VGAM (version 0.8-4.1)

Hzeta: 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, log=FALSE)
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.

log

Logical. If log=TRUE then the logarithm of the density is returned.

Value

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

Details

The probability function is $$f(x) = (2x-1)^{(-\alpha)} - (2x+1)^{(-\alpha)},$$ where $\alpha>0$ and $x=1,2,\ldots$.

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))

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")