2. Locating a Shift in the Mean of a Time Series Melvin J. Hinich Applied Research Laboratories University of Texas at Austin hinich@mail.la.utexas.edu www.la.utexas.edu/~hinich

3. Localizing a Single Change in the Mean A statistical uncertainty principle for the localization of a single change in the The smallest mean squared error for any estimate of the time of change of a bandlimited stationary random process mean GOAL

4. Discrete-Time Sampling It is common in time series analysis to begin Then decimate the filtered output with a discrete-time sample of the time series to obtain the discrete-time sample Apply a linear bandlimited filter to the signal

5. Linear Bandlimited Filter The filter is linear and causal The filter smoothes the input since the filter removed frequency components of the input for. The filter impulse response function is

6. Mean Shift If the mean of the signal has an abrupt shift from at an unknown time The shift in the mean of the output is

7. Integrated Impulse Response

8. Ideal Bandpass Filter Impulse response of the ideal filter - sinc function

9. Meanshift The shift in the mean of x(t n ) is We will now derive the least squares estimate of the location of the shift for

10. Maximum Likelihood Estimate - the least squares estimate of i.i.d. gaussian variates with variance is the value that maximizes the statistic

11. Least Squares Estimate The least squares estimate of  o is the value that maximizes The standard deviation of the estimate is approximately

12. Asymptotic Standard Deviation E - the total energy of the white noise Area under its bandlimited white noise spectrum

13. Hinich Test for a Changing Slope Parameter