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Based on the log-gamma distribution:
lambda = 0: probit (normal)
lambda > 0: F(x) = pgamma(exp(x), shape = lambda, rate = lambda)
lambda < 0: F(x) = 1 - pgamma(exp(-x), shape = -lambda, rate = -lambda)
loggamma_cdf(x, lambda)
CDF values
Numeric vector
Shape parameter
Reference: Prentice (1976) Biometrics