# MaxMC

The goal of MaxMC is to offer a straightforward and easy to use package to perform exact testing using Monte Carlo simulations in the presence of nuisance parameters.

The main functions of MaxMC are mmc and mc. The later function implements the usual Monte Carlo simulations with tie-breakers, while the former implements the Maximized Monte Carlo technique defined by Dufour (2006).

## Installation

You can install the released version of MaxMC from CRAN with:

``install.packages("MaxMC")``

## Examples

These are basic examples which shows you how to use the mmc function:

``````## Example 1
## Exact Unit Root Test
library(fUnitRoots)

# Set seed
set.seed(123)

# Generate an AR(2) process with phi = (-1.5,0.5), and n = 25
y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive")

# Set bounds for the nuisance parameter v
lower <- -1
upper <- 1

# Set the function to generate an AR(2) integrated process
dgp <- function(y, v) {
ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive")
}

# Set the Augmented-Dicky Fuller statistic
statistic <- function(y){
out <- suppressWarnings(adfTest(y, lags = 2, type = "nc"))
return(out@test\$statistic)
}

# Apply the mmc procedure
mmc(y, statistic = statistic , dgp = dgp, lower = lower,
upper = upper, N = 99, type = "leq", method = "GenSA",
control = list(max.time = 2))

## Example 2
## Behrens-Fisher Problem
library(MASS)

# Set seed
set.seed(123)

# Generate sample x1 ~ N(0,1) and x2 ~ N(0,4)
x1 <- rnorm(15, mean = 0, sd = 1)
x2 <- rnorm(25, mean = 0, sd = 2)
data <- list(x1 = x1, x2 = x2)

# Fit a normal distribution on x1 and x2 using maximum likelihood
fit1 <- fitdistr(x1, "normal")
fit2 <- fitdistr(x2, "normal")

# Extract the estimate for the nuisance parameters v = (sd_1, sd_2)
est <- c(fit1\$estimate["sd"], fit2\$estimate["sd"])

# Set the bounds of the nuisance parameters equal to the 99% CI
lower <- est - 2.577 * c(fit2\$sd["sd"], fit1\$sd["sd"])
upper <- est + 2.577 * c(fit2\$sd["sd"], fit1\$sd["sd"])

# Set the function for the DGP under the null (i.e. two population means are equal)
dgp <- function(data, v) {
x1 <- rnorm(length(data\$x1), mean = 0, sd = v)
x2 <- rnorm(length(data\$x2), mean = 0, sd = v)
return(list(x1 = x1, x2 = x2))
}

# Set the statistic function to Welch's t-test
welch <- function(data) {
test <- t.test(data\$x2, data\$x1)
return(test\$statistic)
}

# Apply Welch's t-test
t.test(data\$x2, data\$x1)

# Apply the mmc procedure
mmc(y = data, statistic = welch, dgp = dgp, est = est,
lower = lower, upper = upper, N = 99,   type = "absolute",
method = "pso")``````

This is a basic example which shows you how to use the mc function:

``````## Example 1
## Kolmogorov-Smirnov Test using Monte Carlo

# Set seed
set.seed(999)

# Generate sample data
y <- rgamma(8, shape = 2, rate = 1)

# Set data generating process function
dgp <- function(y)  rgamma(length(y), shape = 2, rate = 1)

# Set the statistic function to the Kolomogorov-Smirnov test for gamma distribution
statistic <- function(y){
out <- ks.test(y, "pgamma", shape = 2, rate = 1)
return(out\$statistic)
}

# Apply the Monte Carlo test with tie-breaker
mc(y, statistic = statistic, dgp = dgp, N = 999, type = "two-tailed")``````

Dufour, J.-M. (2006), Monte Carlo Tests with nuisance parameters: A general approach to finite sample inference and nonstandard asymptotics in econometrics. Journal of Econometrics, 133(2), 443-447.