Nonparametric Models
Last modified: 2026-11-07 21:34:29 (PST)
[Code]
x <- c(4, 1, 7, 3)
n <- length(x)
x_ord <- sort(x)
p_ord <- seq_len(n) / n
ecdf_x_padding <- 0.8
x_min <- min(x_ord) - ecdf_x_padding
x_max <- max(x_ord) + ecdf_x_padding
x_left <- c(x_min, x_ord)
x_right <- c(x_ord, x_max)
f_levels <- c(0, p_ord)
ecdf_pieces <- tibble::tibble(x = x_left, xend = x_right, y = f_levels)
ecdf_open <- tibble::tibble(x = x_ord, y = f_levels[seq_len(n)])
ecdf_closed <- tibble::tibble(x = x_ord, y = f_levels[seq_len(n) + 1])
ggplot2::ggplot() +
ggplot2::geom_segment(
ggplot2::aes(x = x, xend = xend, y = y, yend = y),
data = ecdf_pieces, linewidth = 1, color = "blue"
) +
ggplot2::geom_point(
ggplot2::aes(x = x, y = y), data = ecdf_open, shape = 1, color = "blue"
) +
ggplot2::geom_point(
ggplot2::aes(x = x, y = y), data = ecdf_closed, shape = 19, color = "blue"
) +
ggplot2::coord_cartesian(xlim = c(x_min, x_max), ylim = c(0, 1.05)) +
ggplot2::labs(
x = "Value, t",
y = expression("Empirical CDF, " * hat(plain("F"))(t))
) +
ggplot2::theme_minimal()
[Code]
eqf_y_padding <- 0.5
q_left <- c(0, p_ord[-n])
q_right <- p_ord
eqf_pieces <- tibble::tibble(x = q_left, xend = q_right, y = x_ord)
ggplot2::ggplot(eqf_pieces) +
ggplot2::geom_segment(
ggplot2::aes(x = x, xend = xend, y = y, yend = y),
linewidth = 1, color = "blue"
) +
ggplot2::geom_point(ggplot2::aes(x = x, y = y), shape = 1, color = "blue") +
ggplot2::geom_point(
ggplot2::aes(x = xend, y = y), shape = 19, color = "blue"
) +
ggplot2::coord_cartesian(
xlim = c(0, 1),
ylim = c(min(x_ord) - eqf_y_padding, max(x_ord) + eqf_y_padding)
) +
ggplot2::labs(
x = "Probability, p",
y = expression("Sample quantile, " * hat(Q)(p))
) +
ggplot2::theme_minimal()
Both functions in Figure 1 are step functions, so neither is one-to-one. Each flat piece of one function corresponds to a jump of the other: for example, \(\hat F(t) = 0.5\) for every \(t \in [3, 4)\), and \(\hat Q\) jumps from 3 to 4 at \(p = 0.5\). So \(\hat Q\) is a generalized inverse of \(\hat F\), not an inverse in the usual sense: \(\hat Q(\hat F(t)) \le t\) for every \(t \ge x_{(1)}\), with equality only at the observed values.
These readings look ahead from the empirical CDF to distribution-free tests, which this page does not cover:
- Hogg et al. [7, Sec. 8.5], pp. 392–402 covers some distribution-free tests.
- Rice [5, Sec. 11.2.3], pp. 435–442 covers the Mann–Whitney test.
- Rice [5, Sec. 11.3.2], pp. 448–449 covers the signed rank test.
- Rice [5, Sec. 12.2.3], p. 488 covers the Kruskal–Wallis test.