| Title: | Utility Functions for Davidian Curves |
|---|---|
| Description: | A Davidian curve defines a seminonparametric density, whose shape and flexibility can be tuned by easy to estimate parameters. Since a special case of a Davidian curve is the standard normal density, Davidian curves can be used for relaxing normality assumption in statistical applications (Zhang & Davidian, 2001) <doi:10.1111/j.0006-341X.2001.00795.x>. This package provides the density function, the gradient of the loglikelihood and a random generator for Davidian curves. |
| Authors: | Oğuzhan Öğreden |
| Maintainer: | Oğuzhan Öğreden <[email protected]> |
| License: | GPL-3 |
| Version: | 0.9.2 |
| Built: | 2024-11-13 06:26:49 UTC |
| Source: | CRAN |
Provides the gradient for use in estimation.
dc_grad(x, phi)dc_grad(x, phi)
x |
A vector of observations. |
phi |
phi Davidian curve parameters. A maximum of 10 parameters is allowed. |
Woods & Lin (2009) provide the gradient (Equations 17 and 18). Note that the gradient is not defined for phi = 0.0.
Woods, C. M., & Lin, N. (2009). Item response theory with estimation of the latent density using Davidian curves. Applied Psychological Measurement, 33(2), 102-117. doi:10.1177/0146621608319512
# The loglikelihood of a univariate Davidian curve is given by, dc_LL <- function(phi, dat) { sum(log(ddc(dat, phi))) } # dc_grad can be used for obtaining the gradient of this loglikelihood as follows: dc_LL_GR <- function(phi, dat) { colSums(dc_grad(dat, phi)) } # This can be verified by numerical approximation. # For instance, using numDeriv package: ## Not run: phi <- c(-5, 2.5, 10) d <- runif(10, -5, 5) dc_LL_GR(phi, d) numDeriv::grad(dc_LL, x = phi, dat = d) phi <- c(-5, 0, 10) dc_LL_GR(phi, d) ## End(Not run)# The loglikelihood of a univariate Davidian curve is given by, dc_LL <- function(phi, dat) { sum(log(ddc(dat, phi))) } # dc_grad can be used for obtaining the gradient of this loglikelihood as follows: dc_LL_GR <- function(phi, dat) { colSums(dc_grad(dat, phi)) } # This can be verified by numerical approximation. # For instance, using numDeriv package: ## Not run: phi <- c(-5, 2.5, 10) d <- runif(10, -5, 5) dc_LL_GR(phi, d) numDeriv::grad(dc_LL, x = phi, dat = d) phi <- c(-5, 0, 10) dc_LL_GR(phi, d) ## End(Not run)
Returns the density for a vector of x.
ddc(x, phi)ddc(x, phi)
x |
vector of quantiles. |
phi |
Davidian curve parameters. A maximum of 10 parameters is allowed. |
curve(ddc(x, 1.570789), -6, 6) # Approximately normal. phi <- c(77.32, 78.51, 76.33, 77.16) curve(ddc(x, phi), -6, 6) # A bimodal density. integrate(ddc, phi = phi, lower = -Inf, upper = Inf) # Integrates to 1.curve(ddc(x, 1.570789), -6, 6) # Approximately normal. phi <- c(77.32, 78.51, 76.33, 77.16) curve(ddc(x, phi), -6, 6) # A bimodal density. integrate(ddc, phi = phi, lower = -Inf, upper = Inf) # Integrates to 1.
Returns n samples from a univariate Davidian curve.
rdc(n, phi)rdc(n, phi)
n |
Number of observations to be sampled. |
phi |
Davidian curve parameters. A maximum of 10 parameters is allowed. |
# Sample from the standard normal Davidian curve: hist(rdc(1000, 1.570789), xlim = c(-6, 6), ylim = c(0, 0.5), freq = FALSE, breaks = 20) curve(dnorm(x), -6, 6, col = "blue", lwd = 1, add = TRUE) curve(ddc(x, 1.570789), -6, 6, col = "red", lwd = 2, lty = 3, add = TRUE) # Sample from a bimodal density: phi <- c(77.32, 78.51, 76.33, 77.16) hist(rdc(1000, phi), xlim = c(-6, 6), ylim = c(0, 0.4), freq = FALSE, breaks = "fd") curve(ddc(x, phi), -6, 6, col = "red", lwd = 2, lty = 3, add = TRUE)# Sample from the standard normal Davidian curve: hist(rdc(1000, 1.570789), xlim = c(-6, 6), ylim = c(0, 0.5), freq = FALSE, breaks = 20) curve(dnorm(x), -6, 6, col = "blue", lwd = 1, add = TRUE) curve(ddc(x, 1.570789), -6, 6, col = "red", lwd = 2, lty = 3, add = TRUE) # Sample from a bimodal density: phi <- c(77.32, 78.51, 76.33, 77.16) hist(rdc(1000, phi), xlim = c(-6, 6), ylim = c(0, 0.4), freq = FALSE, breaks = "fd") curve(ddc(x, phi), -6, 6, col = "red", lwd = 2, lty = 3, add = TRUE)