maciekj - 5 months ago 21

R Question

Based on formulas given in the Mathematica UUPDE database

http://blog.wolfram.com/2013/02/01/the-ultimate-univariate-probability-distribution-explorer/

I've plotted the hazard function standard normal distribution in R.

It seems to be correct in certain range, the numerical issues occur for larger values, see attached figure. Below the complete R code.

Any comments would be very appreciated.

See figure

`PDF = function(x) { 1/(sqrt(2*pi))*exp(-x^2/2) }`

erf <- function(x) 2 * pnorm(x * sqrt(2)) - 1

erfc <- function(x) 2 * pnorm(x * sqrt(2), lower = FALSE)

CDF = function(x) { 1/2 * (1 + erf(x/(sqrt(2)))) }

HF = function(x) { sqrt(2/pi)/(exp(x^2/2)*(2-erfc(-x/sqrt(2)))) }

SF = function(x) { 1 - 1/2 *erfc(-x/sqrt(2)) }

par(mar=c(3,3,1.5,0.5), oma=c(0,0,0,0), mgp=c(2,1,0))

par(mfrow = c(2, 2))

x = seq(from = -4,to = 10,by = .001)

a = PDF(x)

plot(x,a,'l',main='',ylab="PDF",xlab="x")

grid(nx = NULL,ny = NULL,col = "grey",lty = "dotted",lwd = par("lwd"),equilogs = TRUE)

##### CDF

a = CDF(x)

plot(x,a,'l',main='',ylab="CDF",xlab="x")

grid(nx = NULL,ny = NULL,col = "grey",lty = "dotted",lwd = par("lwd"),equilogs = TRUE)

##### HF

a = HF(x)

plot(x,a,'l',main='',ylab="HF",xlab="x")

grid(nx = NULL,ny = NULL,col = "grey",lty = "dotted",lwd = par("lwd"),equilogs = TRUE)

##### SF

a = SF(x)

plot(x,a,'l',main='',ylab="SF",xlab="x")

grid(nx = NULL,ny = NULL,col = "grey",lty = "dotted",lwd = par("lwd"),equilogs = TRUE)

Answer

The hazard function is the density function divided by the survivor function. The problem with your code is that you are taking this definition at face value and doing a simple division operation; when both the numerator and the denominator are very small values (on the order of 1e-300), which happens in the tail of the distribution, this operation becomes numerically unstable. For this kind of problem, the more appropriate solution is to compute the *logarithms* of the numerator and denominator (which are moderate-sized negative numbers rather than tiny numbers), subtract the log-denominator from the log-numerator, then exponentiate.

R provides all the pieces you need to do this calculation. You can get the survivor function via `pnorm(x,lower=FALSE)`

; you can get the density and the survivor functions on the log scale by using `log=TRUE`

and `log.p=TRUE`

in `dnorm()`

and `pnorm()`

respectively. So:

```
HF <- function(x) {
exp(dnorm(x,log=TRUE)-pnorm(x,lower=FALSE,log.p=TRUE))
}
curve(HF,from=-4,to=10)
```

This strategy can be generalized to compute the hazard function for any distribution provided the log-density and log-survivor functions are available (in general for a distribution `foo`

R provides density function `dfoo`

and CDF `pfoo`

which can be substituted above).