In R, the procedure density gives a kernel density
estimator (KDE) of data.
In order to get a suitable histogram of the very heavy tailed Cauchy distribution, it is usually necessary to disregard more than a few values in the tails of the sample.
Then in order show the actual Cauchy density curve (black below), it is necessary to adjust the density by dividing by the proportion of the sample plotted in the histogram.
Also, to get a the best KDE (dotted red) you may need to use parameter
adj tp adjust the default bandwidth. (I used the default.) Of course, the histogram would be 'smoother' if it had fewer bins; the KDE is made entirely without
reference to the histogram.
By adjusting the proportion of the sample plotted, the bandwidth of the KDE, and the sample size, you may be able to improve on my plot below. But the agreement of the density function and the KDE in the the plot below is roughly typical for samples of size 500 to 1000.
R code for figure:
k = diff(pt(c(-4,4),1))
set.seed(2022)
w = rt(1000, 1) # whole sample, size 1000
y = w[x > -4 & w < 4] # 861 plotted points
length(y)
[1] 861
hist(y, prob=T, ylim=c(0,.4), br=50, col="skyblue2")
curve(dt(x,1)/k, add=T, lwd=2)
lines(density(y), col="red", lwd=2, lty="dotted")
Note: For moderately large samples, the empirical CDF (ECDF) of the sample matches the density function of the population very well. The test statistic $D$ of the Kolmogorov-Smirnov goodness-of-fit test, is the maximum vertical distance between the two.
plot(ecdf(w), lwd=3, col="red", lty="dotted")
curve(pt(x,1), add=T)
abline(v=0, col="green2")
abline(h=0:1, col="green2")
ks.test(w, pt, 1)
One-sample Kolmogorov-Smirnov test
data: w
D = 0.018508, p-value = 0.8832
alternative hypothesis: two-sided