Confidence Intervals for Quantiles from Histograms and Other Grouped Data

TL;DR

This paper develops methods for estimating confidence intervals for quantiles using only grouped data, such as histograms, by approximating the underlying density with the Generalized Lambda Distribution, and validates these methods through simulations and real data applications.

Contribution

Introduces novel methods for confidence interval estimation of quantiles from grouped data, especially histograms, using the Generalized Lambda Distribution and other approximation techniques.

Findings

Methods achieve excellent coverage across various distributions.

Simulations confirm robustness for skewed distributions like log-normal and Dagum.

Applicable to real-world data summarized by histograms.

Abstract

Interval estimation of quantiles has been treated by many in the literature. However, to the best of our knowledge there has been no consideration for interval estimation when the data are available in grouped format. Motivated by this, we introduce several methods to obtain confidence intervals for quantiles when only grouped data is available. Our preferred method for interval estimation is to approximate the underlying density using the Generalized Lambda Distribution (GLD) to both estimate the quantiles and variance of the quantile estimators. We compare the GLD method with some other methods that we also introduce which are based on a frequency approximation approach and a linear interpolation approximation of the density. Our methods are strongly supported by simulations showing that excellent coverage can be achieved for a wide number of distributions. These distributions include highly-skewed distributions such as the log-normal, Dagum and Singh-Maddala distributions. We also apply our methods to real data and show that inference can be carried out on published outcomes that have been summarized only by a histogram. Our methods are therefore useful for a broad range of applications. We have also created a web application that can be used to conveniently calculate the estimators.