The Inverse-Gamma (IGam) Distribution

Density, distribution function, quantile function and random generation for the inverse-Gamma distribution with parameters a and b.

digam(x, a, b, log = FALSE)

rigam(n, a, b)

pigam(q, a, b, lower.tail = TRUE, log.p = FALSE)

qigam(p, a, b, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q

vector of quantiles.

a, b

parameters a and b. Must be positive.

log, log.p

logical; if TRUE, probabilities/densities \(p\) are returned as \(log(p)\).

n,

number of observations.

lower.tail

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

p

vector of probabilities.

Value

digam gives the density, pigam gives the distribution function, qigam gives the quantile function, and rigam generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is determined by n for rig.

Details

The inverse-Gamma distribution has density given by $$f(x; a, b) = \frac{b^a}{\Gamma(a)}x^{-a-1}\exp( -\frac{b}{x}),$$ where \(x>0\) and \(a,b>0\).

See also

dnig, dgig

Examples

rigam(100, a = 1, b = 1)
#>   [1]   1.9016494   0.6821781   0.8838798  43.2110479   0.9627293   0.3216433
#>   [7]   3.5728557   1.1267577   0.6926790   8.1519486   3.8917801   0.6603720
#>  [13]  96.3678371   0.4465320   1.3857696   1.1778767   2.3916497   0.5945125
#>  [19]   2.9865776   2.3985589   0.9382666   1.2940753   0.4783434   1.8108502
#>  [25]   0.3190744   0.1874310  12.0142534   0.6575729   0.6795114   0.5025854
#>  [31]   2.5502824   0.5485328   7.8547933   2.8733430   3.6970014   0.7854427
#>  [37]   1.7540894   1.3172715   0.5781822  37.8384551   2.4847504   0.7950980
#>  [43]   3.7688737   5.1461741  13.2975012   7.1454609   0.8063314   0.3863400
#>  [49]   2.3339125   0.3122662  64.7762844   2.4377386 162.6405670   3.5126051
#>  [55]   1.1402587   7.5171413   0.4312803   1.3225483   0.7468765   7.3894906
#>  [61]  12.3562802   9.0798559   3.6263994   2.0857028   2.0298622   1.6741318
#>  [67]   1.3181996   0.4923331   0.4009200   0.4967066   0.5959504   0.4628702
#>  [73]   5.5408177   1.5586563   1.0934053   2.1239243   4.5799413   0.4346557
#>  [79]   1.2439756   2.5793766   0.4924331   1.0940504 272.9588328  27.1022658
#>  [85]   0.9459965   4.0250192   1.2334103   0.6109278   0.5588355   0.6637669
#>  [91]  20.2204337   0.6100783   2.8103526   0.5865932   6.1105303   3.0850092
#>  [97]   0.4082117   6.4973408   1.0950702   0.8311981
pigam(0.4, a = 1, b = 1)
#> [1] 0.082085
qigam(0.8, a = 1, b = 1)
#> [1] 4.48142
plot(function(x){digam(x, a = 1, b = 1)}, main =
"Inverse-Gamma density", ylab = "Probability density",
xlim = c(0,10))