---
title: "Introduction to wstdiff"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to wstdiff}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(wstdiff)
```

## Overview

The `wstdiff` package implements the Welch-Satterthwaite approximation for 
differences of non-standardized t-distributed random variables.

## Background

While the classical Welch-Satterthwaite approximation handles combinations of
sample variances, this package extends the framework to t-distribution differences,
which arise in:

- Bayesian inference with t-distributed priors
- Robust statistical modeling
- Uncertainty quantification with heavy-tailed data

## Univariate Example

Consider two independent t-distributed variables with different parameters:

```{r univariate}
# X1 ~ t(mu=0, sigma=1, nu=10)
# X2 ~ t(mu=0, sigma=1.5, nu=15)
result <- ws_tdiff_univariate(
  mu1 = 0, sigma1 = 1, nu1 = 10,
  mu2 = 0, sigma2 = 1.5, nu2 = 15
)

print(result)

# The difference Z = X1 - X2 is approximately:
# Z ~ t(mu_diff, sigma_star^2, nu_star)
```

## Using Distribution Functions

Once we have the approximation, we can compute probabilities and quantiles:

```{r distributions}
# Probability that difference is negative
p_negative <- ptdiff(0, result)
print(paste("P(X1 - X2 < 0) =", round(p_negative, 4)))

# 95% confidence interval for the difference
ci_95 <- qtdiff(c(0.025, 0.975), result)
print(paste("95% CI:", round(ci_95[1], 3), "to", round(ci_95[2], 3)))

# Generate random samples
samples <- rtdiff(1000, result)
hist(samples, breaks = 30, main = "Simulated Difference Distribution",
     xlab = "X1 - X2", probability = TRUE)

# Overlay theoretical density
x_seq <- seq(min(samples), max(samples), length.out = 100)
lines(x_seq, dtdiff(x_seq, result), col = "red", lwd = 2)
```

## Multivariate Example

For multiple independent components:

```{r multivariate}
result_multi <- ws_tdiff_multivariate_independent(
  mu1 = c(0, 1, 2),
  sigma1 = c(1, 1.5, 2),
  nu1 = c(10, 12, 15),
  mu2 = c(0, 0, 0),
  sigma2 = c(1.2, 1, 1.8),
  nu2 = c(15, 20, 25)
)

print(result_multi)
```

## Special Cases

The package handles special cases efficiently:

```{r special}
# When both distributions have identical parameters
result_equal <- ws_tdiff_equal_params(mu = 0, sigma = 1, nu = 10)
print(result_equal)

# Verify: nu_star should be 2*(nu - 4) = 12
stopifnot(result_equal$nu_star == 12)
```

## Approximation Quality

The approximation quality depends on:

1. **Degrees of freedom**: Higher values give better approximations
2. **Parameter similarity**: More similar parameters improve accuracy
3. **Dimension**: In multivariate cases, using a single nu* may be less accurate

```{r quality}
# Low degrees of freedom - use with caution
result_low_df <- ws_tdiff_univariate(0, 1, 5, 0, 1, 6)
print(paste("Warning: nu_star =", round(result_low_df$nu_star, 2)))

# High degrees of freedom - excellent approximation
result_high_df <- ws_tdiff_univariate(0, 1, 100, 0, 1.5, 150)
print(paste("Good: nu_star =", round(result_high_df$nu_star, 2)))
```

## References

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