order_statistics {cpr}

Distribution of Order Statistics

Description

Density of distribution function for the jth order statistics from a sample of size n from a known distribution function.

Usage

d_order_statistic(x, n, j, distribution, ...)

p_order_statistic(q, n, j, distribution, ...)

Arguments

x, q

vector or quantiles

n

sample size

j

jth order statistics

distribution

character string defining the distribution. See Details.

...

additional arguments passed to the density and distribution function

Details

For a known distribution with defined density and distribution functions, e.g., normal (dnorm, pnorm), or chisq (dchisq, pchisq), we define the density function of of the jth order statistic, from a sample of size n, to be

\frac{n!}{(j-1)!(n-j)!} f(x) F(x)^{j-1} (1 - F(x))^{n-j}

.

and the distribution function to be

\sum_{k = j}^{n} \binom{n}{k} \left[F(x)\right]^{k} \left[1-F(x)\right]^{n-k}

.

Value

a numeric vector

References

George Casella and Roger L. Berger (2002). Statistical Inference. 2nd edition. Duxbury Thomson Learning.

Examples

# Example 1
# Find the distribution of the minimum from a sample of size 54 from a
# standard normal distribution

simulated_data <- matrix(rnorm(n = 54 * 5000), ncol = 54)

# find all the minimums for each of the simulated samples of size 54
mins <- apply(simulated_data, 1, min)

# get the density values
x <- seq(-5, 0, length.out = 100)
d <- d_order_statistic(x, n = 54, j = 1, distribution = "norm")

# plot the histogram and density
hist(mins, freq = FALSE)
points(x, d, type = "l", col = "red")

# plot the distribution function
plot(ecdf(mins))
points(x, p_order_statistic(q = x,  n = 54, j = 1, distribution = "norm"), col = "red")

# Example 2
# Find the density and distrubition of the fourth order statistic from a
# sample of size 12 from a chisq distribution with 3 degrees of freedom

simulated_data <- matrix(rchisq(n = 12 * 5000, df = 3), ncol = 12)

os4 <- apply(simulated_data, 1, function(x) sort(x)[4])

x <- seq(min(os4), max(os4), length.out = 100)
d <- d_order_statistic(x, n = 12, j = 4, distribution = "chisq", df = 3)
p <- p_order_statistic(x, n = 12, j = 4, distribution = "chisq", df = 3)

hist(os4, freq = FALSE); points(x, d, type = "l", col = "red")
plot(ecdf(os4)); points(x, p, col = "red")

# Example 3
# For a set of j observations, find the values for each of the j order
# statistics
simulated_data <- matrix(rnorm(n = 6 * 5000), ncol = 6)
simulated_data <- apply(simulated_data, 1, sort)
xs <- apply(simulated_data, 1, range)
xs <- apply(xs, 2, function(x) {seq(x[1], x[2], length.out = 100)})
ds <- apply(xs, 1, d_order_statistic, n = 6, j = 1:6, distribution = "norm")
ps <- apply(xs, 1, p_order_statistic, n = 6, j = 1:6, distribution = "norm")

old_par <- par() # save current settings

par(mfrow = c(2, 3))
for (i in 1:6) {
  hist(simulated_data[i, ]
       , freq = FALSE
       , main = substitute(Density~of~X[(ii)], list(ii = i))
       , xlab = ""
  )
  points(xs[, i], ds[i, ], type = "l", col = "red")
}

for (i in 1:6) {
  plot(ecdf(simulated_data[i, ])
       , main = substitute(CDF~of~X[(ii)], list(ii = i))
       , ylab = ""
       , xlab = ""
       )
  points(xs[, i], ps[i, ], type = "p", col = "red")
}

par(mfrow = c(1, 1))
plot(xs[, 1], ps[1, ], type = "l", col = 1, xlim = range(xs), ylab = "", xlab = "")
for(i in 2:6) {
  points(xs[, i], ps[i, ], type = "l", col = i)
}
legend("topleft", col = 1:6, lty = 1, legend =
       c(
         expression(CDF~of~X[(1)]),
         expression(CDF~of~X[(2)]),
         expression(CDF~of~X[(3)]),
         expression(CDF~of~X[(4)]),
         expression(CDF~of~X[(5)]),
         expression(CDF~of~X[(5)])
         ))

par(old_par) # reset par to setting prior to running this example