gammaDist

This is an R package that implements gamma difference distributions under the null of equal distributions, including the cumulative distribution function (CDF) derived by Klar (2015) and saddlepoint approximations to the CDF and density (Butler, 2007). In particular, for $X_i \sim \text{Gamma}(alpha, \lambda), i=1, \ldots, n_x$ and $Y_j \sim \text{Gamma}(\alpha, \lambda), j=1, \ldots, n_y,$ gammDist computes the distribution and density of $Z = \frac{1}{n_x} \sum_i X_i - \frac{1}{n_y} \sum_j Y_j.$

Due to large gamma and incomplete gamma function values, the true distribution cannot always be computed. However, the saddlepoint approximation is accurate far into the tails (e.g. for left tail probabilities around 10^-200). We use the saddlepoint approximation of Lugannani and Rice (1980) for the cumulative distribution function.

gammaDist also includes a function for computing the maximum likelihood and method of moments estimates for iid gamma random variables.

For additional details, please see Appendix B.2, pages 14-17, in the web-based supplementary materials for Segal et al. (2018), available freely under "supporting information" at doi:10.1111/biom.12731.

Installation

# library("devtools")
install_github("bdsegal/gammaDist")

Examples

library(gammaDist)

# generate data
nx <- 60
ny <- 60
alpha <- 1
lambda <- 4
x <- rgamma(n = nx, shape = alpha, rate = lambda)
y <- rgamma(n = ny, shape = alpha, rate = lambda)

# get parameter estimates for pooled data
fit <- gammaFit(z = c(x,y))
fit$MLE
fit$MoM

# Plot true distribution and saddlepoint approximation
q <- seq(-0.2, 0.2, 0.001)
plot(q, pgammaDif(q = q, nx, ny, alpha, lambda),
     type = "l", lty = 1, lwd = 2, ylab = "F(q)")
lines(q, pgammaDifSaddle(q = q, nx, ny, alpha, lambda),
      col = "red", lty = 2, lwd = 2)
legend("topleft", legend = c("True", "Saddlepoint"), 
       col = c("black", "red"), lty = 1:2, inset = 0.02)

# Plot saddlepoint approximation of the density
plot(q, dgammaDifSaddle(z = q, nx, ny, alpha, lambda),
     type = "l", ylab = "f(z)", xlab = "z")

References

Butler, R. W. (2007). Saddlepoint approximations with applications. Cambridge University Press, Cambridge, UK.

Klar, B. (2015). A note on gamma difference distributions. Journal of Statistical Computation and Simulation 85, 3708–3715.

Lugannani, R. and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Advances in applied probability 12, 475–490.

Segal, B. D., Braun, T., Elliott, M. R. and Jiang, H. (2018). Fast approximation of small p-values in permutation tests by partitioning the permutations. Biometrics, 74(1), 196-206. doi:10.1111/biom.12731.