Minimax distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ minimax(β, γ) is used to indicate that the random variable X has the minimax distribution with positive shape parameters β and γ. A minimax random variable X with parameters β and γ has probability density function

f(x) = βγx^{β−1} ( 1 − x^{β}) ^{γ−1} 0 < x < 1.

The probability density function with three different parameter settings is illustrated below.

PSfrag replacements α = 2, β = 2

α = 1, β = 3
α = 0.5, β = 0.5
x
f(x)

The cumulative distribution on the support of X is

F (x) = P(X ≤ x) = 1 − ( 1 − x^{β}) ^{γ} 0 < x < 1.

The survivor functionx on the support of X is

S(x) = P(X ≥ x) = ( 1 − x^{β}) ^{γ} 0 < x < 1.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{βγx^{β−1}}{1 − x^{β}} 0 < x < 1.

The inverse distribution function of X is

F^{−1} (u) = ( 1 − (1 − u)^{1/γ}) ^{1/β} 0 < u < 1.

The cumulative hazard, moment generating, and characteristic functions on the support of X are mathematically intractable.

The population mean of X is

E[X] = \frac{Γ(γ + 1) Γ((β + 1)/β )}{Γ((βγ + β + 1)/β)} .