moezipf {zipfextR} | R Documentation |
The Marshal-Olkin Extended Zipf Distribution (MOEZipf).
Description
Probability mass function, cumulative distribution function, quantile function and random number
generation for the MOEZipf distribution with parameters and
. The support of the MOEZipf
distribution are the strictly positive integer numbers large or equal than one.
Usage
dmoezipf(x, alpha, beta, log = FALSE)
pmoezipf(q, alpha, beta, log.p = FALSE, lower.tail = TRUE)
qmoezipf(p, alpha, beta, log.p = FALSE, lower.tail = TRUE)
rmoezipf(n, alpha, beta)
Arguments
x , q |
Vector of positive integer values. |
alpha |
Value of the |
beta |
Value of the |
log , log.p |
Logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
Logical; if TRUE (default), probabilities are |
p |
Vector of probabilities. |
n |
Number of random values to return. |
Details
The probability mass function at a positive integer value of the MOEZipf distribution with
parameters
and
is computed as follows:
where is the Riemann-zeta function at
,
is the Hurtwitz zeta function with arguments
and x, and
.
The cumulative distribution function, at a given positive integer value ,
is computed as
, where the survival function
is equal to:
The quantile of the MOEZipf distribution of a given probability value p
is equal to the quantile of the Zipf
distribution at the value:
The quantiles of the Zipf distribution are computed by means of the tolerance
package.
To generate random data from a MOEZipf one applies the quantile function over n values randomly generated from an Uniform distribution in the interval (0, 1).
Value
dmoezipf
gives the probability mass function,
pmoezipf
gives the cumulative distribution function,
qmoezipf
gives the quantile function, and
rmoezipf
generates random values from a MOEZipf distribution.
References
Casellas, A. (2013) La distribució Zipf Estesa segons la transformació Marshall-Olkin. Universitat Politécnica de Catalunya.
Devroye L. (1986) Non-Uniform Random Variate Generation. Springer, New York, NY.
Duarte-López, A., Prat-Pérez, A., & Pérez-Casany, M. (2015). Using the Marshall-Olkin Extended Zipf Distribution in Graph Generation. European Conference on Parallel Processing, pp. 493-502, Springer International Publishing.
Pérez-Casany, M. and Casellas, A. (2013) Marshall-Olkin Extended Zipf Distribution. arXiv preprint arXiv:1304.4540.
Young, D. S. (2010). Tolerance: an R package for estimating tolerance intervals. Journal of Statistical Software, 36(5), 1-39.
Examples
dmoezipf(1:10, 2.5, 1.3)
pmoezipf(1:10, 2.5, 1.3)
qmoezipf(0.56, 2.5, 1.3)
rmoezipf(10, 2.5, 1.3)