# Are sample means for quantiles of sorted data unbiased estimators of the true means?

In studying income inequality, it is very common to look at sample means for deciles or quintiles of the sample, and to assume that the sample means are good estimators of the true means. In this setting, the "deciles" and "quintiles" normally refer, not to the break points, but to the sets of observations divided by the break points.

Suppose that the income values are observed with error, and that the error, or more probably, the percentage error, is independent of the true value.

• Is the mean of the sample quantile, e.g., the top decile, an unbiased estimator of the population mean? I know that with some leptokurtic distributions (e.g., the Pareto) the sample mean understates the population mean. My question refers, not to this, but to any bias that might be induced by the sorting process, because one sorts on the observed values including error, rather than on the true values.

• My intuition is that sample mean of the highest decile/quintile would be biased upwards, because positive errors would be sorted up, and conversely for the lowest. For instance, it seems to me that if incomes were non-negative but observed with a normal error, a large sample would contain some negative values, and a fine enough quantile would gather these negative values together into a lowest group with a negative mean, demonstrating bias since the true mean must be positive. Is this true?

• If the means are biased, is there any good way of correcting for this bias? Am I correct in thinking that the error's independence of the true value does not carry over to independence of the error from the observed value, which includes the error? If so, is there an easy way to at least characterize, and ideally correct, that dependence?

• A commonly used index of inequality is the ration of mean of the incomes in the highest quintile or decile to the mean of the incomes in the lowest. If these means are biased, and the bias is corrected, will the resulting ratios be unbiased estimators of the true ratios?

• "I know that with some leptokurtic distributions (e.g., the Pareto) the sample mean understates the population mean." ... can you demonstrate this theorem, or link to it? it seems to violate properties of expectations. – Glen_b Aug 14 '15 at 12:02

## 1 Answer

For some distributions there is a positive bias due to measurement errors. If you assume the noise has mean $0$, then if you sample people from the top decile, their average measured income will be the average income of the top decile. However, the top decile of your sample will include some people who have displaced people from the top decile. The difference between the measured incomes of the incorrectly included people and the displaced people is always nonnegative, and the average value of this indicates the bias from this source of error.

For some distributions, there is a negative bias due to sampling. I think this is a rare situation which you may be able to ignore based on some assumptions about the income distribution and noise distribution. Here is an artificial distribution which exhibits such a negative bias: Suppose $11\%$ of the population has a job and an income of $1$ unit, while everyone else is unemployed with an income of $0$, and there is no noise. The average income of the top $10\%$ is $1$, but there is a chance that the employment rate in your sample is under $10\%$, so the expected income of the top decile of a sample is less than $1$, so the bias is negative.

If you want to get a ballpark estimate for the size of the bias, you can do a Monte Carlo simulation based on a distribution you fit to your sample and model for noise. There might be more accurate techniques, but this should be fast.

• I've done some numerical experiments that persuade me that the bias in quantile means at the top and bottom is real but probably not very large. But it would certainly be reassuring to get an analytic description of the bias, so that I could know under what circumstances, if any, I should be worrying about it. I am pretty sure that the bias increases when the average absolute error is a larger percentage of the value of the variables. – andrewH Jul 30 '12 at 2:17
• I'm afraid that I do not know how to estimate the error process in this situation. I do not have a model that I am fitting -- I'm just taking mean values for, say, deciles, as descriptive statistics. I could make assumptions about the underlying process, that the data is drawn from a lognormal or gamma or whatever - I think the Generalized Beta type II is thought to be best fitting on U.S. data, at least for the high end of the income spectrum. I would know how to estimate the parameters of the distribution from the sample, but not the parameters of the error process. – andrewH Jul 30 '12 at 2:31