Log-gamma Distribution Family Function
Estimation of the parameter of the standard and nonstandard log-gamma distribution.
lgamma1(lshape = "loglink", ishape = NULL)
lgamma3(llocation = "identitylink", lscale = "loglink", lshape = "loglink",
ilocation = NULL, iscale = NULL, ishape = 1,
zero = c("scale", "shape"))llocation, lscale |
Parameter link function applied to the
location parameter a
and the positive scale parameter b.
See |
lshape |
Parameter link function applied to
the positive shape parameter k.
See |
ishape |
Initial value for k. If given, it must be positive. If failure to converge occurs, try some other value. The default means an initial value is determined internally. |
ilocation, iscale |
Initial value for a and b. The defaults mean an initial value is determined internally for each. |
zero |
An integer-valued vector specifying which
linear/additive predictors are modelled as intercepts only.
The values must be from the set {1,2,3}.
The default value means none are modelled as intercept-only terms.
See |
The probability density function of the standard log-gamma distribution is given by
f(y;k) = exp[ky - exp(y)]/gamma(k),
for parameter k>0 and all real y.
The mean of Y is digamma(k) (returned as
the fitted values) and its variance is trigamma(k).
For the non-standard log-gamma distribution, one replaces y by (y-a)/b, where a is the location parameter and b is the positive scale parameter. Then the density function is
f(y) = exp[k(y-a)/b - exp((y-a)/b)]/(b*gamma(k)).
The mean and variance of Y are a + b*digamma(k) (returned as
the fitted values) and b^2 * trigamma(k), respectively.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
The standard log-gamma distribution can be viewed as a generalization of the standard type 1 extreme value density: when k = 1 the distribution of -Y is the standard type 1 extreme value distribution.
The standard log-gamma distribution is fitted with lgamma1
and the non-standard (3-parameter) log-gamma distribution is fitted
with lgamma3.
T. W. Yee
Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications, pages 48–49, London: Imperial College Press.
Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd edition, Volume 2, p.89, New York: Wiley.
ldata <- data.frame(y = rlgamma(100, shape = exp(1))) fit <- vglm(y ~ 1, lgamma1, data = ldata, trace = TRUE, crit = "coef") summary(fit) coef(fit, matrix = TRUE) Coef(fit) ldata <- data.frame(x2 = runif(nn <- 5000)) # Another example ldata <- transform(ldata, loc = -1 + 2 * x2, Scale = exp(1)) ldata <- transform(ldata, y = rlgamma(nn, loc, scale = Scale, shape = exp(0))) fit2 <- vglm(y ~ x2, lgamma3, data = ldata, trace = TRUE, crit = "c") coef(fit2, matrix = TRUE)
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