View source: R/family.aunivariate.R

betaprime | R Documentation |

Estimation of the two shape parameters of the beta-prime distribution by maximum likelihood estimation.

```
betaprime(lshape = "loglink", ishape1 = 2, ishape2 = NULL,
zero = NULL)
```

`lshape` |
Parameter link function applied to the two (positive) shape
parameters. See |

`ishape1, ishape2, zero` |
See |

The beta-prime distribution is given by

`f(y) = y^{shape1-1} (1+y)^{-shape1-shape2} / B(shape1,shape2)`

for `y > 0`

.
The shape parameters are positive, and
here, `B`

is the beta function.
The mean of `Y`

is `shape1 / (shape2-1)`

provided `shape2>1`

;
these are returned as the fitted values.

If `Y`

has a `Beta(shape1,shape2)`

distribution then
`Y/(1-Y)`

and `(1-Y)/Y`

have a `Betaprime(shape1,shape2)`

and `Betaprime(shape2,shape1)`

distribution respectively.
Also, if `Y_1`

has a `gamma(shape1)`

distribution
and `Y_2`

has a `gamma(shape2)`

distribution
then `Y_1/Y_2`

has a `Betaprime(shape1,shape2)`

distribution.

An object of class `"vglmff"`

(see `vglmff-class`

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

,
`rrvglm`

and `vgam`

.

The response must have positive values only.

The beta-prime distribution is also known as the
*beta distribution of the second kind* or the
*inverted beta distribution*.

Thomas W. Yee

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995).
Chapter 25 of:
*Continuous Univariate Distributions*,
2nd edition,
Volume 2,
New York: Wiley.

`betaff`

,
`Beta`

.

```
nn <- 1000
bdata <- data.frame(shape1 = exp(1), shape2 = exp(3))
bdata <- transform(bdata, yb = rbeta(nn, shape1, shape2))
bdata <- transform(bdata, y1 = (1-yb) / yb,
y2 = yb / (1-yb),
y3 = rgamma(nn, exp(3)) / rgamma(nn, exp(2)))
fit1 <- vglm(y1 ~ 1, betaprime, data = bdata, trace = TRUE)
coef(fit1, matrix = TRUE)
fit2 <- vglm(y2 ~ 1, betaprime, data = bdata, trace = TRUE)
coef(fit2, matrix = TRUE)
fit3 <- vglm(y3 ~ 1, betaprime, data = bdata, trace = TRUE)
coef(fit3, matrix = TRUE)
# Compare the fitted values
with(bdata, mean(y3))
head(fitted(fit3))
Coef(fit3) # Useful for intercept-only models
```

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.

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