# IRMSE: Compute the integrated root mean-square error In SimDesign: Structure for Organizing Monte Carlo Simulation Designs

## Description

Computes the average/cumulative deviation given two continuous functions and an optional function representing the probability density function. Only one-dimensional integration is supported.

## Usage

 1 2 3 4 5 6 7 8 9 IRMSE( estimate, parameter, fn, density = function(theta, ...) 1, lower = -Inf, upper = Inf, ... ) 

## Arguments

 estimate a vector of parameter estimates parameter a vector of population parameters fn a continuous function where the first argument is to be integrated and the second argument is a vector of parameters or parameter estimates. This function represents a implied continuous function which uses the sample estimates or population parameters density (optional) a density function used to marginalize (i.e., average), where the first argument is to be integrated, and must be of the form density(theta, ...) or density(theta, param1, param2), where param1 is a placeholder name for the hyper-parameters associated with the probability density function. If omitted then the cumulative different between the respective functions will be computed instead lower lower bound to begin numerical integration from upper upper bound to finish numerical integration to ... additional parameters to pass to fnest, fnparam, density, and integrate,

## Details

The integrated root mean-square error (IRMSE) is of the form

IRMSE(θ) = √{\int [f(θ, \hat{ψ}) - f(θ, ψ)]^2 g(θ, ...)}

where g(θ, ...) is the density function used to marginalize the continuous sample (f(θ, \hat{ψ})) and population (f(θ, ψ)) functions.

## Value

returns a single numeric term indicating the average/cumulative deviation given the supplied continuous functions

## Author(s)

Phil Chalmers rphilip.chalmers@gmail.com

## References

Chalmers, R. P., & Adkins, M. C. (2020). Writing Effective and Reliable Monte Carlo Simulations with the SimDesign Package. The Quantitative Methods for Psychology, 16(4), 248-280. doi: 10.20982/tqmp.16.4.p248

Sigal, M. J., & Chalmers, R. P. (2016). Play it again: Teaching statistics with Monte Carlo simulation. Journal of Statistics Education, 24(3), 136-156. doi: 10.1080/10691898.2016.1246953

RMSE
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 # logistic regression function with one slope and intercept fn <- function(theta, param) 1 / (1 + exp(-(param[1] + param[2] * theta))) # sample and population sets est <- c(-0.4951, 1.1253) pop <- c(-0.5, 1) theta <- seq(-10,10,length.out=1000) plot(theta, fn(theta, pop), type = 'l', col='red', ylim = c(0,1)) lines(theta, fn(theta, est), col='blue', lty=2) # cumulative result (i.e., standard integral) IRMSE(est, pop, fn) # integrated RMSE result by marginalizing over a N(0,1) distribution den <- function(theta, mean, sd) dnorm(theta, mean=mean, sd=sd) IRMSE(est, pop, fn, den, mean=0, sd=1) # this specification is equivalent to the above den2 <- function(theta, ...) dnorm(theta, ...) IRMSE(est, pop, fn, den2, mean=0, sd=1)