3
$\begingroup$

I have this empirical discrete distribution with the respective percentiles (picture below). I want to know a R function which gives me $P(X\lt0.58)$ instead of $P(X\le 0.58)$ (given by ECDF function).

The ECDF function gives me around $0.89$ and $0.90$ but I want something between $0.83$ and $0.84$ (see the picture below).

Is there any functions that does this?

$\endgroup$
2
  • 1
    $\begingroup$ Hint: how can you use the definition of the cumulative density function to determine the probability in an interval? $\endgroup$
    – Sycorax
    Sep 16, 2020 at 14:26
  • $\begingroup$ @Sycorax I subtract two cdf's $\endgroup$
    – user45523
    Sep 16, 2020 at 14:37

1 Answer 1

4
$\begingroup$

There is a statistical question here.

Let the cumulative distribution function of a random variable be $F_X$, defined as

$$F_X(x) = \Pr(X \le x).$$

What is wanted is a variant of this with a strict inequality,

$$G_X(x) = \Pr(X \lt x).$$

The two differ when there is a nonzero chance that $X=x.$

Notice that the event $X \lt x$ is the same as the event $-X \gt -x,$ which is the complement of the event $-X \le -x.$ Therefore since the chance of an event and the chance of its complement must sum to unity,

$$G_X(x) = 1 - \Pr(-X \le -x) = 1 - F_{-X}(-x).$$

In this way we may find $G_X$ without having to do any more programming simply by computing the empirical CDF for the data $-X.$

In R, for instance, the empirical distribution $F$ of data x would be computed as

f <- ecdf(x)

Consequently we may compute $G$ as

g <- function(a) 1 - (ecdf(-x))(-a)

Here is an illustration of the difference using a dataset much like the one in the question with $x_0=0.58$ the value at which $F_X$ and $G_X$ are to be computed.

Figure

In this example, $F_X(0.58)=0.9$ while $G_X(0.58)=0.8.$ You can see that $F_X$ gives the height of the solid dot at $x_0$ while $G_X$ gives the height of the dot immediately to its left.

The dataset x consists of these 30 numbers:

0.24(7) 0.25(2) 0.26 0.27 0.28 0.29(2) 0.30 0.31 0.37 0.39
0.40 0.41 0.44(2) 0.49 0.52 0.58(3) 0.59 0.61 0.78

The values in parentheses give the counts of duplicated values.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.