Yuewei Liu

Fred J. Hickernell, Lan Jiang, Yuewei Liu et al.

Monte Carlo methods are used to approximate the means, $μ$, of random variables $Y$, whose distributions are not known explicitly. The key idea is that the average of a random sample, $Y_1, ..., Y_n$, tends to $μ$ as $n$ tends to infinity. This article explores how one can reliably construct a confidence interval for $μ$ with a prescribed half-width (or error tolerance) $\varepsilon$. Our proposed two-stage algorithm assumes that the kurtosis of $Y$ does not exceed some user-specified bound. An initial independent and identically distributed (IID) sample is used to confidently estimate the variance of $Y$. A Berry-Esseen inequality then makes it possible to determine the size of the IID sample required to construct the desired confidence interval for $μ$. We discuss the important case where $Y=f(\vX)$ and $\vX$ is a random $d$-vector with probability density function $ρ$. In this case $μ$ can be interpreted as the integral $\int_{\reals^d} f(\vx) ρ(\vx) \dif \vx$, and the Monte Carlo method becomes a method for multidimensional cubature.

Cheng Shen, Yuewei Liu

In industrial settings, surface defects on steel can significantly compromise its service life and elevate potential safety risks. Traditional defect detection methods predominantly rely on manual inspection, which suffers from low efficiency and high costs. Although automated defect detection approaches based on Convolutional Neural Networks(e.g., Mask R-CNN) have advanced rapidly, their reliability remains challenged due to data annotation uncertainties during deep model training and overfitting issues. These limitations may lead to detection deviations when processing the given new test samples, rendering automated detection processes unreliable. To address this challenge, we first evaluate the detection model's practical performance through calibration data that satisfies the independent and identically distributed (i.i.d) condition with test data. Specifically, we define a loss function for each calibration sample to quantify detection error rates, such as the complement of recall rate and false discovery rate. Subsequently, we derive a statistically rigorous threshold based on a user-defined risk level to identify high-probability defective pixels in test images, thereby constructing prediction sets (e.g., defect regions). This methodology ensures that the expected error rate (mean error rate) on the test set remains strictly bounced by the predefined risk level. Additionally, we observe a negative correlation between the average prediction set size and the risk level on the test set, establishing a statistically rigorous metric for assessing detection model uncertainty. Furthermore, our study demonstrates robust and efficient control over the expected test set error rate across varying calibration-to-test partitioning ratios, validating the method's adaptability and operational effectiveness.