Beta Distribution of the Second Kind

Maximum likelihood estimation of the 3-parameter beta II distribution.

betaII(lscale = "loglink", lshape2.p = "loglink",
       lshape3.q = "loglink", iscale = NULL, ishape2.p = NULL,
       ishape3.q = NULL, imethod = 1,
       gscale = exp(-5:5), gshape2.p = exp(-5:5),
       gshape3.q = seq(0.75, 4, by = 0.25),
       probs.y = c(0.25, 0.5, 0.75), zero = "shape")

Arguments

lscale, lshape2.p, lshape3.q

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

iscale, ishape2.p, ishape3.q, imethod, zero

See CommonVGAMffArguments for information.

gscale, gshape2.p, gshape3.q

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

Details

The 3-parameter beta II is the 4-parameter generalized beta II distribution with shape parameter \(a=1\). It is also known as the Pearson VI distribution. Other distributions which are special cases of the 3-parameter beta II include the Lomax (\(p=1\)) and inverse Lomax (\(q=1\)). More details can be found in Kleiber and Kotz (2003).

The beta II distribution has density $$f(y) = y^{p-1} / [b^p B(p,q) \{1 + y/b\}^{p+q}]$$ for \(b > 0\), \(p > 0\), \(q > 0\), \(y \geq 0\). Here, \(b\) is the scale parameter scale, and the others are shape parameters. The mean is $$E(Y) = b \, \Gamma(p + 1) \, \Gamma(q - 1) / (\Gamma(p) \, \Gamma(q))$$ provided \(q > 1\); 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.

References

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

Author

T. W. Yee

Note

See the notes in genbetaII.

See also

betaff, genbetaII, dagum, sinmad, fisk, inv.lomax, lomax, paralogistic, inv.paralogistic.

Examples

bdata <- data.frame(y = rsinmad(2000, shape1.a = 1,
         shape3.q = exp(2), scale = exp(1)))  # Not genuine data!
# fit <- vglm(y ~ 1, betaII, data = bdata, trace = TRUE)
fit <- vglm(y ~ 1, betaII(ishape2.p = 0.7, ishape3.q = 0.7),
            data = bdata, trace = TRUE)
#> Iteration 1: loglikelihood = -403.212452
#> Iteration 2: loglikelihood = -341.043034
#> Iteration 3: loglikelihood = -329.979204
#> Iteration 4: loglikelihood = -326.497757
#> Iteration 5: loglikelihood = -326.072765
#> Iteration 6: loglikelihood = -326.052142
#> Iteration 7: loglikelihood = -326.051897
#> Iteration 8: loglikelihood = -326.051895
#> Iteration 9: loglikelihood = -326.051895
coef(fit, matrix = TRUE)
#>             loglink(scale) loglink(shape2.p) loglink(shape3.q)
#> (Intercept)       1.551698        -0.1000982          2.374101
Coef(fit)
#>      scale   shape2.p   shape3.q 
#>  4.7194769  0.9047485 10.7413555 
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = betaII(ishape2.p = 0.7, ishape3.q = 0.7), 
#>     data = bdata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  1.55170    0.39589   3.919 8.87e-05 ***
#> (Intercept):2 -0.10010    0.03575  -2.800  0.00511 ** 
#> (Intercept):3  2.37410    0.34218   6.938 3.97e-12 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: loglink(scale), loglink(shape2.p), 
#> loglink(shape3.q)
#> 
#> Log-likelihood: -326.0519 on 5997 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 9 
#> 
#> Warning: Hauck-Donner effect detected in the following estimate(s):
#> '(Intercept):3'
#>