本文旨在建立一個用於檢定兩組一維資料是否可視為來自相同分布的兩階段程序。第一階段先比較兩組資料在左右兩端不重疊區域所占的比例，並以單一樣本比例檢定判斷取值範圍是否已有顯著差異；若第一階段未達顯著，第二階段再於共同範圍內進行密度比估計，以兩個方向的 Kullback–Leibler（KL）散度加總形成對稱 KL 統計量，並透過 bootstrap 建立經驗虛無分布與 p 值。
在估計方法上，本文以三階 B-spline 基底表示密度比函數，使未知的分布差異能以有限維參數形式進行估計。為降低共同區間端點附近資料出現機率過小時對密度比估計造成的不穩定影響，本文亦討論加入虛擬樣本後的替代分析結果。模擬研究包含四組情境：均勻分布與常態分布、兩組相同常態分布、相同截斷區間內不同形狀之常態分布，以及外觀相近但理論上仍不同的 Beta 分布與截斷常態分布。
模擬結果顯示，當兩組分布差異明顯時，本文方法能穩定拒絕兩組分布相同的假設；在虛無情境下，未加入虛擬樣本時拒絕比例為 19%，加入後降為 4%，接近設定的顯著水準。對於外觀相近且差異較小的情境，加入虛擬樣本後拒絕比例由 77% 降為 9%，表示此處理具有平滑與保守化效果，但也可能降低對細微差異的敏感度。整體而言，本文流程可作為比較兩組一維資料分布差異的工具，但實際解讀時仍須同時考量樣本數、基底設定、重抽樣次數與虛擬樣本處理方式。

In this thesis, a two-stage procedure is proposed for testing whether two one-dimensional samples can be regarded as being generated from the same distribution. In the first stage, the proportions of observations falling in the non-overlapping regions at the two endpoints are examined using one-sample proportion tests. If no significant endpoint difference is detected, the second stage estimates the density ratio over the common range of the two samples. Two directional Kullback–Leibler (KL) divergence estimates are then combined to form a symmetric KL statistic, and bootstrap resampling is used to construct the empirical null distribution and the corresponding p-values.
The proposed density-ratio estimator is based on approximating the unknown ratio function with cubic B-spline basis functions, allowing distributional differences to be estimated through a finite-dimensional model. To reduce instability caused by very small occurrence probabilities near the endpoints of the common interval, this thesis also investigates an alternative procedure in which additional virtual samples are included for density estimation. Four simulation settings are considered: a uniform distribution versus a normal distribution, two identical normal distributions, two truncated normal distributions with different shapes over the same interval, and a visually similar but theoretically different pair consisting of a beta distribution and a truncated normal distribution.
The simulation results show that the proposed procedure consistently rejects the null hypothesis when the distributional difference is pronounced. Under the null setting, the rejection rate decreases from 19% to 4% after adding uniform virtual samples, which is close to the nominal significance level. For the visually similar setting with a small theoretical difference, the rejection rate decreases from 77% to 9% after adding virtual samples, indicating that the virtual-sample procedure has a smoothing and conservative effect but may also reduce sensitivity to subtle differences. Overall, the proposed workflow provides a practical tool for comparing two one-dimensional distributions, while its conclusions should be interpreted together with the sample size, basis specification, number of bootstrap replications, and the use of virtual samples.

Ali, S. M., & Silvey, S. D. (1966). A general class of coefficients of divergence of one distribution from another. *Journal of the Royal Statistical Society: Series B (Methodological), 28*(1), 131–142. https://doi.org/10.1111/j.2517-6161.1966.tb00626.x

Basseville, M., & Nikiforov, I. V. (1993). *Detection of abrupt changes: Theory and application*. Prentice Hall.

Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. *The Annals of Mathematical Statistics, 23*(4), 493–507. https://doi.org/10.1214/aoms/1177729330

Cover, T. M., & Thomas, J. A. (2006). *Elements of information theory* (2nd ed.). Wiley. https://doi.org/10.1002/047174882X

Cox, M. G. (1972). The numerical evaluation of B-splines. *Journal of the Institute of Mathematics and Its Applications, 10*(2), 134–149. https://doi.org/10.1093/imamat/10.2.134

de Boor, C. (1972). On calculating with B-splines. *Journal of Approximation Theory, 6*(1), 50–62. https://doi.org/10.1016/0021-9045(72)90080-9

de Boor, C. (2001). *A practical guide to splines* (Revised ed.). Springer.

Eilers, P. H. C., & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. *Statistical Science, 11*(2), 89–121. https://doi.org/10.1214/ss/1038425655

Kanamori, T., Suzuki, T., & Sugiyama, M. (2012). f-divergence estimation and two-sample homogeneity test under semiparametric density-ratio models. *IEEE Transactions on Information Theory, 58*(2), 708–720. https://doi.org/10.1109/TIT.2011.2163380

Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. *The Annals of Mathematical Statistics, 22*(1), 79–86. https://doi.org/10.1214/aoms/1177729694

Liu, S., Yamada, M., Collier, N., & Sugiyama, M. (2013). Change-point detection in time-series data by relative density-ratio estimation. *Neural Networks, 43*, 72–83. https://doi.org/10.1016/j.neunet.2013.01.012

Møllersen, K., Dhar, S. S., & Godtliebsen, F. (2016). On data-independent properties for density-based dissimilarity measures in hybrid clustering. *Applied Mathematics, 7*(15), 1674–1706. https://doi.org/10.4236/am.2016.715143

Nguyen, X., Wainwright, M. J., & Jordan, M. I. (2010). Estimating divergence functionals and the likelihood ratio by convex risk minimization. *IEEE Transactions on Information Theory, 56*(11), 5847–5861. https://doi.org/10.1109/TIT.2010.2068870

Sugiyama, M., Nakajima, S., Kashima, H., von Bünau, P., & Kawanabe, M. (2008). Direct importance estimation with model selection and its application to covariate shift adaptation. *Neural Networks, 21*(10), 1396–1405. https://doi.org/10.1016/j.neunet.2008.09.004

Sugiyama, M., Suzuki, T., Itoh, Y., Kanamori, T., & Kimura, M. (2011). Least-squares two-sample test. *Neural Networks, 24*(7), 735–751. https://doi.org/10.1016/j.neunet.2011.04.003

Sugiyama, M., Suzuki, T., & Kanamori, T. (2012). *Density ratio estimation in machine learning*. Cambridge University Press. https://doi.org/10.1017/CBO9781139035613