Test and Estimation for the Structual Change in Econometric Model

得到了独立正态分布随机变量随机停止序列的截断停止时间期望值和样本均值的渐近展开式。该结果可用于获得Cusum检验的统计特征，用于检测平均变化，如误差概率和期望样本量。这些结果可以用来估计在拒绝无平均变化的原假设之后发生变化的时间。我们还对伽马-正态混合模型应用了Cusum检验来检测均值变化，该模型适用于尾部较重的数据，而不是正态分布。最近对金融时间序列的实证研究表明，底层分布趋向于具有较大的峰度。由于方差在实际应用中可能更为重要，我们还应用该方法获得了类似的序列过程来检测方差变化。作为第一步，我们使用布朗运动近似来获得概率。我们现在正努力用（非线性）更新理论来得到更精确的结果，我们将用它来计算边界上的超量的渐近分布。我们相信，在检测独立随机变量序列的均值变化方面，我们已经取得了某种令人满意的结果。但是对于时间序列数据的研究才刚刚开始，还没有得到令人满意的结果。虽然在方差变化的检测方面取得了一些成果，但距离最终目标还有很长的路要走。

We obtained the asymptotic expansions of the expected values of the truncated stopping time and the sample mean of the randomly stopped sequence of independent normally distributed random variables. The results may be applied to obtain the statistical characteristics of the Cusum test for detecting the mean changes, such as error probabilities and the expected sample sizes. The results may be used to estimate the time at which the change occurs after having rejected the null hypothesis of no mean changes. We also applied the Cusum test for the detection of the mean change to the Gamma-mormal mixture model which is suitable for the data with heavier tail than that of the normal distribution. And recent empirical studies of the Financial Time Series show that the underlying distribution tends to have bigger kurtosis. We also applied the method to obtain the similar sequential procedure for detecting the variance change, for the variance may be more important in practice. We used Brownian motion approximation to obtain the probabilities as a first step. We are now working to get the more accurate results by using the (nonlinear) renewal theory with which we will evaluate the asymptotic distribution of the excess over the boundaries. We believe that we have obtained somehow satisfactory results for the detection of the mean change of the sequence of the independent random variables. But the case for the time series data is just started and have not got any satisfactoryresults yet. Although some results concerning the detection of the change of variance was obtained, we are still long way up to the final goal.