Distribution Functions for Approximated t-Difference

Description

Distribution Functions for Approximated t-Difference

Usage

dtdiff(x, ws_result)

ptdiff(q, ws_result)

qtdiff(p, ws_result)

rtdiff(n, ws_result)

Arguments

Value

For dtdiff: Numeric vector of density values. For ptdiff: Numeric vector of cumulative probabilities. For qtdiff: Numeric vector of quantiles. For rtdiff: Numeric vector of random samples from the approximated t-difference distribution.

Examples

result <- ws_tdiff_univariate(0, 1, 10, 0, 1.5, 15)
dtdiff(0, result)
ptdiff(0, result)
qtdiff(c(0.025, 0.975), result)
samples <- rtdiff(100, result)

Validate Welch-Satterthwaite Approximation

Description

Validates the approximation quality by comparing moments of the approximated distribution with the theoretical moments.

Usage

validate_approximation(ws_result, n_sim = 10000, seed = NULL)

Arguments

Value

A list containing validation metrics

Examples

result <- ws_tdiff_univariate(0, 1, 10, 0, 1.5, 15)
validation <- validate_approximation(result)
print(validation)

Equal Parameters Special Case

Description

Computes the Welch-Satterthwaite approximation for the special case where both distributions have identical parameters.

Usage

ws_tdiff_equal_params(mu, sigma, nu)

Arguments

Details

When X1 ~ t(mu, sigma^2, nu) and X2 ~ t(mu, sigma^2, nu) are independent, the difference Z = X1 - X2 simplifies to:

• Location: mu_diff = 0

• Scale: sigma_star = sigma * sqrt(2*nu/(nu-2))

• Degrees of freedom: nu_star = 2*(nu - 4)

This special case provides validation for the general formulas and computational efficiency when parameters are known to be equal.

Value

An S3 object of class "ws_tdiff_univariate" with the simplified parameters

Examples

# Equal parameters case
result <- ws_tdiff_equal_params(mu = 0, sigma = 1, nu = 10)
print(result)
# nu_star should be 2*(10-4) = 12

# Verify against general formula
general <- ws_tdiff_univariate(0, 1, 10, 0, 1, 10)
all.equal(result$nu_star, general$nu_star)

Welch-Satterthwaite Approximation for General Multivariate t-Differences

Description

Approximates the distribution of differences between two independent multivariate t-distributed random vectors with arbitrary covariance structure. This implements Theorem 3 from Yamaguchi et al. (2025).

Usage

ws_tdiff_multivariate_general(
  mu1,
  Sigma1,
  nu1,
  mu2,
  Sigma2,
  nu2,
  max_iter = 10,
  tol = 0.001
)

Arguments

Details

This function handles the general case where components may be correlated within each multivariate t-distribution. The approximation uses a single scalar degrees of freedom parameter to capture the overall tail behavior.

The iterative algorithm (Section 4.3 of the paper):

1. Initialize with sum of covariance matrices

2. Compute effective degrees of freedom using trace formulas

3. Update scale matrix

4. Iterate until convergence

Note: For high dimensions with heterogeneous component behaviors, consider using ws_tdiff_multivariate_independent instead.

Value

An S3 object of class "ws_tdiff_multivariate_general" containing:

Examples

Sigma1 <- matrix(c(1, 0.3, 0.3, 1), 2, 2)
Sigma2 <- matrix(c(1.5, 0.5, 0.5, 1.2), 2, 2)
result <- ws_tdiff_multivariate_general(
  mu1 = c(0, 1), Sigma1 = Sigma1, nu1 = 10,
  mu2 = c(0, 0), Sigma2 = Sigma2, nu2 = 15
)
print(result)

Welch-Satterthwaite Approximation for Multivariate t-Differences (Independent)

Description

Approximates the distribution of differences between two independent p-dimensional vectors with independent t-distributed components.

Usage

ws_tdiff_multivariate_independent(mu1, sigma1, nu1, mu2, sigma2, nu2)

Arguments

Details

This function applies the univariate Welch-Satterthwaite approximation component-wise when all components are mutually independent. Each component difference Zj = X1j - X2j is approximated independently using the univariate method.

This approach is optimal for:

• Marginal inference on specific components

• Cases where components have different tail behaviors

• Maintaining computational efficiency in high dimensions

Value

An S3 object of class "ws_tdiff_multivariate_independent" containing:

See Also

ws_tdiff_multivariate_general for correlated components

Examples

result <- ws_tdiff_multivariate_independent(
  mu1 = c(0, 1), sigma1 = c(1, 1.5), nu1 = c(10, 12),
  mu2 = c(0, 0), sigma2 = c(1.2, 1), nu2 = c(15, 20)
)
print(result)

Welch-Satterthwaite Approximation for Univariate t-Differences

Description

Approximates the distribution of the difference between two independent non-standardized t-distributed random variables using the Welch-Satterthwaite method.

Usage

ws_tdiff_univariate(mu1, sigma1, nu1, mu2, sigma2, nu2)

Arguments

Details

For two independent non-standardized t-distributed random variables:

• X1 ~ t(mu1, sigma1^2, nu1)

• X2 ~ t(mu2, sigma2^2, nu2)

The difference Z = X1 - X2 is approximated as: Z ~ t(mu1 - mu2, sigma_star^2, nu_star)

where the effective parameters are computed through moment matching:

• sigma_star matches the variance of Z

• nu_star is derived from fourth moment matching

The method requires nu1 > 4 and nu2 > 4 for the existence of fourth moments. The approximation quality improves as degrees of freedom increase and approaches exactness as nu -> infinity (normal limit).

Value

An S3 object of class "ws_tdiff_univariate" containing:

References

Yamaguchi, Y., Homma, G., Maruo, K., & Takeda, K. Welch-Satterthwaite Approximation for Difference of Non-Standardized t-Distributed Variables. (unpublished).

See Also

ws_tdiff_equal_params for the special case of equal parameters dtdiff, ptdiff, qtdiff, rtdiff for distribution functions

Examples

# Example 1: Different scale parameters
result <- ws_tdiff_univariate(
  mu1 = 0, sigma1 = 1, nu1 = 10,
  mu2 = 0, sigma2 = 1.5, nu2 = 15
)
print(result)

# Example 2: Equal parameters (special case)
result_equal <- ws_tdiff_univariate(
  mu1 = 5, sigma1 = 2, nu1 = 20,
  mu2 = 3, sigma2 = 2, nu2 = 20
)
# Should match ws_tdiff_equal_params(5-3, 2, 20)