廣義極端值分佈(generalized extreme value distribution)模型廣泛應用於分析各類極端風險事件，但遇到實際的極端值資料可能有斷點(breakpoint)時，研究者需要自行假設斷點位置，將分段常函數(piecewise constant function)置入廣義極端值分佈模型之參數函數內進行參數估計，本研究藉由：方法一、利用逐段迴歸模型(piecewise regression model)最佳化候選斷點模型之目標函數的概念，以廣義極端值分佈的最大概似估計(maximum likelihood estimation)之負對數概似函數為目標函數，在單一斷點假設下建構斷點位置之廣義極端值分佈模型參數估計方法；方法二、藉由模型選擇準則比較含斷點模型與對應無斷點模型分辨模型有無斷點。以此兩方法發展毋需假設斷點位置之廣義極端值分布模型估計，並使用蒙地卡羅方法(Monte Carlo method)，以四種不同位置參數假設下的廣義極端值分佈模型為對象，模擬斷點位置估計值之平均值與均方差，評估方法一之斷點位置參數估計之表現；並計算兩常用模型選擇準則 AIC、BIC 自候選模型選擇正確模型的比例，評估方法二之準則分辨模型有無斷點之表現。

Generalized extreme value distribution is widely used in extreme value analysis. When extreme value data might has breakpoint(change-point), for estimate, we need assuming the location of breakpoint as a known parameter of piecewise constant function and form this function into parameter function of distribution. In this artical we try to approve breakpoint estimate in generalized extreme value distribution. First, by copy idea from piecewise regression, use maximized likelihood in maximum likelihood estimation of all posiable breakpoint as optimize function to estimatie parameter of breakpoint location. Second, use model selection criteria like AIC or BIC to considered the model should have breakpoint or not. To evaluate this two method we selete fore location parameter function as possiable model and use Monte Carlo method. Simulate estimate breakpoint location and investigation mean and mean square error of simulation to evaluate breakpoint location estimation, then use right model selection rate by AIC or BIC to evaluate model selection criteria in breakpoint model.

Akaike, H. (1973). Information Theory and an Extension of the Maximum Likelihood Principle. In Selected Papers of Hirotugu Akaike (pp. 199–213). New York, NY: Springer New York.

Avanesov, V. (2019). Nonparametric Change Point Detection in Regression. arXiv: 1903.02603 [Math, Stat]. Retrieved from http://arxiv.org/abs/1903.02603

Brodsky, E., & Darkhovsky, B. S. (1993). Nonparametric Methods in Change Point Problems. Springer Netherlands.

Chen, S., Li, Y., Kim, J., & Kim, S. W. (2017). Bayesian change point analysis for extreme daily precipitation. International Journal of Climatology, 37(7), 3123–3137. https://doi.org/10/gbnrv8

Cheng, L., AghaKouchak, A., Gilleland, E., & Katz, R. W. (2014). Non-stationary extreme value analysis in a changing climate. Climatic Change, 127(2), 353–369. https://doi.org/10.1007/s10584-014-1254-5

Coles, S. (2001). An introduction to statistical modeling of extreme values. London: Springer.

Dierckx, G., & Teugels, J. L. (2010). Change point analysis of extreme values. Environmetrics, 21(7-8), 661–686. https://doi.org/10/ck6w6p

Ferrari Trecate, G., & Muselli, M. (2002). A New Learning Method for Piecewise Linear Regression (Vol. 2415). https://doi.org/10.1007/3-540-46084-5_72

Gill, R. D. (2014, January). File:GevDensity 2.Svg. wikimedia. https://commons.wikimedia.org/wiki/File:GevDensity_2.svg.

Kim, H., Kim, S., Shin, H., & Heo, J.-H. (2017). Appropriate model selection methods for nonstationary generalized extreme value models. Journal of Hydrology, 547, 557–574. https://doi.org/10.1016/j.jhydrol.2017.02.005

McGee, V. E., & Carleton, W. T. (1970). Piecewise Regression. Journal of the American Statistical Association, 65(331), 1109–1124. https://doi.org/10/dhgwrd

Mohtadi, H., & Murshid, A. P. (2009). Risk of catastrophic terrorism: An extreme value approach. Journal of Applied Econometrics, 24(4), 537–559. https://doi.org/10.1002/jae.1066

Nelder, J. A., & Mead, R. (1965). A Simplex Method for Function Minimization. The Computer Journal, 7(4), 308–313. https://doi.org/10/bkqf

Oosterbaan, R. J., Sharma, D. P., Singh, K. N., & Rao, K. V. G. K. (1990). Crop production and soil salinity: Evaluation of field data from India by segmented linear regression. Symposium on Land Drainage for Salinity Control. Schwarz, G. (1978). Estimating the Dimension of a Model. The Annals of Statistics, 6(2), 461–464. https://doi.org/10/d9mzdb

Stephen, K. (2017, July). Piecewise regression: When one line simply isn’t enough. Piecewise regression: when one line simply isn’t enough. https://www.datadoghq.com/blog/engineering/piecewise-regression/.

Yang, X., Yang, H., Zhang, F., Zhang, L., Fan, X., Ye, Q., & Fu, L. (2019). Piecewise Linear Regression Based on Plane Clustering. IEEE Access, 7, 29845–29855. https://doi.org/10/gf4gv6