小波理論的使用具有可以應用在時間與頻率定義域中，應用在不同的時間點及不同的頻率優點。在本研究中針對受到污染的時間數列來做分析，在估計的過程中發展出頑強性(robust)的估計式，可以用來偵測不同長度的離群值。而在小波的分解過程當中，母小波的特性具有高頻過濾的特性，因此若使用母小波來估計，即使選擇在相當低頻的位置也不會有發散的問題。因此可以用來估計各個波段的關係。在本研究中，我們將估計的模型擴展到非穩定的時間數列，以piecewise B-spline為例，經由蒙地卡羅模擬，驗証虛假概似函數(Pseudo likelihood function)的適用性。 With the potential ability of time-frequency decomposition for a time series analysis, wavelet analysis has triggered wide interest in localization spectrum analysis. This article focuses on the robust estimator and provides an approach to estimate the parameter with a contaminated distribution, in which we lessen the impact of the outliers and detect the eccentric signals with different scales. Similar to an ideal high pass filter, we estimate the model with an integrable spectrum by wavelet coefficients in mild condition rather than whether the spectrum of the original data at origin is bounded or not. Simultaneously, we tune the location for the investigated spectrum by changing the scaling level. In other words, though it is a high pass filter, we still can tune the location at low frequencies, which is conventionally applied in a long-run estimation. When the model extends to a non-stationary process, we can estimate without prior knowledge whether the process is stationary or non-stationary. A well-known wavelet piecewise B-spline is illustrated for estimation. Through Monte Carlo experiments the consistency of the estimator with a pseudo likelihood function, which is used to estimate a non-stationary process, is examined by a growing time length. Through a result analysis of the power tests we see that the estimator is quite successful and powerful.
