Ad-Plot and Ud-Plot

Introduction

Two statistical plots Ad- and Ud-plots derived from the empirical cumulative average deviation function (ecadf), \(U_n(t)\), which is introduced by Wijesuriya (2025) will be illustrated with examples. Suppose that \(X_1,X_2,...,X_n\) is a random sample from a unimodal distribution. Then for a real number \(t\), the ecadf computes the sum of the deviations of the data that are less than or equal to \(t\), divided by the sample size \(n\). The Ad-plot detects critical properties of the distribution such as symmetry, skewness, and outliers of the data. The Ud-plot, a slight modification of the Ad-plot, is prominent on assessing normality. To create these visualizations, let adplots package be installed, including stats and ggplot2 in R.

library(adplots)

Ad-Plot

The first exhibit, Ad-plot consists of \(n\) ordered pairs of the form \((x_i,U_n(x_i))\) for \(i=1,2,...,n\).

Example 1

set.seed(2025)
X<-matrix(rnorm(100, mean = 2 , sd = 5))
adplot(X, title = "Ad-plot", xlab = "x", lcol = "black", rcol = "grey60")

Figure 1. Both left and right points from the sample average in Ad-plot are evenly distributed and hence, it indicates the symmetric property of the data distribution. Further, the leftmost data point appears to be an outlier.

Example 2

set.seed(2025)
X<-matrix(rbeta(100, shape1 = 10, shape2 = 2))
adplot(X, title = "Ad-plot", xlab = "x", lcol = "black", rcol = "grey60")

Figure 2. The points below the average have a wide spread in Ad-plot. Thus, the data distribution is apparently left-skewed.

Example 3

set.seed(2025)
X<-matrix(rf(100, df1 = 10, df2 = 5))
adplot(X, title = "Ad-plot", xlab = "x", lcol = "black", rcol = "grey60")

Figure 3. The points situated above the average have a wide spread in Ad-plot in contrast to Figure 2. Thus, the data distribution is apparently right-skewed.

Ud-Plot

Suppose that the random sample is from a normal distribution with mean \(\mu\) and variance \(\sigma^2\). Then the second illustration, Ud-plot consists of \(n\) ordered pairs of the form \((x_i,U_n(x_i)/[(1-n^{-1})s^{2}])\) for \(i=1,2,...,n\), where \(s^2\) is the sample variance.

Example 4

set.seed(2030)
X<-matrix(rnorm(30, mean = 2, sd = 5))
udplot(X, npdf = FALSE, lcol = "black", rcol = "grey60", pdfcol = "red")

Figure 4. The points in Ud-plot follow a bell-shaped curve evenly distributed about the sample average. Thus, it captures the symmetric property of the data distribution and confirms normality.

Example 5

set.seed(2030)
X<-matrix(rnorm(30, mean = 2, sd = 5))
udplot(X, npdf = TRUE, lcol = "black", rcol = "grey60", pdfcol = "red")

Figure 5. The points in the Ud-plot closely follow the estimated normal density curve, indicating that the data are from normal distribution with mean \(\mu\) and variance \(\sigma^2\) estimated by sample average \(\bar{X}\) and variance \(s^2\), respectively. Further, the \(d\)-value confirms the degree of proximity of Ud-plot to the estimated normal density curve.

Example 6

set.seed(2030)
X<-matrix(rnorm(2025, mean = 2, sd = 5))
udplot(X, npdf = TRUE, lcol = "black", rcol = "grey60", pdfcol = "red")

Figure 6. Ud-plot is indistinguishable from the estimated normal density curve as sample size increases with a higher degree of proximity.