# finding quantiles of a kernel density estimation

I used R to find kernel density estimates of my dataset (for experiment I used 1000 samples generated from a known distribution in this step).

I used code density() to find the kernel density estimate. Now I need to find the cdf of this pdf in order to find quantiles of the distribution. I used these codes for normal:

x=rnorm(1000,8,3)
pdf=density(x)
f=approxfun(pdf$x,pdf$y,yleft=0,yright=0)
FF2=function(x) integrate(f,-Inf,x)$v-0.25 uniroot(FF2,c(2,10))  By the last code I could find the first quantile of the distribution and this is what I want. But my question is about another distribution like gamma. These codes do not run for this distribution as here: x1=rgamma(1000,shape=2,scale=1) pdf1=density(x1) f1=approxfun(pdf1$x,pdf1$y,yleft=0,yright=0) FF1=function(x) integrate(f1,-Inf,x)$value-0.25


The result of the last code is not logical!

• Which quantiles do you want? I see 0.25 in your code and wonder if you want quartiles. (The "first quantile" makes no sense.) If so, or even if not so, why would quartiles from a kernel density estimate be better than, or in any sense preferable to, quartiles from the raw data? There could be a good reason, but please tell us what it is. You could also smooth the quantile function directly. – Nick Cox Jun 15 '18 at 8:21
• To second @NickCox comment, the only way that a density estimate would provide more precision in quantile estimates would be if you had injected prior knowledge into the density estimation process. If density estimation used 3 or less effective degrees of freedom you'd expect it to be more efficient, if the structure imposed on the density happened to be right. – Frank Harrell Jun 15 '18 at 11:15

## 1 Answer

The true value is

> qgamma(.25,shape=2,scale=1)
 0.9612788


so searching over the support $(2,10)$ does not seem like a good idea. For your normal example, the true quantile is

> qnorm(.25,8,3)
 5.976531


so that the support $(2,10)$ contains the solution, explaining why the first example works but the second does not.

Also, the gamma distribution has support on $(0,\infty)$, so that the integration in FF1 ought to be started at 0.

That is,

x1=rgamma(1000,shape=2,scale=1)
pdf1=density(x1)
f1=approxfun(pdf1$x,pdf1$y,yleft=0,yright=0)
FF1=function(x) integrate(f1,0,x)\$value-0.25
uniroot(FF1,c(0,10))


works for me (starting the integration at 0 seems optional, as the distribution has no mass in the negative values).

• thanks for your reply, but I am not searching for the real quantiles of the gamma distribution. I want to find the approximated distribution. I mean I would like to fit a kernel density distribution and then find this density's quantiles so that I can compare them with the real ones. that is why I am doing so. – fattmim Jun 15 '18 at 6:48
• I think I got that - I printed the "real" quantiles in order to emphasize issues I see in your code, namely that the support you specify in uniroot does not seem well-chosen - see also my edit to my answer. – Christoph Hanck Jun 15 '18 at 7:34