| Delaporte {Delaporte} | R Documentation |
The Delaporte Distribution
Description
Density, distribution, quantile, random variate generation, and method of
moments parameter estimation functions for the Delaporte distribution with
parameters alpha, beta, and lambda.
Usage
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)
Arguments
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 |
Details
Definition
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.
Distributional Functions
Density and Distribution
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 double-summation
nature.
Quantile
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 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 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.
Random Variate Generation
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 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.
Method of Moments Fitting
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 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 mean-square error for
non-normal distributions.
Value
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.
Author(s)
Avraham Adler Avraham.Adler@gmail.com
References
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:10.1111/1467-9884.00122
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
See Also
Distributions for standard distributions, including
dnbinom for the negative binomial distribution and
dpois for the Poisson distribution, and
skewness for skew options.
Examples
## 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)