| 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 (non-negative 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 Poisson-negative 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^{n-i}e^{-\lambda}}{\Gamma(\alpha) i! (1+\beta)^{\alpha+i}(n-i)!}
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^{j-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(j-i)!}
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 represenatation. 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 double-summation 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 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 \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 pseudo-Delaporte 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. Ad-hoc testing
indicates that the “exact” method should always 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,
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 \Gamma with shape \alpha and
scale \beta and then n pseudo-Delaporte 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
non-inversion 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 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 \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
vector-based 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 method-of-moments 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. 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
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)