TransformedBeta: The Transformed Beta Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Transformed Beta distribution with parameters shape1, shape2, shape3 and scale.

Usage

dtrbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate,
        log = FALSE)
ptrbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate,
        lower.tail = TRUE, log.p = FALSE)
qtrbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate,
        lower.tail = TRUE, log.p = FALSE)
rtrbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate)
mtrbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate)
levtrbeta(limit, shape1, shape2, shape3, rate = 1, scale = 1/rate,
          order = 1)

Value

dtrbeta gives the density,

ptrbeta gives the distribution function,

qtrbeta gives the quantile function,

rtrbeta generates random deviates,

mtrbeta gives the $k$th raw moment, and

levtrbeta gives the $k$th moment of the limited loss variable.

Invalid arguments will result in return value NaN, with a warning.

Arguments

x, q: vector of quantiles.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
shape1, shape2, shape3, scale: parameters. Must be strictly positive.
rate: an alternative way to specify the scale.
log, log.p: logical; if TRUE, probabilities/densities $p$ are returned as $\log(p)$.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
order: order of the moment.
limit: limit of the loss variable.

Author

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

Details

The transformed beta distribution with parameters shape1 $= \alpha$, shape2 $= \gamma$, shape3 $= \tau$ and scale $= \theta$, has density: $$f(x) = \frac{\Gamma(\alpha + \tau)}{\Gamma(\alpha)\Gamma(\tau)} \frac{\gamma (x/\theta)^{\gamma \tau}}{% x [1 + (x/\theta)^\gamma]^{\alpha + \tau}}$$ for $x > 0$, $\alpha > 0$, $\gamma > 0$, $\tau > 0$ and $\theta > 0$. (Here $\Gamma(\alpha)$ is the function implemented by R's gamma() and defined in its help.)

The transformed beta is the distribution of the random variable $$\theta \left(\frac{X}{1 - X}\right)^{1/\gamma},$$ where $X$ has a beta distribution with parameters $\tau$ and $\alpha$.

The transformed beta distribution defines a family of distributions with the following special cases:

A Burr distribution when shape3 == 1;
A loglogistic distribution when shape1 == shape3 == 1;
A paralogistic distribution when shape3 == 1 and shape2 == shape1;
A generalized Pareto distribution when shape2 == 1;
A Pareto distribution when shape2 == shape3 == 1;
An inverse Burr distribution when shape1 == 1;
An inverse Pareto distribution when shape2 == shape1 == 1;
An inverse paralogistic distribution when shape1 == 1 and shape3 == shape2.

The $k$th raw moment of the random variable $X$ is $E[X^k]$, $-\tau\gamma < k < \alpha\gamma$.

The $k$th limited moment at some limit $d$ is $E[\min(X, d)^k]$, $k > -\tau\gamma$ and $\alpha - k/\gamma$ not a negative integer.

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Examples

Run this code

exp(dtrbeta(2, 2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
ptrbeta(qtrbeta(p, 2, 3, 4, 5), 2, 3, 4, 5)
qpearson6(0.3, 2, 3, 4, 5, lower.tail = FALSE)

## variance
mtrbeta(2, 2, 3, 4, 5) - mtrbeta(1, 2, 3, 4, 5)^2

## case with shape1 - order/shape2 > 0
levtrbeta(10, 2, 3, 4, scale = 1, order = 2)

## case with shape1 - order/shape2 < 0
levtrbeta(10, 1/3, 0.75, 4, scale = 0.5, order = 2)

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