In this thesis, we study the empirical performance of the ANOVA test (Priestley and Subba Rao, 1969) and propose two semiparametric tests for stationarities in time series and spatial processes based on the generalized likelihood ratio test (Fan et al., 2001) in the frequency domain.

We provide suggestions and guidelines for the bandwidth selection in the construction of evolutionary spectra estimator. The advantages and disadvantages of the ANOVA test are studied. In addition, we also provide specific comments on the types of evolutionary spectra estimators to be used under different situations.

The test of stationarity in time series is built on the varying-coefficient fractional exponential model, which is time-adapted from the fractional exponential model of Beran (1993). The proposed test is capable of testing both long-memory and short-memory processes individually or jointly. With the bootstrapped null distribution, this test shows good power performance and satisfactory test sizes in our simulation study.

In the test for spatial nonstationarity, we mainly focus on the isotropic processes. Under similar algorithm to that in the time series test, the simulated results show that a possible direction for enhancement will be on the basis system selection.

在本論文中，我們針對Priestley and Subba Rao (1969) 之ANOVA檢定能力進行實務研究，並在頻譜域中利用Fan et al. (2001) 的廣義概似比檢定 (generalized likelihood ratio test，GLR Test) 提出兩種關於時間序列與空間過程的平穩性檢定方法。 我們提出建構演化譜密度 (evolutionary spectra) 估計量的頻寬選擇指南與建議，並研究該ANOVA檢定法的優缺點。此外，我們亦分析各不同譜密度估計量的適用情形。 在時間序列的平穩性檢定中，我們採用了對Beran (1993) 的分數指數模型 (fractional exponential model) 係數賦予時間變動性而得到的變動時間係數之分數指數模型 (varying-coefficient fractional exponential model) 以建構對立假設。以此模型所得到之GLR檢定可獨自或同時針對長距記憶 (long-memory) 與短距記憶 (short-memory) 過程之平穩性做驗證。模擬與實例分析結果顯示，在使用拔靴法 (bootstrap) 所造出之檢定量虛無分佈下此檢定法具備準確的檢定大小 (test size) 以及良好的檢定力。 而對於空間過程的平穩性檢定，我們主要針對等向性 (isotropic) 空間過程。在和時間序列的GLR檢定相似的方法下，模擬結果顯示在基底系統的選擇上有改善的空間。

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