Log-probability derivation of the minimum statistic of i.i.d. discrete variables

[larryshamalama]

Description

Extensions of #6790 and #6846.

For reference: the survival function $P(X_{(1)} > 1)$ can be written as:

$$ P(X_{(1)} > x) = P(\text{all of } X_1, \dots, X_n > x) = \prod_{i=1}^n P(X_i > x) = \prod_{i=1}^n \big(1 - P(X_i \leq x)\big) = \big(1 - \text{CDF}(x)\big)^n , . $$

The pmf of $X_{(1)}$ can then be derived for any $x$ in $X$'s support as follows:

$$ P(X_{(1)} = x) = P(X_{(1)} \geq x) - P(X_{(1)} > x) = P(X_{(1)} > x - 1) - P(X_{(1)} > x) = \big(1 - \text{CDF}(x - 1)\big)^n - \big(1 - \text{CDF}(x)\big)^n $$

An appropriate test akin to test_max_discrete can be spun from this

CC @Dhruvanshu-Joshi @ricardoV94

[ricardoV94]

Closed via #6968