Abstract
In this paper the statistical hypothesis testing task about the value of the Hurst parameter of fractional Brownian motion {ξ(t), t ∈ (0, 1)} is considered. The estimation problem for the Hurst parameter of a fractional Brownian motion or of Self Similarity Index plays an important role in statistics of stochastic processes. The proposed criterion for testing hypotheses about the value of the Hurst parameter is based on the Baxter sums method. The applications of the Baxter sums method for stochastic processes and fields allows us to build the strongly consistent estimators and to construct the non-asymptotic confidence intervals without the use of classical limit theorems in many models. We want to construct criterion for testing hypotheses about the value of the Hurst parameter α of a fractional Brownian motion H0 : α = α0 with the alternative H1 : α 6= α0, where α0 0 is constructed by using Baxter statistic, the elements of theory of Orlicz space and some inequality for quadratic forms of Gaussian random variables. The null hypotheses is not rejected, if −xp xp} ≤ p. The inequality sets the set of values for variable ˆαn, which will not lead to the rejection of a specific null hypothesis about α = α0. This set of values will be the region of acceptance for a given level of confidence p ∈ (0, 1).
Highlights
The estimation problem for the Hurst parameter of a fractional Brownian motion or of Self Similarity Index plays an important role in statistics of stochastic processes
We want to construct criterion for testing hypotheses about the value of the Hurst parameter α of a fractional Brownian motion H0 : α = α0 with the alternative H1 : α = α0, where α0 < 1, by using the observations of a stochastic process ξ(t) at points k 2n k = 0,
Summary
Дзета-функцiя Рiмана, то iз нерiвностi (6) отримуємо n=1 sup V arSn ≤ Cn (α∗) , α∈(0,α∗]. Розглянемо нульову гiпотезу H0 : α = α0 при альтернативнiй гiпотезi H1 : α = α0, де α0 < 1, про значення параметра α, що входить показником до коварiацiйної функцiї (1) дробового броунiвського руху {ξ(t), t ∈ [0, 1]}. В якостi критерiю для перевiрки нульової гiпотези використаємо статистику. Де оцiнку αn визначено в теоремi 1. За рiвнем значущостi p ∈ (0, 1) знайдемо таке xp, що справедлива нерiвнiсть. З нерiвностi (8) випливає, що pp P {|Kn| > xp} = P {Kn > xp} + P {Kn < −xp} ≤ 2 + 2 = p
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