Working with Order Statistics in the mos Package

Introduction

The mos package provides a suite of tools to work with order statistics, including:

Generating random values of order statistics and censored/record samples,
Analytical computation of moments (mean, variance, skewness, kurtosis),
Simulation-based estimation of moments for some of continuous distributions.

These tools are useful for researchers in statistics, reliability, and simulation studies.

Generating Order Statistics

You can simulate order statistics from continuous distributions using the ros() function. This function accepts either:

A named distribution available in base R (e.g., “norm”, “exp”), or
A user-defined quantile function using the qf argument.

Example 1: Using a built-in distribution

ros(5, r = 2, n = 10, dist = "norm", mean = 0, sd = 1)
#> [1] -1.4874375 -2.1308462 -0.9017272 -0.5325201 -1.2554593

Example 2: Using a custom quantile function (e.g., Gumbel)

qgumbel <- function(p, mu = 0, beta = 1) mu - beta * log(-log(p))
ros(5, r = 3, n = 15, qf = "qgumbel", mu = 0, beta = 1)
#> [1] -0.6914274 -0.6539545 -0.3457394 -0.6877217 -1.1171822

Example 3: Generate multiple order statistics from the uniform distribution

ros(3, 1:5, 5, dist = "unif", min = 1, max = 5)
#>            r1       r2       r3       r4       r5
#> [1,] 1.714890 2.542624 3.671692 3.893994 4.907792
#> [2,] 1.034809 2.775691 3.127737 3.333873 4.308910
#> [3,] 1.377834 2.154740 2.353291 3.164256 3.973773

Moments of Order Statistics

The package provides two categories of moment computation:

Analytical formulas (closed-form), where available
Simulation-based estimation otherwise

Distributions with analytical support

The following distributions have closed-form expressions for moments:

Uniform
Exponential
Weibull
Symmetric Triangular
Topp-Leon
Complementary Beta (for integer b and k <= 2)

Distributions handled via simulation

Normal
Student’s t
Gamma
Beta
Pareto
Kumaraswamy
Complementary Beta (for non-integer b or k > 2)

In all simulation-based cases, the ros() function is used internally to generate the order statistics.

Example: Uniform Distribution (Analytical)

mo_unif(r = 2, n = 5, k = 1)  # Mean of 2nd order statistic
#> [1] 0.3333333

Example: Exponential Distribution (Analytical)

mo_exp(r = 3, n = 6, k = 2, mu = 0, sigma = 1)
#> [1] 0.5105556

Example: Normal Distribution (Simulated)

mo_norm(r = 2, n = 10, k = 1)
#> [1] -1.003343

Example: Beta Distribution (Simulated)

mo_beta(r = 2, n = 5, k = 2, a = 2, b = 3)
#> [1] 0.09487993

In addition to mo_*() functions, the package includes:

varOS() to compute variance
skewOS() to compute skewness
kurtOS() to compute kurtosis

Censored Samples

The mos package supports simulation of censored data using several standard censoring schemes:

Type I Censoring (Time-based)

Observations are censored beyond a fixed time limit.

rcens(n = 10, dist = "exp", type = "I", cens.time = 2, rate = 1)
#>  [1] 0.04965609 1.67364631 1.25681359 0.07160058 0.42816672 0.19903825
#>  [7] 0.02879249 1.12775815 0.20484918 0.32955892

Type II Censoring (Failure-based)

The smallest r values are observed (i.e., censoring after a fixed number of failures).

rcens(n = 10, r = 5, dist = "norm", type = "II", mean = 0, sd = 1)
#> [1] -0.47197938 -0.37010037 -0.08041435 -0.05771255  0.02607693

Progressive Type-II Censoring

Implements progressive removal of remaining items based on a censoring scheme.

rpcens2(n = 10, R = c(2, 1, 2, 0, 0), dist = "exp", rate = 1)
#> [1] 0.02910825 0.15546476 0.31029202 0.47766020 0.86184076

This is useful in reliability experiments where censoring occurs at each failure stage.

Record Values

The rkrec() function generates upper and lower k-records from continuous distributions.

Upper k-Records

rkrec(size = 5, k = 2, record = "upper", dist = "norm", mean = 0, sd = 1)
#> [1] -0.1252028 -1.2616253 -0.5911243 -1.6933552  0.2620640

Lower k-Records

rkrec(size = 5, k = 3, record = "lower", dist = "exp", rate = 1)
#> [1] 0.3086123 1.9005258 0.8497093 2.1867893 0.8350877

An Application of the Package in Graphical Goodness-of-Fit

This section compares analytical and simulated values of the second moment, variance, skewness, and kurtosis of order statistics from the Uniform(0,1) distribution:

if (requireNamespace("moments", quietly = TRUE)) {
  library(moments)
  set.seed(123)
  x <- ros(1e4, r = 1:25, n = 25, dist = "unif")
  observed <- colMeans(x^2)
  expected <- mo_unif(r = 1:25, n = 25, k = 2, a = 0, b = 1)
  oldpar <- par(mfrow = c(2, 2))
  on.exit(par(oldpar))
  plot(observed, expected, main = "2nd Moment of U(0,1)", xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)

  var_obs <- apply(x, 2, var)
  var_expc <- varOS(1:25, 25, dist = "unif", a = 0, b = 1)
  plot(var_obs, var_expc, main = "Variance of Order Statistics from U(0,1)",
       xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)

  skew_obs <- apply(x, 2, moments::skewness)
  skew_expc <- skewOS(1:25, 25, dist = "unif", a = 0, b = 1)
  plot(skew_obs, skew_expc, main = "Skewness of Order Statistics from U(0,1)",
       xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)

  kurt_obs <- apply(x, 2, moments::kurtosis)
  kurt_expc <- kurtOS(1:25, 25, dist = "unif", a = 0, b = 1)
  plot(kurt_obs, kurt_expc, main = "Kurtosis of Order Statistics from U(0,1)",
       xlab = "Observed", ylab = "Expected")
  abline(0, 1, col = 2)
}

The plots show a strong alignment between the observed and expected values along the identity line. This suggests that the order statistics simulated from the Uniform(0,1) distribution closely follow their theoretical moments. Such alignment in graphical comparisons is often taken as visual evidence of a good fit between observed data and the theoretical model.

Conclusion

The mos package integrates analytical and simulation-based methods for working with order statistics, censored data, and record values. It serves as a powerful and flexible toolkit for statisticians, particularly those involved in reliability analysis and the study of ordered data.