(a) For each of the four AR(2) time series displayed in plots (a) to (d) of Figure 34 (downloadable from the “Data” part of the website for the book), compute MSE(m), NMSE(m), MSLE(m), KL(m), LK(m) and KLLK(m) as per Equations (293), (296a), (296b), (297c), (345b) and (345c) for Parzen lag window estimates with m = 16, 32 and finally 64. Compare these measures to their corresponding expected values (see Equations (296d), (296c) and (297d) and parts (c) and (d) of Comment briefly on your findings.
(b) Generate a large number, say NR, of simulated time series of length N = 1024 from the AR(2) process of Equation (34) (take “large” to be at least 1000 – see Exercise [597] for a description of how to generate realizations from this process). For each simulated series, compute MSE(m) for Parzen lag window estimates with m = 16, 32 and finally 64. Do the same for NMSE(m), MSLE(m), KL(m), LK(m) and KLLK(m). Create a scatterplot of the NR values for NMSE(16) versus those for MSE(16) and then also for m = 32 and m = 64. Do the same for the remaining 14 pairings of the six measures. For a given m, compute the sample correlation coefficients corresponding to the 15 scatterplots. The sample correlation coefficients for one of the 15 pairings should be almost indistinguishable from unity for all three settings of m – offer an explanation as to why this happens. For each m, average all NR values of MSE(m). Compare this average with its expected value. Do the same for the remaining five measures. Comment briefly on your findings.