The Zeta Distribution
Density, distribution function, quantile function and random generation for the zeta distribution.
dzeta(x, shape, log = FALSE) pzeta(q, shape, lower.tail = TRUE) qzeta(p, shape) rzeta(n, shape)
The density function of the zeta distribution is given by
y^(-s-1) / zeta(s+1)
where s>0, y=1,2,…, and zeta is Riemann's zeta function.
dzeta
gives the density,
pzeta
gives the distribution function,
qzeta
gives the quantile function, and
rzeta
generates random deviates.
qzeta()
runs slower and slower as shape
approaches 0
and p
approaches 1.
The VGAM family function zetaff
estimates the
shape parameter s.
T. W. Yee
Johnson N. L., Kotz S., and Balakrishnan N. (1993). Univariate Discrete Distributions, 2nd ed. New York: Wiley.
dzeta(1:20, shape = 2) myshape <- 0.5 max(abs(pzeta(1:200, myshape) - cumsum(1/(1:200)^(1+myshape)) / zeta(myshape+1))) # Should be 0 ## Not run: plot(1:6, dzeta(1:6, 2), type = "h", las = 1, col = "orange", ylab = "Probability", main = "zeta probability function; orange: shape = 2; blue: shape = 1") points(0.10 + 1:6, dzeta(1:6, 1), type = "h", col = "blue") ## End(Not run)
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