本論文提出一種以加權核累積分布函數為基礎的經驗概似比檢定方法，用於在未知母體分布下對平均數與分位數等統計特徵量進行無母數推論。傳統參數推論通常需要事先假設特定的分布模型，當模型假設不適當時，推論結果可能產生偏誤。為避免對母體分布作過強假設，本研究結合核平滑分布估計與經驗概似方法，將統計特徵量在虛無假設下的限制條件透過加權核累積分布函數表示，並進一步建構經驗概似比檢定統計量。
本研究主要考慮平均數與分位數兩類統計特徵量。對於平均數，本文利用加權高斯核累積分布估計量所導出的積分表示式建立限制條件；對於分位數，則利用分位數之定義，將虛無假設轉換為加權核累積分布函數上的限制。這些限制條件可進一步改寫為估計方程式的形式，進而建構經驗概似比檢定統計量；在適當的正則條件下，該檢定統計量於虛無假設成立時漸近服從自由度為一的卡方分布。此外，所提出的檢定統計量具有封閉形式的漸近表示，可由估計函數的樣本平均與二階樣本動差直接計算；其中，估計函數的樣本平均亦可依其定義，反映樣本資料所呈現之統計特徵量相對於虛無假設目標值的偏離方向。
本文透過模擬研究評估所提出方法在有限樣本下的表現，並與既有的加權核密度概似方法進行比較。模擬結果顯示，將第一型錯誤率控制在合理範圍內所需的樣本數，會隨母體分布型態與檢定目標而有所不同；整體而言，當樣本數增加時，所提出方法的經驗第一型錯誤率會趨近於名目水準。對於中位數，所提出方法在多數非正態分布下具有與比較方法相近或略高的檢定力；對於平均數，當第一型錯誤率已接近名目水準時，兩種方法大多呈現相近的檢定力表現。此外，實際資料分析結果顯示，所提出方法可應用於身體質量指數資料之中位數檢定，亦可延伸至品管中的第二階段製程監控問題。整體而言，本研究建立一個以加權核累積分布函數連結核平滑分布估計與經驗概似推論的無母數檢定架構，可應用於平均數與分位數等統計特徵量之假設檢定。此外，估計函數的樣本平均可依其定義，反映樣本資料所呈現之統計特徵量相對於目標值的偏離方向。

This thesis proposes an empirical likelihood ratio testing method based on the weighted kernel cumulative distribution function for nonparametric inference on statistical functionals, such as the mean and quantiles, when the underlying population distribution is unknown. Conventional parametric inference usually requires a specified distributional model, and the resulting inference may be biased or unreliable if the assumed model is inappropriate. To avoid imposing strong assumptions on the population distribution, this study combines kernel-smoothed distribution estimation with empirical likelihood inference. The main idea is to express the null constraint for a statistical functional through the weighted kernel cumulative distribution function and then construct the corresponding empirical likelihood ratio test statistic.
This study mainly considers two types of statistical functionals: the mean and quantiles. For the mean, the constraint is derived from the integral representation obtained from the weighted Gaussian kernel cumulative distribution estimator. For quantiles, the null hypothesis is transformed into a constraint on the weighted kernel cumulative distribution function by using the definition of a quantile. These constraints can be further written in the form of estimating equations, which are then used to construct the empirical likelihood ratio test statistic. Under suitable regularity conditions, the proposed test statistic converges in distribution to a chi-square distribution with one degree of freedom under the null hypothesis. In addition, the proposed test statistic admits a closed-form asymptotic representation that can be computed directly from the sample mean and sample second moment of the estimating function. Depending on its specific definition, the sample mean of the estimating function can also indicate the direction in which the statistical functional represented by the sample data deviates from the target value under the null hypothesis.
Simulation studies are conducted to evaluate the finite-sample performance of the proposed method and to compare it with an existing weighted kernel density-based likelihood method. The simulation results show that the sample size required for the empirical Type I error rate to fall within the approximate 95% Monte Carlo error interval around the nominal level depends on the shape of the underlying distribution and the testing target. As the sample size increases, the empirical Type I error rate of the proposed method generally becomes closer to the nominal level and eventually falls within the Monte Carlo error interval in the considered settings. For the median, the proposed method provides power comparable to or slightly higher than the competing method under most non-normal distributions. For the mean, at the sample sizes where the empirical Type I error rates fall within the Monte Carlo error interval, the two methods generally show similar power performance. In addition, the real data analyses show that the proposed method can be applied to median testing for body mass index data and can also be extended to Phase II process monitoring in quality control. Overall, this study establishes a nonparametric testing framework that connects kernel-smoothed distribution estimation and empirical likelihood inference through the weighted kernel cumulative distribution function and can be applied to hypothesis testing for statistical functionals such as the mean and quantiles. In addition, depending on its definition, the sample mean of the estimating function can reflect the direction in which the statistical functional represented by the sample data deviates from the target value.

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