Oilog {VGAMdata}

One-Inflated Logarithmic Distribution

Description

Density, distribution function, quantile function and random generation for the one-inflated logarithmic distribution with parameter pstr1.

Usage

doilog(x, shape, pstr1 = 0, log = FALSE)
poilog(q, shape, pstr1 = 0)
qoilog(p, shape, pstr1 = 0)
roilog(n, shape, pstr1 = 0)

Arguments

x, q, p, n

Same as Uniform.

shape

Vector of parameters that lie in (0,1).

pstr1

Probability of a structural one (i.e., ignoring the logarithmic distribution), called \phi. The default value of \phi = 0 corresponds to the response having an ordinary logarithmic distribution.

log

Same as Uniform.

Details

The probability function of Y is 1 with probability \phi, and Logarithmic(prob) with probability 1-\phi. Thus

P(Y=1) =\phi + (1-\phi) P(W=1)

where W is distributed as a Logarithmic(shape) random variable. The VGAM family function oilog estimates \phi by MLE.

Value

doilog gives the density, poilog gives the distribution function, qoilog gives the quantile function, and roilog generates random deviates.

Note

The argument pstr1 is recycled to the required length, and usually has values which lie in the interval [0,1].

These functions actually allow for the zero-deflated logarithmic distribution. Here, pstr1 is also permitted to lie in the interval [-dlog(1, shape) / (1 - dlog(1, shape)), 0]. The resulting probability of a unit count is less than the nominal logarithmic value, and the use of pstr1 to stand for the probability of a structural 1 loses its meaning. When pstr1 equals -dlog(1, shape) / (1 - dlog(1, shape)) this corresponds to the 1-truncated logarithmic distribution.

Author(s)

T. W. Yee

See Also

Gaitdlog, oilog, rlog, logff, Otlog.

Examples

shape <- 0.5; pstr1 <- 0.3; x <- (-1):7
(ii <- doilog(x, shape, pstr1 = pstr1))
max(abs(poilog(1:200, shape) -
  cumsum(shape^(1:200) / (-(1:200) * log1p(-shape)))))  # Should be 0

## Not run:  x <- 0:10
par(mfrow = c(2, 1))  # One-Inflated logarithmic
barplot(rbind(doilog(x, shape, pstr1 = pstr1), dlog(x, shape)),
   beside = TRUE, col = c("blue", "orange"),
   main = paste0("OILogff(", shape, ", pstr1 = ", pstr1,
                 ") (blue) vs Logff(", shape, ") (orange)"),
   names.arg = as.character(x))

deflat.limit <- -dlog(1, shape) / plog(1, shape, lower.tail = FALSE)
newpstr1 <- round(deflat.limit, 3) + 0.001  # Near the boundary
barplot(rbind(doilog(x, shape, pstr1 = newpstr1),
                dlog(x, shape)),
   beside = TRUE, col = c("blue","orange"),
   main = paste0("ODLogff(", shape, ", pstr1 = ", newpstr1,
                 ") (blue) vs Logff(", shape, ") (orange)"),
   names.arg = as.character(x)) 
## End(Not run)