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.
integer; 1L will return g1, 2L will return G1, and 3L will return b1, as per
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.
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^(j-i) exp(-λ)] / [Γ(α) i! (1+β)^(α+i) (j-i)!]
for n = 0, 1, 2, …; α, β, λ > 0. If only singleton values for the parameters are passed in, the function uses a shortcut. It identifies the largest value passed to it, computes a vector of CDF values for all integers up to and including that value, and reads the remaining results 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, 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 pseudo-Delaporte 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. Ad-hoc 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 pseudo-Delaporte 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 non-inversion method is most 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
nloptr. If the data is clustered near 0, there are times when method of moments would result in a non-positive parameter. In these cases
MoMdelap will throw an error. For the sample skew, the user has the choice to select g1, G1, or b1 as defined in Joanes & Gill (1997) and found in
skewness. The selection defaults to option 2, G1, which Joanes & Gill found to have the least mean-square error for non-normal distributions.
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 or equal to 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.
MoMdelap returns a triplet comprising a method-of-moments based estimate of α, β, and λ.
Avraham Adler [email protected]
Joanes, D. N. and Gill, C. A. (1998) Comparing Measures of Sample Skewness and Kurtosis. Journal of the Royal Statistical Society. Series D (The Statistician) 47(1), 183–189. https://www.jstor.org/stable/2988433
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. https://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
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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
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