# Introduction to mmtdiff

    library(mmtdiff)

## Overview

The `mmtdiff` package implements the moment-matching approximation for differences of non-standardized t-distributed random variables.

## Background

This package apply the moment-matching approximation 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:

    # X1 ~ t(mu=0, sigma=1, nu=10)
    # X2 ~ t(mu=0, sigma=1.5, nu=15)
    result <- mm_tdiff_univariate(
      mu1 = 0, sigma1 = 1, nu1 = 10,
      mu2 = 0, sigma2 = 1.5, nu2 = 15
    )

    print(result)
    #> Moment-Matching Approximation (Univariate)
    #> ==============================================
    #> Location (mu1 - mu2): 0.0000
    #> Scale (sigma*): 1.8652
    #> df (nu*): 20.9421

    # 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:

    # Probability that difference is negative
    p_negative <- ptdiff(0, result)
    print(paste("P(X1 - X2 < 0) =", round(p_negative, 4)))
    #> [1] "P(X1 - X2 < 0) = 0.5"

    # 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)))
    #> [1] "95% CI: -3.879 to 3.879"

    # 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:

    result_multi <- mm_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)
    #> Moment-Matching Approximation (Multivariate Independent)
    #> =============================================================
    #> Location difference:
    #> [1] 0 1 2
    #>
    #> Scale:
    #> [1] 1.621278 1.844829 2.756493
    #>
    #> df:
    #> [1] 20.57649 18.69487 30.20081