随机变量和的估计：例子与信息界

m相依随机变量和的离散逼近

Discrete approximations for sums of m-dependent random variables

1相依整数值随机变量的和用复合Poisson分布、负二项分布、二项分布和带符号复合Poisson测度来近似。估计得到的全变差和局部度量。然后将结果应用于$m$依赖性$(k_1,k_2)$事件和2次运行的统计。海因里希的方法和卷积的光滑性质用于证明。

Efficiency of maximum likelihood estimation for a multinomial distribution with known probability sums

For a multinomial distribution, suppose that we have prior knowledge of the sum of the probabilities of some categories. This allows us to construct a submodel in a full (i.e., no-restriction) model. Maximum likelihood estimation (MLE) under this submodel is expected to have better estimation efficiency than MLE under the full model. This article presents the asymptotic expansion of the risk of MLE with respect to Kullback--Leibler divergence for both the full model and submodel. The results reveal that, using the submodel, the reduction of the risk is quite small in some cases. Furthermore, w…

Robust and efficient mean estimation: an approach based on the properties of self-normalized sums

Let $X$ be a random variable with unknown mean and finite variance. We present a new estimator of the mean of $X$ that is robust with respect to the possible presence of outliers in the sample, provides tight sub-Gaussian deviation guarantees without any additional assumptions on the shape or tails of the distribution, and moreover is asymptotically efficient. This is the first estimator that provably combines all these qualities in one package. Our construction is inspired by robustness properties possessed by the self-normalized sums. Theoretical findings are supplemented by numerical simula…

$m$ -因变量和的复合Poisson收敛定理

A Compound Poisson Convergence Theorem for Sums of $m$-Dependent Variables

本文证明了$m$相依随机变量的和$S_n$的Simons-Johnson定理，其中$S_n$具有指数权和极限复合泊松分布$\CP(s,λ)$。更精确地说，我们给出了$\sum_{k=0}^\infty\ee^{hk}\ab{P(S_n=k)-\CP(s,λ)\{k\}}\to 0$的充分条件，并提供了一个估计的收敛速度。证明了Simons-Johnson定理对加权Wasserstein范数也成立。定义在%不同格上的两个泊松变量的%极限和。然后显示$N(n;k_1,k_2)$和$k$运行统计量的结果。

关于独立Rademacher随机变量正规化和尾的上确界

On the supremum of the tails of normalized sums of independent Rademacher random variables

证明了一个著名的关于独立Rademacher随机变量正规化和尾的上确界的猜想是错误的。一个相关的猜想，最近也反驳，进行了讨论。

相依过程部分和的方差估计

Estimation of the variance of partial sums of dependent processes

研究了平稳随机过程$(X_k)_{k\geq 1}$的部分和的极限方差\[ σ^2=Var(X_1)+2 \sum_{k=2}^\infty Cov(X_1,X_k) \]的次抽样估计。我们建立了一个非重叠块恢复方法的$L_2$ -相容性。我们的结果适用于可以表示为泛函的强混合过程的过程。出于最近的应用程序的秩检验，我们还研究了系列$Var(F(X_1))+2 \sum_{k=2}^\infty Cov(F(X_1),F(X_k))$，其中$F$是分布函数的$X_1$的估计。仿真结果表明，建议的估计和均方误差的最佳规则的选择的分组长度的有用性。

On large deviations for sums of discrete m-dependent random variables

The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.

可交换随机变量加权和的Hoeffding与Bernstein不等式

Hoeffding and Bernstein inequalities for weighted sums of exchangeable random variables

本文为可交换随机变量的加权和建立了Hoeffding和Bernstein型集中不等式，统一了独立同分布与可交换两种情形。

自然规范化下独立中心化随机变量之和的精确指数尾部估计，并应用于 U-统计量理论

Exact exponential tail estimation for sums of independent centered random variables, under natural norming, with applications to the theory of U-statistics

本文利用 GLS 理论推导自然规范化独立中心化随机变量之和的精确指数尾部，并给出 U 统计量中的精细尾部估计。

一类序独立同分布部分和的泛函中心极限定理。随机变量

A functional central limit theorem for the partial sums of sorted i.i.d. random variables

令$(X_i,i\geq 1)$是i.i.d.具有$[0,1]$和$f$中的值的随机变量是使得$`E(f(X_1)^2)<+\infty$的函数。给出了过程$t\mapsto \sum_{i=1}^n f(X_i)1_{X_i\leq t}$的一个泛函中心极限定理.

利用残差平方和的惩罚项估计病态模型

Estimation of ill-conditioned models using penalized sums of squares of the residuals

针对病态计量经济模型，本文提出将残差平方和惩罚为向简单回归系数估计收缩的估计量，并证明其包含ridge特例，在bootstrap推断和数值稳定性上更优。

可交换随机变量加权和的更紧Hoeffding界

A Sharper Hoeffding Bound for Weighted Sums of Exchangeable Random Variables

本文针对有界可交换随机变量加权和，证明了更紧的Hoeffding型矩生成函数界，改进了现有膨胀因子的阶并给出显式常数，还证明了逆总体大小膨胀的不可避免性。

相依无穷方差随机变量和的稳定极限

Stable limits for sums of dependent infinite variance random variables

本文给出了强平稳过程的仿射变换部分和按分布收敛于无穷方差稳定分布的条件。这种收敛保持的条件在文献中是已知的。然而，大多数这些结果是定性的意义上说，极限分布的参数表示的一些极限点过程。在本文中，我们将能够确定的参数的限制稳定分布的一些尾部特征的平稳序列。我们将把我们的结果应用到一些标准的时间序列模型中，包括GARCH（1，1）过程及其平方，随机波动率模型和随机递归方程的解。

超广义中心极限定理：非同分布随机变量求和的极限分布

Super Generalized Central Limit Theorem: Limit distributions for sums of non-identical random variables with power-laws

本文研究非同分布随机变量求和的极限分布，证明其收敛于唯一稳定分布，解释了稳定分布的普遍性。