Hzeta {VGAM}R Documentation

Haight's Zeta Function

Description

Density, distribution function, quantile function and random generation for Haight's Zeta function distribution with parameter alpha.

Usage

dhzeta(x, alpha)
phzeta(q, alpha)
qhzeta(p, alpha)
rhzeta(n, alpha)

Arguments

x, q Vector of quantiles. For the density, it should be a vector with positive integer values in order for the probabilities to be positive.
p vector of probabilities.
n number of observations. A single positive integer.
alpha The parameter value. Must contain positive values and is recycled to the length of x or p or q if necessary.

Details

The probability function is

f(x) = (2x-1)^(-alpha) - (2x+1)^(-alpha),

where alpha>0 and x=1,2,....

Value

dhzeta gives the density, phzeta gives the distribution function, qhzeta gives the quantile function, and rhzeta generates random deviates.

Note

Given some response data, the VGAM family function hzeta estimates the parameter alpha.

Author(s)

T. W. Yee

References

Page 533 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

See Also

hzeta, zeta, zetaff.

Examples

dhzeta(1:20, 2.1)
rhzeta(20, 2.1)

round(1000 * dhzeta(1:8, 2))
table(rhzeta(1000, 2))

## Not run: 
alpha = 1.1; x = 1:10
plot(x, dhzeta(x, alpha=alpha), type="h", ylim=0:1, lwd=2,
     sub=paste("alpha =", alpha), las=1, col="blue", ylab="Probability",
     main="Haight's zeta: blue=density; red=distribution function")
lines(x+0.1, phzeta(x, alpha=alpha), col="red", lty=3, lwd=2, type="h")
## End(Not run)

[Package VGAM version 0.7-7 Index]