in our method.

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confidence: 72%

Nonparametric estimation of infinite order regression and its application to the risk-return tradeoff

Hong

Linton

2020

Journal of Econometrics

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“…(1,2) k the eigenvalues of the problems (3). Using the Li comparison theorem (see [12,14]) and Theorem 1, we obtain…”

Section: Proof Denote By µmentioning

confidence: 99%

Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms

Nazarov

Pusev

2014

St. Petersburg Math. J.

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“…Lemma 2.2 (Lemma 5 in [3]). Suppose {c n } is a sequence of real numbers such that ∞ n=1 c n converges, and g has total variation D on [0, ∞).…”

Section: Proofs Of Theorem 21 and Theorem 22mentioning

confidence: 99%

“…Gaussian random variables N (α, β 2 ); see Theorem 2.1 and Theorem 2.2. This is motivated by [3] in which the following exact level comparison theorems for small deviations were obtained: as r → 0, P { ∞ n=1 a n |ξ n | ≤ r} ∼ cP { ∞ n=1 b n |ξ n | ≤ r} for i.i.d. random variables {ξ n } whose common distribution satisfies several weak assumptions in the vicinity of zero.…”

Section: Introductionmentioning

confidence: 99%

Comparison for upper tail probabilities of random series

Gao

Liu

Yang

2013

Journal of the Korean Statistical Society

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Let {ξ n } be a sequence of independent and identically distributed random variables. In this paper we study the comparison for two upper tail probabilities P { ∞ n=1 a n |ξ n | p ≥ r} and P { ∞ n=1 b n |ξ n | p ≥ r} as r → ∞ with two different real series {a n } and {b n }. The first result is for Gaussian random variables {ξ n }, and in this case these two probabilities are equivalent after suitable scaling. The second result is for more general random variables, thus a weaker form of equivalence (namely, logarithmic level) is proved.

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Comparison Theorems for Small Deviations of Random Series

Abstract: Let {ξ n } be a sequence of i.i.d. positive random variables with common distribution

Cited by 18 publications

References 10 publications

Nonparametric estimation of infinite order regression and its application to the risk-return tradeoff

Nonparametric estimation of infinite order regression and its application to the risk-return tradeoff

Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms

Comparison for upper tail probabilities of random series

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