统计/概率学术速递[1.10]
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math.ST统计，共计1篇
math.PR概率，共计5篇
1.math.ST统计:
【1】 Power and Sample Size Calculations for Rerandomized Experiments
标题：随机试验的功率和样本量计算
链接：https://arxiv.org/abs/2201.02486
备注：20 pages, 4 figures
摘要：Power is an important aspect of experimental design, because it allows
researchers to understand the chance of detecting causal effects if they exist.
It is common to specify a desired level of power, and then compute the sample
size necessary to obtain that level of power; thus, power calculations help
determine how experiments are conducted in practice. Power and sample size
calculations are readily available for completely randomized experiments;
however, there can be many benefits to using other experimental designs. For
example, in recent years it has been established that rerandomized designs,
where subjects are randomized until a prespecified level of covariate balance
is obtained, increase the precision of causal effect estimators. This work
establishes the statistical power of rerandomized treatment-control
experiments, thereby allowing for sample size calculators. Our theoretical
results also clarify how power and sample size are affected by treatment effect
heterogeneity, a quantity that is often ignored in power analyses. Via
simulation, we confirm our theoretical results and find that rerandomization
can lead to substantial sample size reductions; e.g., in many realistic
scenarios, rerandomization can lead to a 25% or even 50% reduction in sample
size for a fixed level of power, compared to complete randomization. Power and
sample size calculators based on our results are in the R package rerandPower
on CRAN.
2.math.PR概率:
【1】 Cauchy, normal and correlations versus heavy tails
标题：柯西，正态和相关性与重尾
链接：https://arxiv.org/abs/2201.02498
摘要：A surprising result of Pillai and Meng (2016) showed that a transformation
$\sum_{j=1}^n w_j X_j/Y_j$ of two iid centered normal random vectors,
$(X_1,\ldots, X_n)$ and $(Y_1,\ldots, Y_n)$, $n>1$, for any weights $0\leq
w_j\leq 1$, $ j=1,\ldots, n$, $\sum_{j=1}^n w_j=1$, has a Cauchy distribution
regardless of any correlations within the normal vectors. The correlations
appear to lose out in the competition with the heavy tails. To clarify how
extensive this phenomenon is, we analyze two other transformations of two iid
centered normal random vectors. These transformations are similar in spirit to
the transformation considered by Pillai and Meng (2016). One transformation
involves absolute values: $\sum_{j=1}^n w_j X_j/|Y_j|$. The second involves
randomly stopped Brownian motions: $\sum_{j=1}^n w_j X_j\bigl(Y_j^{-2}\bigr)$,
where $\bigl\{\bigl( X_1(t),\ldots, X_n(t)\bigr), \, t\geq 0\bigr\},\ n>1,$ is
a Brownian motion with positive variances; $(Y_1,\ldots, Y_n)$ is a centered
normal random vector with the same law as $( X_1(1),\ldots, X_n(1))$ and
independent of it; and $X(Y^{-2})$ is the value of the Brownian motion $X(t)$
evaluated at the random time $t=Y^{-2}$. All three transformations result in a
Cauchy distribution if the covariance matrix of the normal components is
diagonal, or if all the correlations implied by the covariance matrix equal 1.
However, while the transformation Pillai and Meng (2016) considered produces a
Cauchy distribution regardless of the normal covariance matrix. the
transformations we consider here do not always produce a Cauchy distribution.
The correlations between jointly normal random variables are not always
overwhelmed by the heaviness of the marginal tails. The mysteries of the
connections between normal and Cauchy laws remain to be understood.
【2】 de Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products
标题：关于拟局部代数和无限费米张量积的de Finetti型定理
链接：https://arxiv.org/abs/2201.02488
备注：31 pages
摘要：Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on
$\mathbb{N}$, on quasi-local algebras are defined and proved to be
$\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local
actions are automatically even, and extreme invariant states are strongly
clustering. Tail algebras of invariant states are shown to obey a form of the
Hewitt and Savage theorem, in that they coincide with the fixed-point von
Neumann algebra. Infinite graded tensor products of $C^*$-algebras, which
include the CAR algebra, are then addressed as particular examples of
quasi-local algebras acted upon $\mathbb{P}_\mathbb{N}$ in a natural way.
Extreme invariant states are characterized as infinite products of a single
even state, and a de Finetti theorem is established. Finally, infinite products
of factorial even states are shown to be factorial by applying a twisted
version of the tensor product commutation theorem, which is also derived here.
【3】 Spatial integral of the solution to hyperbolic Anderson model with time-independent noise
标题：带时间无关噪声的双曲型Anderson模型解的空间积分
链接：https://arxiv.org/abs/2201.02319
备注：43 pages
摘要：In this article, we study the asymptotic behavior of the spatial integral of
the solution to the hyperbolic Anderson model in dimension $d\leq 2$, as the
domain of the integral gets large (for fixed time $t$). This equation is driven
by a spatially homogeneous Gaussian noise, whose covariance function is either
integrable, or is given by the Riesz kernel. The novelty is that the noise does
not depend on time, which means that It\^o's martingale theory for stochastic
integration cannot be used. Using a combination of Malliavin calculus with
Stein's method, we show that with proper normalization and centering, the
spatial integral of the solution converges to a standard normal distribution,
by estimating the speed of this convergence in the total variation distance. We
also prove the corresponding functional limit theorem for the spatial integral
process.
【4】 Pricing Bermudan options using regression trees/random forests
标题：使用回归树/随机森林为百慕大期权定价
链接：https://arxiv.org/abs/2201.02587
摘要：The value of an American option is the maximized value of the discounted cash
flows from the option. At each time step, one needs to compare the immediate
exercise value with the continuation value and decide to exercise as soon as
the exercise value is strictly greater than the continuation value. We can
formulate this problem as a dynamic programming equation, where the main
difficulty comes from the computation of the conditional expectations
representing the continuation values at each time step. In (Longstaff and
Schwartz, 2001), these conditional expectations were estimated using
regressions on a finite-dimensional vector space (typically a polynomial
basis). In this paper, we follow the same algorithm; only the conditional
expectations are estimated using Regression trees or Random forests. We discuss
the convergence of the LS algorithm when the standard least squares regression
is replaced with regression trees. Finally, we expose some numerical results
with regression trees and random forests. The random forest algorithm gives
excellent results in high dimensions.
【5】 Invasion of cooperative parasites in moderately structured host populations
标题：合作寄生虫在中等结构宿主群体中的入侵
链接：https://arxiv.org/abs/2201.02249
摘要：Certain defense mechanisms of phages against the immune system of their
bacterial host rely on cooperation of phages. Motivated by this example we
analyse invasion probabilities of cooperative parasites in host populations
that are moderately structured. More precisely we assume that hosts are
arranged on the vertices of a configuration model and that offspring of
parasites move to nearest neighbours sites to infect new hosts. We consider
parasites that generate many offspring at reproduction, but do this (usually)
only when infecting a host simultaneously. In this regime we identify and
analyse the spatial scale of the population structure at which invasion of
parasites turns from being an unlikely to an highly probable event.
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