stats (version 3.3.2)

ecdf: Empirical Cumulative Distribution Function

Description

Compute an empirical cumulative distribution function, with several methods for plotting, printing and computing with such an “ecdf” object.

Usage

ecdf(x)
# S3 method for ecdf
plot(x, …, ylab="Fn(x)", verticals = FALSE,
     col.01line = "gray70", pch = 19)
# S3 method for ecdf
print(x, digits= getOption("digits") - 2, …)
# S3 method for ecdf
summary(object, …)
# S3 method for ecdf
quantile(x, …)

Arguments

x, object

numeric vector of the observations for ecdf; for the methods, an object inheriting from class "ecdf".

…

arguments to be passed to subsequent methods, e.g., plot.stepfun for the plot method.

ylab

label for the y-axis.

verticals

see plot.stepfun.

col.01line

numeric or character specifying the color of the horizontal lines at y = 0 and 1, see colors.

pch

plotting character.

digits

number of significant digits to use, see print.

Value

For ecdf, a function of class "ecdf", inheriting from the "stepfun" class, and hence inheriting a knots() method. For the summary method, a summary of the knots of object with a "header" attribute. The quantile(obj, ...) method computes the same quantiles as quantile(x, ...) would where x is the original sample.

Details

The e.c.d.f. (empirical cumulative distribution function) \(F_n\) is a step function with jumps \(i/n\) at observation values, where \(i\) is the number of tied observations at that value. Missing values are ignored. For observations x\(= (\)\(x_1,x_2\), … \(x_n)\), \(F_n\) is the fraction of observations less or equal to \(t\), i.e., $$F_n(t) = \#\{x_i\le t\}\ / n = \frac1 n\sum_{i=1}^n \mathbf{1}_{[x_i \le t]}.$$ The function plot.ecdf which implements the plot method for ecdf objects, is implemented via a call to plot.stepfun; see its documentation.

See Also

stepfun, the more general class of step functions, approxfun and splinefun.

Examples

##-- Simple didactical  ecdf  example :
x <- rnorm(12)
Fn <- ecdf(x)
Fn     # a *function*
Fn(x)  # returns the percentiles for x
tt <- seq(-2, 2, by = 0.1)
12 * Fn(tt) # Fn is a 'simple' function {with values k/12}
summary(Fn)
##--> see below for graphics
knots(Fn)  # the unique data values {12 of them if there were no ties}

y <- round(rnorm(12), 1); y[3] <- y[1]
Fn12 <- ecdf(y)
Fn12
knots(Fn12) # unique values (always less than 12!)
summary(Fn12)
summary.stepfun(Fn12)

## Advanced: What's inside the function closure?
ls(environment(Fn12))
##[1] "f"  "method"  "n"  "x"  "y"  "yleft"  "yright"
utils::ls.str(environment(Fn12))
stopifnot(all.equal(quantile(Fn12), quantile(y)))

###----------------- Plotting --------------------------
require(graphics)

op <- par(mfrow = c(3, 1), mgp = c(1.5, 0.8, 0), mar =  .1+c(3,3,2,1))

F10 <- ecdf(rnorm(10))
summary(F10)

plot(F10)
plot(F10, verticals = TRUE, do.points = FALSE)

plot(Fn12 , lwd = 2) ; mtext("lwd = 2", adj = 1)
xx <- unique(sort(c(seq(-3, 2, length = 201), knots(Fn12))))
lines(xx, Fn12(xx), col = "blue")
abline(v = knots(Fn12), lty = 2, col = "gray70")

plot(xx, Fn12(xx), type = "o", cex = .1)  #- plot.default {ugly}
plot(Fn12, col.hor = "red", add =  TRUE)  #- plot method
abline(v = knots(Fn12), lty = 2, col = "gray70")
## luxury plot
plot(Fn12, verticals = TRUE, col.points = "blue",
     col.hor = "red", col.vert = "bisque")

##-- this works too (automatic call to  ecdf(.)):
plot.ecdf(rnorm(24))
title("via  simple  plot.ecdf(x)", adj = 1)

par(op)