Polono {VGAM}R Documentation

The Poisson Lognormal Distribution

Description

Density, and random generation for the Poisson lognormal distribution.

Usage

dpolono(x, meanlog=0, sdlog=1, bigx=Inf, ...)
rpolono(n, meanlog=0, sdlog=1)

Arguments

x vector of quantiles.
n number of observations. Must be a positive integer of length 1.
meanlog, sdlog the mean and standard deviation of the normal distribution (on the log scale). They match the arguments in Lognormal.
bigx Numeric. This argument is for handling large values of x and/or when integrate fails. A first order Taylor series approximation [Equation (7) of Bulmer (1974)] is used at values of x that are greater or equal to this argument. For bigx=10, he showed that the approximation has a relative error less than 0.001 for values of meanlog and sdlog ``likely to be encountered in practice''. The default value means that this approximation is not used. Setting something like bigx=100 may be a good idea.
... Arguments passed into integrate.

Details

The Poisson lognormal distribution is similar to the negative binomial in that it can be motivated by a Poisson distribution whose mean parameter comes from a right skewed distribution (gamma for the negative binomial and lognormal for the Poisson lognormal distribution).

Value

dpolono gives the density, and rpolono generates random deviates.

Note

By default, dpolono involves numerical integration that is performed using integrate. Consequently, computations are very slow and numerical problems may occur (if so then the use of ... may be needed). Alternatively, for extreme values of x, meanlog, sdlog, etc., the use of bigx avoids the call to integrate; however the answer may be a little inaccurate.

For the maximum likelihood estimation of the 2 parameters a VGAM family function called polono, say, has not been written yet.

Author(s)

T. W. Yee

References

Bulmer, M. G. (1974) On fitting the Poisson lognormal distribution to species-abundance data. Biometrics, 30, 101–110.

See Also

lognormal, poissonff, negbinomial.

Examples

meanlog = 0.5; sdlog = 0.5
y = 0:19
proby = dpolono(y, m=meanlog, sd=sdlog)
sum(proby)  # Should be 1
## Not run: 
opar = par(no.readonly = TRUE)
par(mfrow=c(2,2))
plot(y, proby, type="h", col="blue", ylab="P[Y=y]", log="",
     main=paste("Poisson lognormal(meanlog=",meanlog,", sdlog=",sdlog,")",
                sep=""))

# More extreme values; use the approximation and plot on a log scale
# Notice the kink at bigx.
y = 0:190
proby = dpolono(y, m=meanlog, sd=sdlog, bigx=100)
sum(proby)  # Should be 1
plot(y, proby, type="h", col="blue", ylab="P[Y=y]", log="y",
     main=paste("Poisson lognormal(meanlog=",meanlog,", sdlog=",sdlog,")"))

# Random number generation
table(y <- rpolono(n=1000, m=meanlog, sd=sdlog))
hist(y, breaks=((-1):max(y))+0.5, prob=TRUE, border="blue")
par(opar)
## End(Not run)

[Package VGAM version 0.7-7 Index]