The Garman-Klass unbiased estimator of the variance per unit time of zero-drift Brownian Motion, is quadratic in the range-based financial-type data CLOSE-OPEN, MAX-OPEN, OPEN-MIN reported on regular time windows. Its variance, 7.4 times smaller than that of the common estimator (CLOSE-OPEN)2, is widely believed to be the minimal possible variance of unbiased estimators. The current report disproves this belief by exhibiting an unbiased estimator in which 7.4 becomes 7.7322. The essence of the improvement lies in data compression to a more stringent sufficient statistic. The Maximum Likelihood Estimator, known to be more efficient, attains asymptotically the Craḿer-Rao upper bound 8.471, unattainable by unbiased estimators because the distribution is not of exponential type. Beyond Brownian Motion, regression-fitted (mean-1) quadratic functions of the more stringent statistic increasingly out-perform those of CLOSE-OPEN, MAX-OPEN, OPEN-MIN when applied to random walks with heavier-tail distributed increments.
|Number of pages||14|
|Journal||Revstat Statistical Journal|
|State||Published - Nov 2011|
- Brownian motion