Uniform Lower Bound Research Articles

Estimates on the dimension of self‐similar measures with overlaps

In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any β ∈ ( 1 , 2 ) $ \beta \in (1,2)$ , the dimension of the Bernoulli convolution μ β $ \mu _{\beta }$ satisfies dim ( μ β ) ⩾ 0.98040856 , \begin{equation*}\hskip7pc \dim (\mu _{\beta })\geqslant 0.98040856,\hskip-7pc\vspace*{-6pt} \end{equation*} which improves a previous uniform lower bound 0.82 obtained by Hare and Sidorov (Exp. Math. 27 (2018), no. 4, 414–418). This new uniform lower bound is very close to the known numerical approximation 0.98040931953 ± 10 − 11 $ 0.98040931953\pm 10^{-11}$ for dim μ β 3 $\dim \mu _{\beta _3}$ , where β 3 ≈ 1.839286755214161 $ \beta _{3} \approx 1.839286755214161$ is the largest root of the polynomial x 3 − x 2 − x − 1 $ x^{3}-x^{2}-x-1$ . Moreover, the infimum inf β ∈ ( 1 , 2 ) dim ( μ β ) $\inf _{\beta \in (1,2)}\dim (\mu _{\beta })$ is attained at a parameter β ∗ $\beta _*$ in a small interval ( β 3 − 10 − 8 , β 3 + 10 − 8 ) . \begin{equation*}\hskip7pc (\beta _{3} -10^{-8}, \beta _{3} + 10^{-8}).\hskip-7pc \end{equation*} When β $\beta$ is a Pisot number, we express dim ( μ β ) $\dim (\mu _{\beta })$ in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate dim ( μ β ) $\dim (\mu _\beta )$ from above as well.

О равномерной оценке снизу решений нелинейной системы дифференциальных уравнений

О равномерной оценке снизу решений нелинейной системы дифференциальных уравнений

Linearized Ridge Regression Estimator Under the Mean Squared Error Criterion in a Linear Regression Model

This article mainly aims to study the superiority of the notion of linearized ridge regression estimator (LRRE) under the mean squared error criterion in a linear regression model. Firstly, we derive uniform lower bound of MSE for the class of the generalized shrinkage estimator (GSE), based on which it is shown that the optimal LRRE is the best estimator in the class of GSE's. Secondly, we propose the notion of the almost unbiased completeness and show that LRRE possesses such a property. Thirdly, the simulation study is given, from which it indicates that the LRRE performs desirably. Finally, the main results are applied to the well known Hald data.