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

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

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

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

and `vgam`

.

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).

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.

# NOT RUN { 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) # }

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