Density, distribution, quantile, random variate generation, and method of moments parameter estimation functions for the Delaporte distribution with parameters alpha
, beta
, and lambda
.
1 2 3 4 5 6 
x 
vector of (nonnegative integer) quantiles. 
q 
vector of quantiles. 
p 
vector of probabilities. 
n 
number of observations. 
alpha 
vector of alpha parameters of the gamma portion of the Delaporte distribution. Must be strictly positive, but need not be integer. 
beta 
vector of beta parameters of the gamma portion of the Delaporte distribution. Must be strictly positive, but need not be integer. 
lambda 
vector of lambda parameters of the Poisson portion of the Delaporte distribution. Must be strictly positive, but need not be integer. 
log, log.p 
logical; if TRUE, probabilities p are given as log(p). 
lower.tail 
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. 
exact 
logical; if TRUE uses double summation to generate quantiles or random variates. Otherwise uses Poissonnegative binomial approximation. 
The Delaporte distribution with parameters alpha
, beta
, and lambda
is a discrete probability distribution which can be considered the convolution of a negative binomial distribution with a Poisson distribution. Alternatively, it can be considered a counting distribution with both Poisson and negative binomial components. The Delaporte's probability mass function, called via ddelap
, is:
p(n) = ∑ (i=0:n) [Γ(α+i) β^i λ^(ni) exp(λ)] / [Γ(α) i! (1+β)^(α+i) (ni)!]
for n = 0, 1, 2, …; α, β, λ > 0.
If an element of x is not integer, the result of ddelap
is zero, with a warning.
The Delaporte's cumulative distribution function, pdelap
, is calculated through double summation:
CDF(n) = ∑(j=0:n) ∑(i=0:j) [Γ(α+i) β^i λ^(ji) exp(λ)] / [Γ(α) i! (1+β)^(α+i) (ji)!]
for n = 0, 1, 2, …; α, β, λ > 0. If only singleton values for the parameters are passed in, the function uses the shortcut of identifying the largest value passed to it, computes a vector of CDF values for all integers up to and including that value, and having the remaining results read from this vector. This requires only one double summation instead of length(q)
such summations. If at least one of the parameters is itself a vector of length greater than 1, the function has to build the double summation for each entry in q
.
The quantile function, qdelap
, is right continuous: qdelap(q, alpha, beta, lambda)
is the smallest integer x such that P(X ≤ x) ≥ q. This has function has two versions. When exact = TRUE
, the function builds a CDF vector and the first value for which the CDF is greater than or equal to q
is returned as the quantile. While this procedure is accurate, for sufficiently large α, β, or λ it can take a very long time. Therefore, when dealing with singleton parameters, exact = FALSE
can be passed to take advantage of the Delaporte's definition as a counting distribution with both a Poisson and a negative binomial component. Based on Karlis & Xekalaki (2005) it will generate n
gamma variates Γ with shape α and scale β and then n
pseudoDelaporte variates as Poisson random variables with parameter λ + Γ, finally calling the quantile
function on the result. The “exact” method is always more accurate and is also significantly faster for reasonable values of the parameters. Also, the “exact” method must be used when passing parameter vectors, as the pooling would become intractable. Adhoc testing indicates that the “exact” method should always be used until αβ + λ ~ 2500. Both versions return NaN
for quantiles < 0, 0 for quantiles = 0, and Inf
for quantiles ≥ 1.
The random variate generator, rdelap
, also has multiple versions. When exact = TRUE
, it uses inversion by creating a vector of n
uniformly distributed random variates between 0 and 1. If all the parameters are singletons, a single CDF vector is constructed as per the quantile function, and the entries corresponding to the uniform variates are read off of the constructed vector. If the parameters are themselves vectors, then it passes the entire uniform variate vector to qdelap
, which is slower. When exact = FALSE
, regardless of the length of the parameters, it generates n
gamma variates Γ with shape α and scale β and then n
pseudoDelaporte variates as Poisson random variables with parameter λ + Γ. As there is no pooling, each individual random variate reflects the parameter triplet which generated it. The noninversion method is pretty much always faster.
MoMdelap
uses the definition of the Delaporte's mean, variance, and skew to calculate the method of moments estimates of α, β, and λ, which it returns as a numeric vector. This estimate is also a reasonable starting point for maximum likelihood estimation using nonlinear optimizers such as optim
or nloptr
. If the data is clustered near 0, there are times when method of moments would result in a nonpositive parameter. In these cases MoMdelap
will throw an error.
ddelap
gives the probability mass function, pdelap
gives the cumulative distribution function, qdelap
gives the quantile function, and rdelap
generates random deviates. Values close to 0 (less than machine epsilon) for α, β or λ will return NaN
for that particular entry. Proper triplets within a set of vectors should still return valid values. For the approximate versions of qdelap
and rdelap
, having a value close to 0 will trip an error, sending the user to the exact version which currently properly handles vectorbased inputs which contain 0.
Invalid quantiles passed to qdelap
will result in return values of NaN
or Inf
as appropriate.
The length of the result is determined by x
for ddelap
, q
for pdelap
, p
for qdelap
, and n
for rdelap
. The distributional parameters (α, β, λ) are recycled as necessary to the length of the result.
When using the lower.tail = FALSE
or log / log.p = TRUE
options, some accuracy may be lost at knot points or the tail ends of the distributions due to the limitations of floating point representation.
Avraham Adler Avraham.Adler@gmail.com
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005) Univariate discrete distributions (Third ed.). John Wiley & Sons. pp. 241–242. ISBN 9780471272465.
Karlis, D. and Xekalaki, E. (2005) Mixed Poisson Distributions. International Statistical Review 73(1), 35–58. http://projecteuclid.org/euclid.isr/1112304811
Vose, D. (2008) Risk analysis: a quantitative guide (Third, illustrated ed.). John Wiley & Sons. pp. 618–619 ISBN 9780470512845
Distributions for standard distributions, including dnbinom
for the negative binomial distribution
and dpois
for the Poisson distribution.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  ## Density and distribution
A < c(0, seq_len(50))
PMF < ddelap(A, alpha = 3, beta = 4, lambda = 10)
CDF < pdelap(A, alpha = 3, beta = 4, lambda = 10)
##Quantile
A < seq(0,.95, .05)
qdelap(A, alpha = 3, beta = 4, lambda = 10)
A < c(1, A, 1, 2)
qdelap(A, alpha = 3, beta = 4, lambda = 10)
## Compare a Poisson, negative binomial, and three Delaporte distributions with the same mean:
P < rpois(25000, 25) ## Will have the tightest spread
DP1 < rdelap(10000, alpha = 2, beta = 2, lambda = 21) ## Close to the Poisson
DP2 < rdelap(10000, alpha = 3, beta = 4, lambda = 13) ## In between
DP3 < rdelap(10000, alpha = 4, beta = 5, lambda = 5) ## Close to the Negative Binomial
NB < rnbinom(10000, size = 5, mu = 25) ## Will have the widest spread
mean(P);mean(NB);mean(DP1);mean(DP2);mean(DP3) ## Means should all be near 25
MoMdelap(DP1);MoMdelap(DP2);MoMdelap(DP3) ## Estimates should be close to originals

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