Type:	Package
Title:	Simulation and Moment Computation for Order Statistics
Version:	0.1.3
Description:	Provides a comprehensive set of tools for working with order statistics, including functions for simulating order statistics, censored samples (Type I and Type II), and record values from various continuous distributions. Additionally, it offers functions to compute moments (mean, variance, skewness, kurtosis) of order statistics for several continuous distributions. These tools assist researchers and statisticians in understanding and analyzing the properties of order statistics and related data. The methods and algorithms implemented in this package are based on several published works, including Ahsanullah et al (2013, ISBN:9789491216831), Arnold and Balakrishnan (2012, ISBN:1461236444), Harter and Balakrishnan (1996, ISBN:9780849394522), Balakrishnan and Sandhu (1995) <doi:10.1080/00031305.1995.10476150>, Genç (2012) <doi:10.1007/s00362-010-0320-y>, Makouei et al (2021) <doi:10.1016/j.cam.2021.113386> and Nagaraja (2013) <doi:10.1016/j.spl.2013.06.028>.
License:	GPL-3
Encoding:	UTF-8
Imports:	stats, hypergeo2
Suggests:	knitr, rmarkdown, moments
RoxygenNote:	7.3.2
Config/testthat/edition:	3
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2025-06-13 05:49:28 UTC; Saeed
Author:	Reyhaneh Arafeh [aut, cre], Mahdi Salehi [aut]
Maintainer:	Reyhaneh Arafeh <reyhane.arafe@gmail.com>
Repository:	CRAN
Date/Publication:	2025-06-16 11:00:02 UTC

Recursive Computation of Moments from Complementary Beta Distribution

Description

This internal function computes raw moments based on a recursive relation given in Makouei et al. (2021). It is used by the main moment functions.

Usage

.Ms_recursive(a, b, k, s)

Arguments

References

Makouei, R., Khamnei, H. J., & Salehi, M. (2021). Moments of order statistics and k-record values arising from the complementary beta distribution with application. Journal of Computational and Applied Mathematics, 390, 113386.

Incomplete Beta Function

Description

Incomplete Beta Function

Usage

.ibeta(x, a, b)

Check for Natural Numbers

Description

Check for Natural Numbers

Usage

.is.natural(x)

Quantile Function for Complementary Beta Distribution

Description

Quantile Function for Complementary Beta Distribution

Usage

.qcompbeta(p, a, b)

Quantile Function for Kumaraswamy Distribution

Description

Quantile Function for Kumaraswamy Distribution

Usage

.qkumar(p, a, b)

Quantile Function for Pareto Distribution

Description

Quantile Function for Pareto Distribution

Usage

.qpareto(p, scale, shape)

Kurtosis of Order Statistics

Description

This function computes the kurtosis of order statistics for a given distribution.

Usage

kurtOS(r, n, dist = c("unif", "exp", "weibull", "tri"), ...)

Arguments

Details

The kurtosis of the rth order statistic is calculated using the formula:

\text{kurtosis}(X_{r:n}) = \text{E}(\frac{X_{r:n}-\mu_{r:n}} {\sigma_{r:n}})^4

where \mu_{r:n} and \sigma_{r:n} are the mean and standard deviation of the rth order statistic, respectively.

Value

The kurtosis of the rth order statistic.

See Also

varOS, skewOS

Examples

# Compute the kurtosis of the 3rd order statistic from a sample of size 10
kurtOS(r = 3, n = 10, dist = "unif")

Moments of Order Statistics from the Beta Distribution (Simulated)

Description

This function computes the moments of order statistics from the beta distribution using simulation.

Usage

mo_beta(r, n, k = 1, a, b, rep = 1e+05, seed = 42)

Arguments

Details

This function estimates the kth moment of the rth order statistic in a sample of size n drawn from a beta distribution with specified shape parameters. The estimation is done via Monte Carlo simulation using the formula:

\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,

where X_i are the simulated order statistics from the beta distribution.

The function relies on the ros() function to generate order statistics.

Value

The estimated kth moment of the rth order statistic from a beta distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

See Also

ros for generating random samples of order statistics.

Examples

# Compute the first moment of the 2nd order statistic from Beta(3, 4) with sample size 5
mo_beta(r = 2, n = 5, k = 1, a = 3, b = 4)

# Compute the second moment with 10000 simulations
mo_beta(r = 2, n = 5, k = 2, a = 2, b = 2.5, rep = 1e4)

Moments of Order Statistics from the Complementary Beta Distribution

Description

This function computes the moments of order statistics from the complementary beta (CB) distribution. For small values of k and integer b, a closed-form formula is used; otherwise, Monte Carlo simulation is applied.

Usage

mo_compbeta(r, n, k = 1, a, b, rep = 1e+05, seed = 42, verbose = TRUE)

Arguments

Details

The computation method varies depending on b and k:

• For integer b and k = 1, 2: The function calculates the moments using the closed-form expression derived in Makouei et al. (2021):

  \text{E}[X_{r:n}^s] = \frac{1}{B(r, n - r + 1)} \sum_{j=0}^{n-r} \binom{n - r}{j} (-1)^j \mathcal{M}^{(s)}(a, b, r + j - 1),

  Here

  \mathcal{M}^{(s)}(a, b, k) = \frac{1}{k + 1} \left[1 - \frac{s}{B(a, b)} \sum_{j=0}^{\infty} \binom{b-1}{j} (-1)^j \mathcal{M}^{(s-1)}(a, b, a + k + j) \right], \quad s \geq 1,

  with the starting point

  \mathcal{M}^{(1)}(a, b, k) = \frac{B(a + k + 1, b + 1)}{a B(a, b)} \cdot {_3F_2}\left(a + b, 1, a + k + 1; a + 1, a + b + k + 2; 1\right),

  where B(a, b) is the beta function, _3F_2 is the generalized hypergeometric function, and the upper limit of the summation stops at j = b - 1 if b is an integer.

• For non-integer b or k > 2: When b is non-integer or k is greater than 2 the function employs Monte Carlo simulation using the following formula:

  \text{E}[X^s] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^s,

  where X_i are the simulated order statistics obtained from the complementary beta distribution. The method relies on the ros() function to generate order statistics.

When verbose = TRUE, the function prints a message only if Monte Carlo simulation is used (i.e., when k > 2 or b is non-integer).

Value

The estimated or exact kth moment of the rth order statistic from a complementary beta distribution.

Note

The closed-form formula is only available for small values of k and integer b. Monte Carlo simulation is used otherwise, and results may vary slightly depending on rep.

References

Makouei, R., Khamnei, H. J., & Salehi, M. (2021). Moments of order statistics and k-record values arising from the complementary beta distribution with application. Journal of Computational and Applied Mathematics, 390, 113386.

See Also

ros for generating random samples of order statistics.

Examples

# Exact moment when k = 1
mo_compbeta(r = 2, n = 15, k = 1, a = 0.5, b = 2)

# Simulation when k > 2 or b is non-integer
mo_compbeta(r = 2, n = 15, k = 3, a = 2.5, b = 3.7, rep = 1e4)

Moments Of Order Statistics from the Exponential Distribution

Description

This function computes the moments of order statistics from the exponential distribution.

Usage

mo_exp(r, n, k = 1, mu = 0, sigma = 1)

Arguments

Details

The function calculates the kth moment using the following relationship:

\text{E}[X_{r:n}^k] = \frac{n!}{(r-1)!(n-r)!} \cdot \sum_{j=0}^{r-1} (-1)^j \binom{r-1}{j} \frac{\Gamma(k+1)}{(n-r+j+1)^{k+1}}.

For non-standard exponential distributions with \mu and \sigma parameters, the transformation X^*=\mu + \sigma X is used.

Value

The kth moment of the rth order statistic from an exponential distribution.

References

Ahsanullah, M., Nevzorov, V. B., & Shakil, M. (2013). An introduction to order statistics (Vol. 8). Paris: Atlantis Press.

Examples

# First moment (mean) of the 2nd order statistic from a sample of size 5
mo_exp(2, 5, k = 1, mu = 0, sigma = 1)

# Second moment of the 3rd order statistic from an exponential distribution
# with mu = 2 and sigma = 3
mo_exp(3, 7, k = 2, mu = 2, sigma = 3)

Moments of Order Statistics from the Gamma Distribution (Simulated)

Description

This function computes the moments of order statistics from the gamma distribution using simulation.

Usage

mo_gamma(r, n, k = 1, shape, rate, rep = 1e+05, seed = 42)

Arguments

Details

This function estimates the kth moment of the rth order statistic in a sample of size n drawn from a gamma distribution with specified shape and rate parameters. The estimation is done via Monte Carlo simulation using the formula:

\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,

where X_i are the simulated order statistics from the gamma distribution.

The function relies on the ros() function to generate order statistics.

Value

The estimated kth moment of the rth order statistic from a gamma distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

See Also

ros

Examples

# Compute the first moment (mean) of the 3rd order statistic from a sample of size 10
mo_gamma(r = 3, n = 10, shape = 2, rate = 1, k = 1)

# Compute the second moment with 10000 simulations
mo_gamma(r = 2, n = 10, shape = 2, rate = 0.5, k = 2, rep = 1e4)

Moments of Order Statistics from the Kumaraswamy Distribution (Simulated)

Description

This function computes the moments of order statistics from the kumaraswamy distribution using simulation.

Usage

mo_kumar(r, n, k = 1, a, b, rep = 1e+05, seed = 42)

Arguments

Details

This function estimates the kth moment of the rth order statistic in a sample of size n drawn from a kumaraswamy distribution with specified shape parameters. The estimation is done via Monte Carlo simulation using the formula:

\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,

where X_i are the simulated order statistics from the kumaraswamy distribution.

The function relies on the ros() function to generate order statistics.

Value

The estimated kth moment of the rth order statistic from a kumaraswamy distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

See Also

ros for generating random samples of order statistics.

Examples

# Compute the 2nd moment of the 3rd order statistic from Kumaraswamy(2, 3) with sample size 10
mo_kumar(r = 3, n = 10, k = 2, a = 2, b = 3)

# Compute the first moment with 10000 simulations
mo_kumar(r = 2, n = 5, k = 1, a = 2, b = 2.5, rep = 1e4)

Moments of Order Statistics from the Normal Distribution (Simulated)

Description

This function computes the moments of order statistics from the normal distribution using simulation.

Usage

mo_norm(r, n, k = 1, mean = 0, sd = 1, rep = 1e+05, seed = 42)

Arguments

Details

This function estimates the kth moment of the rth order statistic in a sample of size n drawn from a normal distribution with the specified mean and standard deviation. The estimation is done via Monte Carlo simulation using the formula:

\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,

where X_i are the simulated order statistics obtained from the normal distribution.

The function relies on the ros() function to generate order statistics.

Value

The estimated kth moment of the rth order statistic from a normal distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

See Also

ros

Examples

# Compute the first moment (mean) of the 3rd order statistic from a sample of size 10
mo_norm(r = 3, n = 10, k = 1, mean = 0, sd = 1)

# Compute the second moment of the 2nd order statistic with 1 million simulations
mo_norm(r = 2, n = 10, k = 2, rep = 1e6)

Moments of Order Statistics from the Pareto Distribution (Simulated)

Description

This function computes the kth moment of order statistics from the pareto distribution using simulation.

Usage

mo_pareto(r, n, k = 1, scale, shape, rep = 1e+05, seed = 42)

Arguments

Details

This function estimates the kth moment of the rth order statistic in a sample of size n drawn from a pareto distribution with specified scale and shape parameters. The estimation is done via Monte Carlo simulation using the formula:

\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,

where X_i are the simulated order statistics from the pareto distribution.

The function relies on the ros() function to generate order statistics.

Value

The estimated kth moment of the rth order statistic from a pareto distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

See Also

ros

Examples

# Compute the first moment (mean) of the 3rd order statistic from a sample of size 10
mo_pareto(r = 3, n = 10, scale = 2, shape = 3, k = 1)

# Compute the second moment with 1 million simulations
mo_pareto(r = 2, n = 10, scale = 1, shape = 2, k = 2, rep = 1e6)

Moments of Order Statistics from the Student's t-Distribution (Simulated)

Description

This function computes the moments of order statistics from the student's t-distribution using simulation.

Usage

mo_t(r, n, k = 1, df, rep = 1e+05, seed = 42)

Arguments

Details

This function estimates the kth moment of the rth order statistic in a sample of size n drawn from a student's t-distribution with the specified degrees of freedom (df). The estimation is done via Monte Carlo simulation using the formula:

\text{E}[X^k] \approx \frac{1}{\mathrm{rep}} \sum_{i=1}^{\mathrm{rep}} X_i^k,

where X_i are the simulated order statistics obtained from the student's t-distribution.

The function relies on the ros() function to generate order statistics.

Value

The estimated kth moment of the rth order statistic from a student's t-distribution.

Note

The accuracy of the estimated moment depends on the number of simulations (rep). The default value rep = 1e5 provides a reasonable trade-off between speed and accuracy for most practical cases. For higher order moments or when greater precision is required, users are encouraged to increase rep (e.g. 1e6).

See Also

ros

Examples

# Compute the first moment (mean) of the 3rd order statistic from a sample of size 10
mo_t(r = 3, n = 10, df = 5, k = 1)

# Compute the second moment of the 2nd order statistic with 10000 simulations
mo_t(r = 2, n = 10, df = 10, k = 2, rep = 1e4)

Moments of Order Statistics from the Topp-Leone Distribution

Description

This function computes the moments of order statistic from the topp-leone distribution, based on the formula presented in Genç, A. İ. (2012).

Usage

mo_topple(r, n, k = 1, a, b = 1)

Arguments

Details

This function implements the exact formula for moments of order statistics from the topp-leone distribution as provided in Genç, A. İ. (2012):

\text{E}[X_{r:n}^k] = \frac{n!(ab^k)}{(r-1)!(n-r)!} \sum_{j=0}^{n-r} \binom{n-r}{j} (-1)^j 2^{k + 2a(r+j)} \left[ B_{1/2}(k + a(r+j), a(r+j)) - 2 B_{1/2}(k + a(r+j) + 1, a(r+j)) \right]

Here, B_x(., .) is the incomplete Beta function.

Value

The kth moment of the rth order statistic from a topp-leone distribution.

References

Genç, A. İ. (2012). Moments of order statistics of Topp–Leone distribution. Statistical Papers, 53, 117-131.

Examples

# Compute the first moment of the first order statistic for n=5, a=2, b=1
mo_topple(1, 5, 1, 2)

# Compute the second moment of the second order statistic for n=10, a=1.5, b=2
mo_topple(2, 10, 2, 1.5, 2)

Moments of Order Statistics from the Symmetric Triangular Distribution

Description

This function computes the moments of order statistics from the symmetric triangular distribution.

Usage

mo_tri(r, n, k = 1)

Arguments

Details

The function implements the following relationship from Nagaraja (2013) for the symmetric triangular distribution:

\text{E}[X_{r:n}^k] = \frac{n!}{(r-1)!(n-r)!} \left\{ (\frac{1}{2})^{k/2} B\left(\frac{1}{2}; \frac{k}{2} + r, n - r + 1\right) + \sum_{j=0}^k (-1)^j \binom{k}{j} (\frac{1}{2})^{j/2} B\left(\frac{1}{2}; \frac{j}{2} + n - r + 1, r\right) \right\}.

Here, B(x; a, b) is the incomplete Beta function.

Value

The kth moment of the rth order statistic from a symmetric triangular distribution.

References

Nagaraja, H. N. (2013). Moments of order statistics and L-moments for the symmetric triangular distribution. Statistics & Probability Letters, 83(10), 2357-2363.

Examples

# Compute the 2nd moment of the 3rd order statistic for n=5
mo_tri(3, 5, 2)

Moments of Order Statistics from the Uniform Distribution

Description

This function computes the moments of order statistics for the uniform distribution based on the relationship described by Arnold and Balakrishnan (2012).

Usage

mo_unif(r, n, k = 1, a = 0, b = 1)

Arguments

Details

The function calculates the kth moment based on the formula:

\text{E}[U_{r,n}^k] = \frac{B(k + r, n - r + 1)}{B(r, n - r + 1)},

where B(a, b) is the complete beta function. When a \neq 0 or b \neq 1, the transformation U^* = a + (b - a)U is used.

Value

The kth moment of the rth order statistic from a uniform distribution.

References

Arnold, B. C., & Balakrishnan, N. (2012). Relations, bounds and approximations for order statistics (Vol. 53). Springer Science & Business Media.

Examples

# Example 1: First moment (mean) of the 2nd order statistic from a sample of size 5
mo_unif(2, 5, k = 1, a = 0, b = 1)

# Example 2: Second moment of the 3rd order statistic from a uniform distribution on [2, 5]
mo_unif(3, 7, k = 2, a = 2, b = 5)

Moments of Order Statistics from the Weibull Distribution

Description

This function computes the moments of order statistics from the weibull distribution.

Usage

mo_weibull(r, n, k = 1, shape, scale = 1)

Arguments

Details

The function calculates the kth moment using the formula:

\text{E}[X_{r:n}^k] = \frac{n!}{(r-1)!(n-r)!} \Gamma\left(1 + \frac{k}{\text{shape}}\right) \sum_{j=0}^{r-1} (-1)^j \binom{r-1}{j} \frac{1}{(n-r+1+j)^{1 + \frac{k}{\text{shape}}}}

For non-standard weibull distributions (scale not equal to 1), the relationship \text{E}[Z_{r:n}^k] = \text{scale}^k \text{E}[X_{r:n}^k] is used.

Value

The kth moment of the rth order statistic from a weibull distribution.

References

Harter, H. L., & Balakrishnan, N. (1996). CRC handbook of tables for the use of order statistics in estimation. CRC press.

Examples

# Example 1: Standard weibull distribution (shape = 2, scale = 1)
mo_weibull(r = 2, n = 5, k = 1, shape = 2)

# Example 2: Non-standard weibull distribution (shape = 2, scale = 3)
mo_weibull(r = 3, n = 6, k = 2, shape = 2, scale = 3)

Generate Censored Samples (Type I or Type II)

Description

This function generates censored samples from a specified distribution, using Type I (time-based) or Type II (failure-based) censoring schemes.

Usage

rcens(n, r = NULL, dist, type = c("I", "II"), cens.time = NULL, ...)

Arguments

Details

This function implements two types of censoring schemes:

1. Type I censoring: Observations are censored if they exceed a specified cens.time. The function returns all observations less than cens.time.

2. Type II censoring: The smallest r observations are returned, simulating a situation where the experiment stops after r failures.

Value

A numeric vector of censored samples.

See Also

rpcens2

Examples

# Type I censoring: Exponential distribution with rate = 1, censored at time 2
rcens(n = 10, dist = "exp", type = "I", cens.time = 2, rate = 1)

# Type II censoring: Normal distribution, smallest 5 values
rcens(n = 10, r = 5, dist = "norm", type = "II", mean = 0, sd = 1)

Generate Upper and Lower k-Records from Continuous Distributions

Description

This function generates k-records (upper or lower) from a specified continuous distribution.

Usage

rkrec(size, k, record = c("upper", "lower"), dist, ...)

Arguments

Details

Note: Setting k = 1 generates standard (1-)records.

Value

A numeric vector of size size, representing the simulated k-records.

See Also

ros

Examples

# Generate 5 upper 2-records from the normal distribution
rkrec(size = 5, k = 2, record = "upper", dist = "norm", mean = 0, sd = 1)

# Generate 5 lower 3-records from the exponential distribution
rkrec(size = 5, k = 3, record = "lower", dist = "exp", rate = 1)

Generate Random Data from Order Statistics

Description

This function generates random data from the order statistics of a specified distribution. The user can specify a known distribution in R or provide a custom quantile function.

Usage

ros(size, r, n, dist = NULL, qf = NULL, ...)

Arguments

Details

The ros function generates random data from order statistics using two approaches:

1. Using a Known Distribution: When dist is provided, random data is generated from a known distribution in R.

2. Using a Custom Quantile Function: When qf is provided, ros applies the user-provided quantile function to generate random data.

Value

A numeric vector or matrix containing the generated random data from the specified order statistics. If a single rank is provided (i.e., scalar r), a numeric vector of size size is returned. If multiple ranks are provided (i.e., vector r), a matrix is returned with size rows and length(r) columns, where each row corresponds to a simulation and each column to an order statistic.

Examples

# Example 1: Generate from the normal distribution
ros(5, 3, 15, "norm", mean = 4, sd = 2)

# Example 2: Using a custom quantile function for the Pareto distribution
ros(3, 2, 10, qf = function(p, scale, shape) scale * (1 - p)^(-1 / shape), scale = 3, shape = 2)

# Example 3: Generate multiple order statistics from the uniform distribution
# In this example, first through 5th order statistics are generated from a sample size of 5.
ros(3, 1:5, 5, dist = "unif")

Generate Progressive Type-II Censored Samples

Description

This function generates progressive Type-II censored samples based on the algorithm provided by Balakrishnan and Sandhu (1995).

Usage

rpcens2(n, R, dist, ...)

Arguments

Details

This function implements the algorithm described by Balakrishnan and Sandhu (1995) to simulate progressive Type-II censored samples. Progressive Type-II censoring is a common scheme in reliability and life-testing experiments, where items are progressively removed (censored) during the testing process.

Value

A numeric vector of size m, representing the failure times of the observed items.

References

Balakrishnan, N., & Sandhu, R. A. (1995). A simple simulational algorithm for generating progressive Type-II censored samples. The American Statistician, 49(2), 229-230.

See Also

rcens

Examples

# Generate a progressive Type-II censored sample from the normal distribution
n <- 10
R <- c(2, 1, 2, 0, 0)
rpcens2(n, R, dist = "norm", mean = 0, sd = 1)

# Generate a progressive Type-II censored sample from the exponential distribution
rpcens2(n = 10, R = c(2, 2, 1, 0, 0), dist = "exp", rate = 1)

Skewness of Order Statistics

Description

This function computes the skewness of the order statistics for a given distribution.

Usage

skewOS(r, n, dist = c("unif", "exp", "weibull", "tri"), ...)

Arguments

Details

The skewness of the rth order statistic is calculated using the formula:

\text{Skewness}(X_{r:n}) = \text{E}(\frac{X_{r:n}-\mu_{r:n}} {\sigma_{r:n}})^3

where \mu_{r:n} and \sigma_{r:n} are the mean and standard deviation of the rth order statistic, respectively.

Value

The skewness of the rth order statistic.

See Also

varOS, kurtOS

Examples

# Skewness of the 3rd order statistic for a uniform distribution
skewOS(3, 10, "unif")

Variance of Order Statistics

Description

This function computes the variance of order statistics for a given distribution.

Usage

varOS(r, n, dist = c("unif", "exp", "weibull", "tri", "topple"), ...)

Arguments

Details

This function computes the variance of the rth order statistic (X_{r:n}) for a given sample size (n) and distribution (dist). The variance is calculated using:

\text{Var}(X_{r:n}) = \text{E}(X^2_{r:n}) - (\text{E}(X_{r:n}))^2

Value

The variance of the rth order statistic.

Examples

# Variance of the 3rd order statistic in a sample of size 10 from a uniform distribution
varOS(r = 3, n = 10, dist = "unif")