fisk: Fisk Distribution family function

Description Usage Arguments Details Value Note Author(s) References See Also Examples

View source: R/family.actuary.R

Description

Maximum likelihood estimation of the 2-parameter Fisk distribution.

Usage

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fisk(lscale = "loglink", lshape1.a = "loglink", iscale = NULL,
    ishape1.a = NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5),
    gshape1.a = seq(0.75, 4, by = 0.25), probs.y = c(0.25, 0.5, 0.75),
    zero = "shape")

Arguments

lss

See CommonVGAMffArguments for important information.

lshape1.a, lscale

Parameter link functions applied to the (positive) parameters a and scale. See Links for more choices.

iscale, ishape1.a, imethod, zero

See CommonVGAMffArguments for information. For imethod = 2 a good initial value for iscale is needed to obtain a good estimate for the other parameter.

gscale, gshape1.a

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

Details

The 2-parameter Fisk (aka log-logistic) distribution is the 4-parameter generalized beta II distribution with shape parameter q=p=1. It is also the 3-parameter Singh-Maddala distribution with shape parameter q=1, as well as the Dagum distribution with p=1. More details can be found in Kleiber and Kotz (2003).

The Fisk distribution has density

f(y) = a y^(a-1) / [b^a (1 + (y/b)^a)^2]

for a > 0, b > 0, y >= 0. Here, b is the scale parameter scale, and a is a shape parameter. The cumulative distribution function is

F(y) = 1 - [1 + (y/b)^a]^(-1) = [1 + (y/b)^(-a)]^(-1).

The mean is

E(Y) = b gamma(1 + 1/a) gamma(1 - 1/a)

provided a > 1; these are returned as the fitted values. This family function handles multiple responses.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Note

See the notes in genbetaII.

Author(s)

T. W. Yee

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

See Also

Fisk, genbetaII, betaII, dagum, sinmad, inv.lomax, lomax, paralogistic, inv.paralogistic, simulate.vlm.

Examples

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fdata <- data.frame(y = rfisk(n = 200, shape = exp(1), scale = exp(2)))
fit <- vglm(y ~ 1, fisk(lss = FALSE), data = fdata, trace = TRUE)
fit <- vglm(y ~ 1, fisk(ishape1.a = exp(2)), data = fdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)

Example output

Loading required package: stats4
Loading required package: splines
VGLM    linear loop  1 :  loglikelihood = -627.60791
VGLM    linear loop  2 :  loglikelihood = -627.60786
VGLM    linear loop  3 :  loglikelihood = -627.60786
VGLM    linear loop  1 :  loglikelihood = -717.9889
VGLM    linear loop  2 :  loglikelihood = -647.18601
VGLM    linear loop  3 :  loglikelihood = -628.49268
VGLM    linear loop  4 :  loglikelihood = -627.6093
VGLM    linear loop  5 :  loglikelihood = -627.60786
VGLM    linear loop  6 :  loglikelihood = -627.60786
            loglink(scale) loglink(shape1.a)
(Intercept)       2.020942         0.9049797
   scale shape1.a 
7.545433 2.471882 

Call:
vglm(formula = y ~ 1, family = fisk(ishape1.a = exp(2)), data = fdata, 
    trace = TRUE)

Pearson residuals:
                     Min      1Q   Median     3Q    Max
loglink(scale)    -1.681 -0.9244 0.005981 0.9394 1.7301
loglink(shape1.a) -5.375 -0.4263 0.291090 0.7389 0.8363

Coefficients: 
              Estimate Std. Error z value Pr(>|z|)    
(Intercept):1  2.02094    0.04955   40.79   <2e-16 ***
(Intercept):2  0.90498    0.05913   15.30   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: loglink(scale), loglink(shape1.a)

Log-likelihood: -627.6079 on 398 degrees of freedom

Number of Fisher scoring iterations: 6 

No Hauck-Donner effect found in any of the estimates

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.