在數值實驗部分，本文透過 Monte Carlo simulation 比較不同區間方法在各樣本數下的平均區間長度、coverage rate、Monte Carlo standard error（MCSE）與 scaling behavior。結果顯示，median-based CI、one-step estimator CI 及 PIT-based back-transformed interval 的平均區間長度皆呈現近似 O(n⁻¹ᐟ²) 的縮減趨勢。其中，median-based CI 與 one-step estimator CI 的 coverage rate 大致接近信賴水準（confidence level）。
整體而言，本文從一維 FAP-Cauchy 穩健統計推論出發，透過二維 Cauchy-type 分類模型之建立，再進一步延伸至含漂移的 FAP 分析，建立了一套適用於重尾 FAP 模型之參數推論與分類分析框架。;At the nanoscale， conventional electromagnetic communication may be constrained by device size， energy consumption， and propagation media. Molecular communication (MC)， which uses chemical molecules as information carriers， has therefore attracted increasing attention. In diffusion-based molecular communication systems， molecular motion can be described by stochastic processes. By analyzing the first-arrival position (FAP) of molecules recorded at the receiver， information regarding the transmitter location can be inferred， thereby providing a basis for source localization， parameter estimation， and source classification.
This thesis is theoretically based on the FAP-Cauchy model induced by drift-free Brownian motion and a linear absorbing receiver， and further investigates statistical inference under the heavy-tailed Cauchy distribution. In a two-dimensional diffusion space with a one-dimensional linear absorbing receiver， the position at which a molecule first reaches the absorbing boundary can be described by a Cauchy distribution. Because the expectation and variance of the Cauchy distribution do not exist， conventional inference methods based on the sample mean and sample variance are highly sensitive to extreme observations. Therefore， this thesis adopts the sample median as an estimator of the location parameter and uses half of the interquartile range (IQR)， referred to as the half-IQR， as an estimator of the scale parameter.
In addition， the probability integral transform (PIT) is used to transform the original Cauchy observations into a bounded uniform domain. On this basis， several confidence interval (CI) and transformation-based interval procedures are compared， including the median-based CI， the one-step estimator CI， the sample mean CI， and the PIT-based back-transformed interval. Their interval lengths， coverage performance， and finite-sample properties are then analyzed.
In the numerical experiments， Monte Carlo simulations are conducted to compare the average interval length， coverage probability， Monte Carlo standard error (MCSE)， and scaling behavior of the different interval procedures across various sample sizes. The results show that the average interval lengths of the median-based CI， the one-step estimator CI， and the PIT-based back-transformed interval decrease at an approximately O(n⁻¹ᐟ²) rate. Among these methods， the coverage probabilities of the median-based CI and the one-step estimator CI are generally close to the nominal confidence level.
Under the known-parameter oracle benchmark， the coverage probability of the PIT-based back-transformed interval is also close to 0.95， indicating that the PIT transformation provides stable coverage performance under ideal parameter conditions. However， when the location and scale parameters are estimated from the same sample and directly substituted into the PIT and inverse-transformation procedures， the naive plug-in interval exhibits a coverage probability substantially below the nominal level. This result indicates that the reduction in coverage is mainly attributable to the uncertainty introduced by parameter estimation rather than to a failure of the PIT transformation itself.
Based on the preceding FAP-Cauchy model and PIT results， this thesis further develops a PIT-based hypothesis-testing framework and its corresponding power curve to evaluate the ability of the receiver to detect changes in the transmitter location. The results show that， as the distance between the location parameter under the alternative hypothesis and that under the null hypothesis increases， the probability of rejecting the null hypothesis also increases substantially. This finding demonstrates that the PIT-based test can reflect changes in the FAP distribution caused by shifts in the source location.
After completing the statistical inference analysis for the one-dimensional FAP-Cauchy model， this thesis further extends the study to a two-dimensional source-classification problem. A two-dimensional Cauchy-type density function is adopted as a statistical approximation model to construct a density-based classification rule and its corresponding decision boundary. By solving the density equality condition f₁(x， y) = f₂(x， y)， the geometric structure of the decision boundary in a two-source model is analyzed， and the center-line intersection behavior of the Gaussian and Cauchy-type models is compared.
The numerical results show that， owing to its faster exponential tail decay， the center-line intersection point of the Gaussian model is more sensitive to changes in the scale ratio. In contrast， because the Cauchy-type model has a more slowly decaying heavy tail， its intersection point varies more smoothly. In addition， finite-grid numerical integration is used to approximate the classification error rates under different source-center distances and scale parameters， and the overall variation is presented using heat maps.
Finally， this thesis extends the analysis to an FAP model with a vertical drift component and investigates the effects of vertical drift on the FAP density function， decision boundary， and center-line intersection point. The results show that vertical drift changes the relative density levels of the two sources and consequently shifts the decision boundary. Moreover， as the drift intensity increases， the center-line intersection point becomes more sensitive to changes in the scale ratio.
Overall， this thesis begins with robust statistical inference for the one-dimensional FAP-Cauchy model， extends the analysis to a two-dimensional Cauchy-type classification model， and further incorporates an FAP model with vertical drift. Through these analyses， a parameter-inference and classification framework applicable to heavy-tailed FAP models is established.
