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The log-gamma distribution

lgamma.Rd

Density, distribution function and gradient of density for the log-gamma distribution. These are implemented in C for speed and care is taken that the correct results are provided for values of NA, NaN, Inf, -Inf or just extremely small or large values.

The log-gamma is a flexible location-scale distribution on the real line with an extra parameter, \(\lambda\). For \(\lambda = 0\) the distribution equals the normal or Gaussian distribution, and for \(\lambda\) equal to 1 and -1, the Gumbel minimum and maximum distributions are obtained.

Usage

plgamma(q, lambda, lower.tail = TRUE)

dlgamma(x, lambda, log = FALSE)

glgamma(x, lambda)

Arguments

x,q

numeric vector of quantiles.

lambda

numerical scalar

lower.tail

logical; if TRUE (default), probabilities are \(P[X \leq x]\) otherwise, \(P[X > x]\).

log

logical; if TRUE, probabilities p are given as log(p).

Details

If \(\lambda < 0\) the distribution is right skew, if \(\lambda = 0\) the distribution is symmetric (and equals the normal distribution), and if \(\lambda > 0\) the distribution is left skew.

These distribution functions, densities and gradients are used in the Newton-Raphson algorithms in fitting cumulative link models with clm2 and cumulative link mixed models with clmm2 using the log-gamma link.

Value

plgamma gives the distribution function, dlgamma gives the density and glgamma gives the gradient of the density.

References

Genter, F. C. and Farewell, V. T. (1985) Goodness-of-link testing in ordinal regression models. The Canadian Journal of Statistics, 13(1), 37-44.

See also

Gradients of densities are also implemented for the normal, logistic, cauchy, cf. gfun and the Gumbel distribution, cf. gumbel.

Author

Rune Haubo B Christensen

Examples


## Illustrating the link to other distribution functions: 
x <- -5:5
plgamma(x, lambda = 0) == pnorm(x)
#>  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
all.equal(plgamma(x, lambda = -1), pgumbel(x)) ## TRUE, but:
#> [1] TRUE
plgamma(x, lambda = -1) == pgumbel(x)
#>  [1] FALSE FALSE FALSE FALSE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE
plgamma(x, lambda = 1) == pgumbel(x, max = FALSE)
#>  [1] FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE

dlgamma(x, lambda = 0) == dnorm(x)
#>  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
dlgamma(x, lambda = -1) == dgumbel(x)
#>  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
dlgamma(x, lambda = 1) == dgumbel(x, max = FALSE)
#>  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

glgamma(x, lambda = 0) == gnorm(x)
#>  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
all.equal(glgamma(x, lambda = -1), ggumbel(x)) ## TRUE, but:
#> [1] TRUE
glgamma(x, lambda = -1) == ggumbel(x)
#>  [1]  TRUE FALSE FALSE  TRUE  TRUE  TRUE FALSE FALSE FALSE  TRUE FALSE
all.equal(glgamma(x, lambda = 1), ggumbel(x, max = FALSE)) ## TRUE, but:
#> [1] TRUE
glgamma(x, lambda = 1) == ggumbel(x, max = FALSE)
#>  [1] FALSE  TRUE FALSE FALSE FALSE  TRUE  TRUE  TRUE FALSE FALSE  TRUE
## There is a loss of accuracy, but the difference is very small: 
glgamma(x, lambda = 1) - ggumbel(x, max = FALSE)
#>  [1] -8.673617e-19  0.000000e+00 -1.387779e-17 -1.387779e-17 -2.775558e-17
#>  [6]  0.000000e+00  0.000000e+00  0.000000e+00  1.058791e-22 -7.523164e-37
#> [11]  0.000000e+00

## More examples:
x <- -5:5
plgamma(x, lambda = .5)
#>  [1] 0.0003729435 0.0023299245 0.0130924716 0.0621186950 0.2267546348
#>  [6] 0.5665298796 0.8945151191 0.9945917346 0.9999813218 0.9999999993
#> [11] 1.0000000000
dlgamma(x, lambda = .5)
#>  [1] 6.974573e-04 4.164869e-03 2.166083e-02 8.970337e-02 2.551632e-01
#>  [6] 3.907336e-01 2.155388e-01 2.208500e-02 1.410374e-04 9.274904e-09
#> [11] 3.227334e-16
glgamma(x, lambda = .5)
#>  [1]  1.280413e-03  7.202431e-03  3.365529e-02  1.134067e-01  2.007978e-01
#>  [6]  0.000000e+00 -2.796492e-01 -7.589652e-02 -9.820967e-04 -1.185158e-07
#> [11] -7.217929e-15