Delaporte {Delaporte}  R Documentation 
Density, distribution, quantile, random variate generation, and method of
moments parameter estimation functions for the Delaporte distribution with
parameters alpha
, beta
, and lambda
.
ddelap(x, alpha, beta, lambda, log = FALSE)
pdelap(q, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE)
qdelap(p, alpha, beta, lambda, lower.tail = TRUE, log.p = FALSE, exact = TRUE)
rdelap(n, alpha, beta, lambda, exact = TRUE)
MoMdelap(x, type = 2L)
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

exact 
logical; if TRUE uses double summation to generate quantiles or random variates. Otherwise uses Poissonnegative binomial approximation. 
type 
integer; 1L will return g1, 2L will return G1,
and 3L will return b1, as per 
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) = \sum_{i=0}^n\frac{\Gamma(\alpha+i)\beta^i\lambda^{ni}
e^{\lambda}}{\Gamma(\alpha) i! (1+\beta)^{\alpha+i}(ni)!}
for n = 0, 1, 2, \ldots
; \alpha, \beta, \lambda > 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) = \sum_{j=0}^n \sum_{i=0}^j\frac{\Gamma(\alpha+i)\beta^i
\lambda^{ji}e^{\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(ji)!}
for n = 0, 1, 2, \ldots
; \alpha, \beta, \lambda > 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 q
.
ddelap
will return 0 for all values > 2^{31}
whereas
pdelap
will not run at all, due to the limitations of integer
representation. Also, for values > 2^{15}
, pdelap
will ask
for positive input from the user to continue, as otherwise, depending on the
parameters, the function can take hours to complete given its doublesummation
nature.
The quantile function, qdelap
, is right continuous:
qdelap(q, alpha, beta, lambda)
is the smallest integer x
such that
P(X \le x) \ge q
. This 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 \alpha, \beta
, or \lambda
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 \Gamma
with shape \alpha
and scale \beta
and then n
pseudoDelaporte
variates as Poisson random variables with parameter \lambda + \Gamma
,
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 be
used until \alpha\beta + \lambda \approx 2500
. Both versions return NaN
for quantiles < 0
,
0
for quantiles = 0
, and Inf
for quantiles \ge 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,
it then 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 \Gamma
with shape \alpha
and
scale \beta
and then n
pseudoDelaporte variates as Poisson random
variables with parameter \lambda + \Gamma
. As there is no pooling, each
individual random variate reflects the parameter triplet which generated it. The
noninversion method is usually faster.
MoMdelap
uses the definition of the Delaporte's mean, variance, and skew
to calculate the method of moments estimates of \alpha
, \beta
, and
\lambda
, 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.
For the sample skew, the user has the choice to select g_1
,
G_1
, or b_1
as defined in Joanes & Gill (1997) and found in
skewness
. The selection defaults to option 2,
G_1
, which Joanes & Gill found to have the least meansquare error for
nonnormal 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 \alpha, \beta
or \lambda
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 (\alpha, \beta, \lambda
) 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 methodofmoments based
estimate of \alpha
, \beta
, and \lambda
.
Avraham Adler Avraham.Adler@gmail.com
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. doi:10.1111/14679884.00122
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. https://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, and
skewness
for skew options.
## 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
## Not run:
plot(density(P), col = "black", lwd = 2, main = "Distribution Comparison",
xlab = "Value", xlim = c(0, 80))
lines(density(DP1), col = "blue", lwd = 2)
lines(density(DP2), col = "green3", lwd = 2)
lines(density(DP3), col = "orange3", lwd = 2)
lines(density(NB), col = "red", lwd = 2)
legend(x = "topright", legend = c("Poisson {l=25}", "DP {a=2, b=2, l=21}",
"DP {a=3, b=4, l=13}", "DP {a=4, b=5, l=5}", "NegBinom {a=5, b=5}"),
col=c("black", "blue", "green3","orange3", "red"), lwd = 2)
## End(Not run)