Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes |
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Authors: | Manfred Mudelsee |
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Institution: | 1. Institute of Meteorology, University of Leipzig, Stephanstraβe 3, 04103, Leipzig, Germany
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Abstract: | We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use
over the entire range of the autocorrelation coefficient ρ. The least-squares estimator ∑
n
−1
i
=1ε
i
ε
i
+1 / ∑
n
−1
i
=1ε2
i
is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ
the arithmetic-geometric series instead of replacing partial Cesàro sums. In case of the mean we derive Marriott and Pope's
(1954) formula, with (n− 1)−1 instead of (n)−1, and an additional term α (n− 1)−2. This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett's
(1946) formula results, with (n− 1)−1 instead of (n)−1. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean
is more accurate than the higher-order approximation of White (1961), for |ρ| > 0.88 and n≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes.
Received: November 30, 1999; revised version: July 3, 2000 |
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Keywords: | |
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