paralogistic: Paralogistic 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 paralogistic distribution.

Usage

1
2
3
4
paralogistic(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 ishape1.a is needed to obtain good estimates for the other parameter.

gscale, gshape1.a

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

Details

The 2-parameter paralogistic distribution is the 4-parameter generalized beta II distribution with shape parameter p=1 and a=q. It is the 3-parameter Singh-Maddala distribution with a=q. More details can be found in Kleiber and Kotz (2003).

The 2-parameter paralogistic has density

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

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

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

Paralogistic, sinmad, genbetaII, betaII, dagum, fisk, inv.lomax, lomax, inv.paralogistic.

Examples

1
2
3
4
5
6
7
pdata <- data.frame(y = rparalogistic(n = 3000, exp(1), scale = exp(1)))
fit <- vglm(y ~ 1, paralogistic(lss = FALSE), data = pdata, trace = TRUE)
fit <- vglm(y ~ 1, paralogistic(ishape1.a = 2.3, iscale = 5),
            data = pdata, 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 = -3616.63447
VGLM    linear loop  2 :  loglikelihood = -3616.47705
VGLM    linear loop  3 :  loglikelihood = -3616.47344
VGLM    linear loop  4 :  loglikelihood = -3616.47344
Taking a modified step.
VGLM    linear loop  4 :  loglikelihood = -3616.47339
VGLM    linear loop  5 :  loglikelihood = -3616.47342
Taking a modified step....................
Warning messages:
1: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
  iterations terminated because half-step sizes are very small
2: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
  some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
VGLM    linear loop  1 :  loglikelihood = -4733.7391
VGLM    linear loop  2 :  loglikelihood = -4061.44235
VGLM    linear loop  3 :  loglikelihood = -3650.61063
VGLM    linear loop  4 :  loglikelihood = -3631.24299
VGLM    linear loop  5 :  loglikelihood = -3616.97948
VGLM    linear loop  6 :  loglikelihood = -3616.55755
VGLM    linear loop  7 :  loglikelihood = -3616.47613
VGLM    linear loop  8 :  loglikelihood = -3616.47341
VGLM    linear loop  9 :  loglikelihood = -3616.47343
Taking a modified step.
VGLM    linear loop  9 :  loglikelihood = -3616.47339
VGLM    linear loop  10 :  loglikelihood = -3616.47342
Taking a modified step....................
Warning messages:
1: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
  iterations terminated because half-step sizes are very small
2: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
  some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
            loge(scale) loge(shape1.a)
(Intercept)   0.9881697       1.021211
   scale shape1.a 
2.686313 2.776556 

Call:
vglm(formula = y ~ 1, family = paralogistic(ishape1.a = 2.3, 
    iscale = 5), data = pdata, trace = TRUE)


Pearson residuals:
                   Min      1Q  Median     3Q    Max
loge(scale)    -0.9544 -0.8029 -0.3514 0.4935 4.6489
loge(shape1.a) -6.6837 -0.3512  0.3532 0.7234 0.8357

Coefficients: 
              Estimate Std. Error z value Pr(>|z|)    
(Intercept):1  0.98817    0.00953  103.69   <2e-16 ***
(Intercept):2  1.02121    0.01466   69.66   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2 

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

Log-likelihood: -3616.473 on 5998 degrees of freedom

Number of iterations: 10 

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