Exponentially weighted Poisson distribution functions

Density, distribution, quantile, and random generation for the exponentially weighted Poisson of Ridout and Besbeas (2004), in its three-parameter form (EWP3) and, with beta2 = beta1, its two-parameter form (EWP2): a Poisson with centrality lambda reweighted by a two-sided exponential in the distance from it, $$P(Y = y) \propto w_y e^{-\lambda} \lambda^y / y!, \quad w_y = e^{-\beta_1 (\lambda - y)} \ (y \le \lambda), \quad w_y = e^{-\beta_2 (y - \lambda)} \ (y > \lambda).$$ beta1 and beta2 greater than zero thin the lower and upper tails (underdispersion), values below zero thicken them, and beta1 = beta2 = 0 is the Poisson. lambda is not the mean; the normalizer and the exact mean and variance are sums over the support (no truncation of the normalizing sum at a fixed count), and the exact mean is what count_reg(family = "ewp3") and count_reg(family = "ewp2") report through predict() and fitted().

Usage

dewp(x, lambda, beta1, beta2 = beta1, log = FALSE)

pewp(q, lambda, beta1, beta2 = beta1, lower.tail = TRUE, log.p = FALSE)

qewp(p, lambda, beta1, beta2 = beta1, lower.tail = TRUE, log.p = FALSE)

rewp(n, lambda, beta1, beta2 = beta1)

Arguments

x, q

Vector of quantiles (non-negative integers).

lambda

Centrality parameter (scalar or vector, recycled).

beta1, beta2

Lower- and upper-tail dispersion parameters (scalars); beta2 defaults to beta1, the two-parameter EWP.

log, log.p

Return log probabilities.

lower.tail

If TRUE (default), \(P(X \le x)\).

p

Vector of probabilities.

n

Number of draws.

Value

dewp a density, pewp a CDF, qewp a quantile, rewp a numeric vector of count draws.

References

Ridout, M. S. and Besbeas, P. (2004). An empirical model for underdispersed count data. Statistical Modelling, 4(1), 77-89.

See also

count_reg()

Examples

dewp(0:6, lambda = 3, beta1 = 1, beta2 = 1)
#> [1] 0.006081605 0.049594547 0.202217932 0.549685330 0.151663449 0.033476319
#> [7] 0.006157625
var(rewp(2000, lambda = 3, beta1 = 1, beta2 = 1))
#> [1] 0.8357929