VGAM (version 1.1-9)

geometric: Geometric (Truncated and Untruncated) Distributions

Description

Maximum likelihood estimation for the geometric and truncated geometric distributions.

Usage

geometric(link = "logitlink", expected = TRUE, imethod = 1,
          iprob = NULL, zero = NULL)
truncgeometric(upper.limit = Inf,
               link = "logitlink", expected = TRUE, imethod = 1,
               iprob = NULL, zero = NULL)

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Arguments

link

Parameter link function applied to the probability parameter \(p\), which lies in the unit interval. See Links for more choices.

expected

Logical. Fisher scoring is used if expected = TRUE, else Newton-Raphson.

iprob, imethod, zero

See CommonVGAMffArguments for details.

upper.limit

Numeric. Upper values. As a vector, it is recycled across responses first. The default value means both family functions should give the same result.

Author

T. W. Yee. Help from Viet Hoang Quoc is gratefully acknowledged.

Details

A random variable \(Y\) has a 1-parameter geometric distribution if \(P(Y=y) = p (1-p)^y\) for \(y=0,1,2,\ldots\). Here, \(p\) is the probability of success, and \(Y\) is the number of (independent) trials that are fails until a success occurs. Thus the response \(Y\) should be a non-negative integer. The mean of \(Y\) is \(E(Y) = (1-p)/p\) and its variance is \(Var(Y) = (1-p)/p^2\). The geometric distribution is a special case of the negative binomial distribution (see negbinomial). The geometric distribution is also a special case of the Borel distribution, which is a Lagrangian distribution. If \(Y\) has a geometric distribution with parameter \(p\) then \(Y+1\) has a positive-geometric distribution with the same parameter. Multiple responses are permitted.

For truncgeometric(), the (upper) truncated geometric distribution can have response integer values from 0 to upper.limit. It has density prob * (1 - prob)^y / [1-(1-prob)^(1+upper.limit)].

For a generalized truncated geometric distribution with integer values \(L\) to \(U\), say, subtract \(L\) from the response and feed in \(U-L\) as the upper limit.

References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

See Also

negbinomial, Geometric, betageometric, expgeometric, zageometric, zigeometric, rbetageom, simulate.vlm.

Examples

gdata <- data.frame(x2 = runif(nn <- 1000) - 0.5)
gdata <- transform(gdata, x3 = runif(nn) - 0.5,
                          x4 = runif(nn) - 0.5)
gdata <- transform(gdata, eta  = -1.0 - 1.0 * x2 + 2.0 * x3)
gdata <- transform(gdata, prob = logitlink(eta, inverse = TRUE))
gdata <- transform(gdata, y1 = rgeom(nn, prob))
with(gdata, table(y1))
fit1 <- vglm(y1 ~ x2 + x3 + x4, geometric, data = gdata, trace = TRUE)
coef(fit1, matrix = TRUE)
summary(fit1)

# Truncated geometric (between 0 and upper.limit)
upper.limit <- 5
tdata <- subset(gdata, y1 <= upper.limit)
nrow(tdata)  # Less than nn
fit2 <- vglm(y1 ~ x2 + x3 + x4, truncgeometric(upper.limit),
             data = tdata, trace = TRUE)
coef(fit2, matrix = TRUE)

# Generalized truncated geometric (between lower.limit and upper.limit)
lower.limit <- 1
upper.limit <- 8
gtdata <- subset(gdata, lower.limit <= y1 & y1 <= upper.limit)
with(gtdata, table(y1))
nrow(gtdata)  # Less than nn
fit3 <- vglm(y1 - lower.limit ~ x2 + x3 + x4,
             truncgeometric(upper.limit - lower.limit),
             data = gtdata, trace = TRUE)
coef(fit3, matrix = TRUE)