Introduction

In this chapter, we delve into the concept of the quantile function and its significance in statistical analysis. The quantile function provides valuable insights into the distribution of a random variable and plays a crucial role in constructing quantile-based estimators. We’ll explore how quantiles help us summarize and interpret data, and we’ll discuss the application of quantile-based methods in various scenarios.

Quantile Function: Distribution Percentiles

The quantile function, also known as the inverse cumulative distribution function (CDF), is a powerful tool used to characterize the distribution of a random variable. It provides a way to determine the value below which a certain proportion of observations fall. Mathematically, for a given probability value $0<p<1$, the $p$-th quantile $Q(p)$ is defined as the value $x$ for which the cumulative distribution function $F(x)$ equals $p$ :

$$Q(p)=F^{-1}(p)$$

For instance, the median of a distribution corresponds to the 50th quantile, as it’s the value that divides the data into two halves.

Quantile-Based Estimators: Robust Measures of Location

Quantile-based estimators provide robust alternatives to traditional mean-based estimators. Instead of relying on the mean, which is sensitive to outliers, quantile-based estimators focus on specific quantiles of the distribution. For example, the first quartile (​$Q_1$) is a robust measure of central tendency that represents the value below which 25% of the data lies.

Quantile-based estimators are particularly useful when dealing with skewed or heavy-tailed distributions. They provide a more accurate representation of the center and spread of the data in such cases.

Application in Statistical Inference

Quantile-based methods play a significant role in statistical inference. Confidence intervals constructed using quantiles offer robust estimates for population parameters, even in the presence of outliers. Moreover, quantile regression is a powerful technique for modeling the conditional distribution of a response variable.

Example: Consider a dataset representing household incomes in a region. Calculating the median (50th quantile) of this dataset provides a more reliable measure of central tendency than the mean, as outliers have less impact on the median.