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?