Package 'mos'

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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
dist	a character string specifying the name of a distribution. Supported values are: "unif": Uniform distribution "exp": Exponential distribution "weibull": Weibull distribution "tri": Triangular distribution
...	further arguments to be passed to dist.

Details

The kurtosis of the $r$th 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 $r$th order statistic, respectively.

Value

The kurtosis of the $r$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
a, b	non-negative parameters of the beta distribution.
rep	number of simulations (default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the $k$th moment of the $r$th 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 $k$th moment of the $r$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
a, b	positive parameters of the complementary beta distribution.
rep	number of simulations (used when b is non-integer, default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (used when b is non-integer, default is 42).
verbose	logical; if TRUE, prints a message when Monte Carlo simulation is used.

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 $k$th moment of the $r$th 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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
mu	location parameter of the exponential distribution (default is 0).
sigma	scale parameter of the exponential distribution (default is 1).

Details

The function calculates the $k$th 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 $k$th moment of the $r$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
shape	shape parameter of the gamma distribution.
rate	rate parameter of the gamma distribution.
rep	number of simulations (default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the $k$th moment of the $r$th 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 $k$th moment of the $r$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
a, b	positive parameters of the kumaraswamy distribution.
rep	number of simulations (default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the $k$th moment of the $r$th 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 $k$th moment of the $r$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
mean	mean of the normal distribution (default is 0).
sd	standard deviation of the normal distribution (default is 1).
rep	number of simulations (default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the $k$th moment of the $r$th 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 $k$th moment of the $r$th 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 $k$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
scale, shape	non-negative parameters of the pareto distribution.
rep	Number of simulations (default is 1e5).
seed	Optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the $k$th moment of the $r$th 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 $k$th moment of the $r$th 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

r	rank of the desired order statistic (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
df	degrees of freedom for the student's t-distribution.
rep	number of simulations (default is 1e5).
seed	optional seed for random number generation to ensure reproducibility (default is 42).

Details

This function estimates the $k$th moment of the $r$th 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 $k$th moment of the $r$th 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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
a	shape parameter of the topp-leone distribution (a > 0).
b	scale parameter of the topp-leone distribution (default is 1, b > 0).

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 $k$th moment of the $r$th 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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).

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 $k$th moment of the $r$th 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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
a, b	lower and upper limits of the distribution. Must be finite.

Details

The function calculates the $k$th 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 $k$th moment of the $r$th 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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
k	order of the moment to compute (default is 1).
shape	shape parameter of the weibull distribution.
scale	scale parameter of the weibull distribution (default is 1).

Details

The function calculates the $k$th 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 $k$th moment of the $r$th 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

n	total number of items in the sample
r	number of uncensored observations (only for Type II censoring).
dist	a character string specifying the name of the distribution (e.g., "norm" for the normal distribution, "exp" for the exponential distribution).
type	type of censoring: "I" for Type I (time-based) or "II" for Type II (failure-based).
cens.time	censoring time for Type I censoring.
...	further arguments to be passed to dist.

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

size	number of k-records to generate.
k	the rank of the record to generate ($k$-record).
record	the type of record to generate: "upper" for upper k-records, "lower" for lower k-records. Default is "upper".
dist	a character string specifying the name of the continuous distribution (e.g., "norm", "exp", "gamma").
...	further arguments to be passed to dist.

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

size	number of observations.
r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
dist	a character string specifying the name of a known distribution in R (e.g. 'norm', 'exp'). Default is NULL.
qf	a custom quantile function, either as a name (string) or directly as a function. Default is NULL.
...	further arguments to be passed to dist or qf.

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

n	total number of items in the sample.
R	a vector of non-negative integers representing the number of items censored at each stage of the experiment. The length of R determines the number of failure stages (m), and the sum of R plus m must equal n.
dist	a character string specifying the name of the distribution (e.g., "norm" for the normal distribution, "exp" for the exponential distribution).
...	further arguments to be passed to dist.

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)

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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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
dist	a character string specifying the name of a distribution. Supported values are: "unif": Uniform distribution "exp": Exponential distribution "weibull": Weibull distribution "tri": Triangular distribution
...	further arguments to be passed to dist.

Details

The skewness of the $r$th 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 $r$th order statistic, respectively.

Value

The skewness of the $r$th order statistic.

See Also

varOS, kurtOS

Examples

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

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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

r	rank(s) of the desired order statistic(s) (e.g., 1 for the smallest order statistic).
n	sample size from which the order statistic is derived.
dist	a character string specifying the name of a distribution. Supported values are: "unif": Uniform distribution "exp": Exponential distribution "weibull": Weibull distribution "tri": Triangular distribution "topple": Topp-Leon distribution
...	further arguments to be passed to dist.

Details

This function computes the variance of the $r$th 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 $r$th 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")

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