Estimation of the shape parameters of the two-parameter beta distribution.
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An argument which is fed into
Lower and upper limits of the distribution. The defaults correspond to the standard beta distribution where the response lies between 0 and 1.
The two-parameter beta distribution is given by f(y) =
(y-A)^(shape1-1) * (B-y)^(shape2-1) / [Beta(shape1,shape2) * (B-A)^(shape1+shape2-1)]
for A < y < B, and Beta(.,.) is the beta function
The shape parameters are positive, and
here, the limits A and B are known.
The mean of Y is E(Y) = A + (B-A) * shape1 /
(shape1 + shape2), and these are the fitted values of the object.
For the standard beta distribution the variance of Y is shape1 * shape2 / ((1+shape1+shape2) * (shape1+shape2)^2). If σ^2= 1 / (1+shape1+shape2) then the variance of Y can be written mu*(1-mu)*sigma^2 where mu=shape1 / (shape1 + shape2) is the mean of Y.
Another parameterization of the beta distribution involving the mean
and a precision parameter is implemented in
An object of class
The object is used by modelling functions such as
The response must have values in the interval (A, B).
VGAM 0.7-4 and prior called this function
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.
Gupta, A. K. and Nadarajah, S. (2004). Handbook of Beta Distribution and Its Applications, New York: Marcel Dekker.
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bdata <- data.frame(y = rbeta(n = 1000, shape1 = exp(0), shape2 = exp(1))) fit <- vglm(y ~ 1, betaR(lshape1 = "identitylink", lshape2 = "identitylink"), data = bdata, trace = TRUE, crit = "coef") fit <- vglm(y ~ 1, betaR, data = bdata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) # Useful for intercept-only models bdata <- transform(bdata, Y = 5 + 8 * y) # From 5 to 13, not 0 to 1 fit <- vglm(Y ~ 1, betaR(A = 5, B = 13), data = bdata, trace = TRUE) Coef(fit) c(meanY = with(bdata, mean(Y)), head(fitted(fit),2))
Loading required package: stats4 Loading required package: splines VGLM linear loop 1 : coefficients = 0.99453982, 2.66042022 VGLM linear loop 2 : coefficients = 1.0100043, 2.6876497 VGLM linear loop 3 : coefficients = 1.0102765, 2.6881580 VGLM linear loop 4 : coefficients = 1.0102766, 2.6881582 VGLM linear loop 1 : coefficients = 0.00023757603, 0.97934360189 VGLM linear loop 2 : coefficients = 0.010165579, 0.988799243 VGLM linear loop 3 : coefficients = 0.010224105, 0.988856275 VGLM linear loop 4 : coefficients = 0.010224107, 0.988856277 VGLM linear loop 5 : coefficients = 0.010224107, 0.988856277 loge(shape1) loge(shape2) (Intercept) 0.01022411 0.9888563 shape1 shape2 1.010277 2.688158 VGLM linear loop 1 : loglikelihood = -1723.9357 VGLM linear loop 2 : loglikelihood = -1723.9035 VGLM linear loop 3 : loglikelihood = -1723.9035 shape1 shape2 1.010277 2.688158 meanY 7.181428 7.185306 7.185306
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