The Delaporte Distribution
Density, distribution, quantile, random variate generation, and method of moments parameter estimation functions for the Delaporte distribution with parameters
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vector of (non-negative integer) quantiles.
vector of quantiles.
vector of probabilities.
number of observations.
vector of alpha parameters of the gamma portion of the Delaporte distribution. Must be strictly positive, but need not be integer.
vector of beta parameters of the gamma portion of the Delaporte distribution. Must be strictly positive, but need not be integer.
vector of lambda parameters of the Poisson portion of the Delaporte distribution. Must be strictly positive, but need not be integer.
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].
logical; if TRUE uses double summation to generate quantiles or random variates. Otherwise uses Poisson-negative binomial approximation.
logical; if TRUE uses older and slower approximation code. Otherwise uses Shifted-Gamma approximation.
The Delaporte distribution with parameters
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
p(n) = ∑ (i=0:n) [Γ(α+i) β^i λ^(n-i) exp(-λ)] / [Γ(α) i! (1+β)^(α+i) (n-i)!]
for n = 0, 1, 2, …; α, β, λ > 0.
Its cumulative distribution function,
pdelap, is calculated through double summation:
CDF(n) = ∑(j=0:n) ∑(i=0:j) [Γ(α+i) β^i λ^(j-i) exp(-λ)] / [Γ(α) i! (1+β)^(α+i) (j-i)!]
for n = 0, 1, 2, …; α, β, λ > 0. For both the probability mass and distribution calculations, if a non-integer value is passed into the function, it is rounded up to the next integer. 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
The quantile function,
qdelap, 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
exact = FALSE, the function takes advantage of the Delaporte's definition as a counting distribution with both a Poisson and negative binomial component. Based on Karlis & Xekalaki (2005) it will generate
n gamma variates Γ with shape α and scale β and then
n psuedo-Delaporte variates as Possion random variables with parameter λ + Γ, finally calling the
quantile function on the result. This is significantly faster than the old, now deprecated, method. For a while, to help with repeatability, the old method can be used by passing
exact = FALSE, old = TRUE. In this case,
qdelap will generate up to 10^7 variates from a negative binomial distribution with shape α and scale β (size = α, mean = αβ), and the same number of variates from a Poisson distribution with the mean λ. It then sums the two sets of variates and calls the
quantile function on the result. The “exact” method is always more accurate and is also significantly faster for reasonable values of the parameters. Ad-hoc testing indicates that the “exact” method should always be used until αβ + λ ~ 5000. 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 psuedo-Delaporte variates as Possion random variables with parameter λ + Γ. This is significantly faster than the old, now deprecated, method. For a while, to help with repeatability, the old method can be used by passing
exact = FALSE, old = TRUE, upon which the larger of
n or 10^7 variates from both a Poisson and negative binomial with the appropriate parameters are generated and summed. If
n < 10^7, sampling with replacement is used to generate the
n samples from the pool of 10^7 pseudo-Delaporte variates.
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
nloptr. If the data is clustered near 0, there are times when method of moments would result in a non-positive parameter (usually λ). 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
rdelap, having a value close to 0 will trip an error, sending the user to the exact version which currently properly handles vector-based inputs which contain 0.
Invalid quantiles passed to
qdelap will result in return values of
Inf as appropriate.
The length of the result is determined by
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 978-0-471-27246-5.
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 978-0-470-51284-5
Distributions for standard distributions, including
dnbinom for the negative binomial distribution
dpois for the Poisson distribution.
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## 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