SimDesign (version 2.18)

IRMSE: Compute the integrated root mean-square error

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

IRMSE(
  estimate,
  parameter,
  fn,
  density = function(theta, ...) 1,
  lower = -Inf,
  upper = Inf,
  ...
)

Value

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

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,

Author

Phil Chalmers

Details

The integrated root mean-square error (IRMSE) is of the form $$IRMSE(\theta) = \sqrt{\int [f(\theta, \hat{\psi}) - f(\theta, \psi)]^2 g(\theta, ...)}$$ where \(g(\theta, ...)\) is the density function used to marginalize the continuous sample (\(f(\theta, \hat{\psi})\)) and population (\(f(\theta, \psi)\)) functions.

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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1080/10691898.2016.1246953")

See Also

RMSE

Examples

# 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)