arXiv subjects

Yi-Ching Yao

Publications and source records attributed to Yi-Ching Yao.

5 recordsLinked to original sources

arXiv2025

Stochastic Ordering for Bernoulli and Normal Random Walks

Let $(S_n^p)_{n\geq 0}$ be a Bernoulli random walk where each of the independent increments is either $1$ or $-1$ with probabilities $p$ and $1-p$. For $p'$ and $p'' \in [0,1]$ with $|p'-1/2|>|p''-1/2|$, we show that $(|S_n^{p''}|)_{n\geq 0}$ is stochastically smaller than $(|S_n^{p'}|)_{n\geq 0}$. In other words, $(|S_n^{p}|)_{n\geq 0}$ is stochastically decreasing in $p \in [0,1/2]$ and increasing in $p\in [1/2,1]$. An analogous result is also given for the family of normal random walks indexed by $μ\in R$ where each of the independent increments is normally distributed with common mean $μ$ and variance $1$. Extension to Brownian motion then follows by a limiting argument. As an application, these results are used to easily derive stochastic ordering properties for stopping times of Bernoulli and normal random walks.

math.PR↗

arXiv2017

Optimal selection of the $k$-th best candidate

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of $n$ candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the $k$-th best candidate for $k\ge 2$. While the optimal stopping rule for $k=1,2$ (and all $n\ge 2$) is known to be of threshold type (involving one threshold), we solve the case $k=3$ (and all $n\ge 3$) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for $p(k,n)$, the maximum probability of selecting the $k$-th best of $n$ candidates. It is shown that (i) $p(1,n)=p(n,n)>p(k,n)$ for $1<k<n$, (ii) $p(k,n)\ge p(k,n+1)$, (iii) $p(k,n)\ge p(k+1,n+1)$, and (iv) $p(k,\infty):=\lim_{n\to \infty} p(k,n)$ is decreasing in $k$.

math.PR↗

arXiv2017

One-sided solutions for optimal stopping problems with logconcave reward functions

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including $(x^+)^ν$, $(e^x-K)^+$, $(K-e^{-x})^+$, $x\in\mathbb{R}$, $ν\in(0,\infty)$ and $K>0$) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All of such reward functions are continuous, increasing and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper, we show that all optimal stopping problems with increasing, logconcave and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

math.PR↗

arXiv2016

Corrected Discrete Approximations for the Conditional and Unconditional Distributions of the Continuous Scan Statistic

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). With the help of a change-of-measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.

math.PR↗

arXiv2007

Some results on the Gittins index for a normal reward process

We consider the Gittins index for a normal distribution with unknown mean $θ$ and known variance where $θ$ has a normal prior. In addition to presenting some monotonicity properties of the Gittins index, we derive an approximation to the Gittins index by embedding the (discrete-time) normal setting into the continuous-time Wiener process setting in which the Gittins index is determined by the stopping boundary for an optimal stopping problem. By an application of Chernoff's continuity correction in optimal stopping, the approximation includes a correction term which accounts for the difference between the discrete and continuous-time stopping boundaries. Numerical results are also given to assess the performance of this simple approximation.

math.ST↗