4) Hence, from Equation (20), the geometric average is always biased downward for [Rho] [less than] 0, even as N [approaches] T.
As N [approaches] T, more weight is given to the geometric average.
Notice from Table 1 that for any investment horizon and stationary variance, the geometric average is always biased downward.
Overall, the geometric average is the most efficient estimator, and the overlapping average is the least efficient.
Here, we describe the return generating process and derive the biases in the arithmetic and geometric averages.
We use simulations to assess the severity of the biases in the arithmetic and geometric averages.
If we compare both Panel A's in Tables 1 and 2, we see that the arithmetic and geometric averages are more upward- and less downward-biased, respectively, and that both averages are less efficient.
We show that both the arithmetic and geometric averages are biased estimates of long-run expected returns, and the bias increases with the length of the investment horizons.
The horizon-weighted average of the arithmetic and geometric averages, proposed by Blume (1974), is an alternative estimate of long-run expected returns.
After studying mean value of factors involved in water resources deterioration, it's indicated that ground water table decrease index with a geometric average of 3.
Analyzing the mean value of three effective indices on soil degradation presents that soil EC index is the most effective factor in increasing soil degradation intensity by the geometric average.
Finally the value of each criterion was obtained as geometric average of scores of single indices according to the formula: