kumar {VGAM} | R Documentation

## Kumaraswamy Distribution Family Function

### Description

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

### Usage

    kumar(lshape1 = "loge", lshape2 = "loge",
          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

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 * Beta(1+1/a,b) (returned as the fitted values) and the variance is b * Beta(1+2/a,b) - (b * 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`.

### Author(s)

T. W. Yee

### 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.

### See Also

`dkumar`, `betaff`, `simulate.vlm`.

### 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)
    c(with(kdata, mean(y)), head(fitted(fit), 1))
    coef(fit, matrix = TRUE)
    Coef(fit)
    summary(fit)

[Package VGAM version 1.0-4 Index]