Question: For two joint normally distributed time series \(\mathrm{x}_{\mathrm{i}}\) and \(\mathrm{y}_{\mathrm{t}}\) which have a record length \(\mathrm{N}\) which is long compared to the auto and cross correlation time scales of \(\mathrm{x}\) and \(\mathrm{y}\), the effective number of independent samples (effective degrees of freedom) is

where \(\mathrm{I}_{\mathrm{xx}}\) and \(\mathrm{T}_{\mathrm{yy}}\) are the autocorrelation and \(\mathrm{I}_{\mathrm{xy}}\) and \(\mathrm{r}_{\mathrm{yx}}\) are the cross correlations. The autocorrelations for a stationary \(1^{\text {st }}\) order Gauss-Markov process with white noise are given by

and

where \(1 / \beta_{\mathrm{x}}\) and \(1 / \beta_{\mathrm{y}}\) are the correlation times for the data \(\mathrm{x}\) and \(\mathrm{y}\) and \(\mathrm{t}\) is the time lag.

1. Assuming that \(\mathrm{x}_{\mathrm{t}}\) and \(\mathrm{y}_{\mathrm{t}}\) have the auto correlations given above, but that \(\mathrm{x}_{\mathrm{t}}\) and \(\mathrm{y}_{\mathrm{t}}\) are not well correlated with each other evaluate an expression for \(\mathrm{N}^{*}\) in terms of \(\mathrm{N}, \beta_{\mathrm{x}}\), and \(\beta_{y}\). (Hint: Use the geometric series)
2. If a parameter is estimated by the data \(x_{t}\) and \(y_{t}\) each with a record size \(\mathrm{N}\) and assumed to be normally distributed, what is the effective reduction in the confidence interval for a parameter estimate due to the auto correlations of \(\mathrm{x}_{\mathrm{t}}\) and \(\mathrm{y}_{\mathrm{t}}\) if \(1 / \beta_{\mathrm{x}}=3\) and \(1 / \beta_{\mathrm{y}}=5 ?\) (Hint: You don't need to use a table.)

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