Kumaraswamy Regression Family Function

Estimates the two parameters of the Kumaraswamy distribution by maximum likelihood estimation.

Usage

kumar(lshape1 = "loglink", lshape2 = "loglink",
      ishape1 = NULL,   ishape2 = NULL,
      gshape1 = exp(2*ppoints(5) - 1), tol12 = 1.0e-4, zero = NULL)

Arguments

lshape1, lshape2

Link function for the two positive shape parameters, respectively, called $a$ and $b$ below. See Links for more choices.

ishape1, ishape2

Numeric. Optional initial values for the two positive shape parameters.

tol12

Numeric and positive. Tolerance for testing whether the second shape parameter is either 1 or 2. If so then the working weights need to handle these singularities.

gshape1

Values for a grid search for the first shape parameter. See CommonVGAMffArguments for more information.

zero

See CommonVGAMffArguments.

Details

The Kumaraswamy distribution has density function $$f(y;a = shape1,b = shape2) = a b y^{a-1} (1-y^{a})^{b-1}$$ where $0 < y < 1$ and the two shape parameters, $a$ and $b$, are positive. The mean is $b \times Beta(1+1/a,b)$ (returned as the fitted values) and the variance is $b \times Beta(1+2/a,b) - (b \times Beta(1+1/a,b))^2$. Applications of the Kumaraswamy distribution include the storage volume of a water reservoir. Fisher scoring is implemented. Handles multiple responses (matrix input).

Value

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

References

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46, 79–88.

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6, 70–81.

Author

T. W. Yee

Examples

shape1 <- exp(1); shape2 <- exp(2)
kdata <- data.frame(y = rkumar(n = 1000, shape1, shape2))
fit <- vglm(y ~ 1, kumar, data = kdata, trace = TRUE)
#> Iteration 1: loglikelihood = 457.21842
#> Iteration 2: loglikelihood = 459.90823
#> Iteration 3: loglikelihood = 459.92136
#> Iteration 4: loglikelihood = 459.92136
#> Iteration 5: loglikelihood = 459.92136
c(with(kdata, mean(y)), head(fitted(fit), 1))
#> [1] 0.4190108 0.4192859
coef(fit, matrix = TRUE)
#>             loglink(shape1) loglink(shape2)
#> (Intercept)        1.007726        1.969458
Coef(fit)
#>   shape1   shape2 
#> 2.739364 7.166789 
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = kumar, data = kdata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  1.00773    0.02906   34.67   <2e-16 ***
#> (Intercept):2  1.96946    0.06305   31.24   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: loglink(shape1), loglink(shape2)
#> 
#> Log-likelihood: 459.9214 on 1998 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 5 
#>