Geometric mean appropriateness with bimodally distributed data

Question

I am trying to find out whether the performance of the geometric mean of a distribution as a measure of its central tendency would be impaired by the distribution being multimodal.

For example, imagine I want to estimate the most likely central value for the probability distribution of stock market returns, which is unknown. I draw a random sample of equities in order to use sample moments to estimate the population moments. I can do this using sample metrics such as arithmetic / geometric mean, median, mode etc.

If I use a geometric mean to estimate the most likely central value for the population pdf, and I have bimodal / multimodal data, will the geometric mean of that sample be more biased as an estimate of the central tendency of the population than it would be were the data unimodal? For instance, I know that the arithmetic mean of both unimodal and bimodal samples here would produce the same value. What if the whole population followed a bimodal distribution? Would a geometric mean be inappropriate to use to estimate population mean from a sample mean in this case?

Sorry if this is unclear. I am still new to statistics.

Answer 1

I am new to statistics myself.

Here's a numeric example pinpointing the constituents of above question:

import numpy as np
import pandas as pd
from scipy.stats import gmean

s = pd.Series(np.r_[np.random.normal(5, 1, size=1000),
                    np.random.normal(13, 2, size=1000)], name='f(x)')
ax = s.plot.hist(bins=50)
_, ymax = ax.get_ylim()
ax.axvline(s.mean(), ymax=ymax, color='red', label='arithmetic mean')
ax.axvline(gmean(s), ymax=ymax, color='blue', label='geometric mean')
ax.legend()

While in the above example geometric mean tends closer to the sample distribution peak, it is certainly unwise to use it as a general replacement for $E[X]$ or $\mu$ in equations, unless you already know what you are doing.