NBlevy: Negative Binomial Levy process

In Renext: Renewal Method for Extreme Values Extrapolation

Negative Binomial Levy process

Description

Negative Binomial Lévy process estimation from partial observations (counts)

Usage

NBlevy(N,
       gamma = NA,
       prob = NA,
       w = rep(1, length(N)),
       sum.w = sum(w),
       interval = c(0.01, 1000),
       optim = TRUE,
       plot = FALSE, ...)

Arguments

N	Vector of counts, one count by time period.
gamma	The gamma parameter if known (NOT IMPLEMENTED YET).
prob	The prob parameter if known (NOT IMPLEMENTED YET).
w	Vector of time length (durations).
sum.w	NOT IMPLEMENTED YET. The effective duration. If sum.w is strictly inferior to sum(w), it is to be understood that missing periods occur within the counts period. This can be taken into account with a suitable algorithm (Expectation Maximisation, etc.)
interval	Interval giving min and max values for gamma.
optim	If TRUE a one-dimensional optimisation is used. Else the zero of the derivative of the (concentrated) log-likelihood is searched for.
plot	Should a plot be drawn? May be removed in the future.
...	Arguments passed to plot.

Details

The vector \mathbf{N} contains counts for events occurring on non-overlapping time periods with lengths given in \mathbf{w}. Under the NB Lévy process assumptions, the observed counts (i.e. elements of \mathbf{N}) are independent random variables, each following a negative binomial distribution. The size parameter r_k for N_k is r_k = \gamma w_k and the probability parameter p is prob. The vector \boldsymbol{\mu} of the expected counts has elements

\mu_k=\mathrm{E}(N_k)=\frac{1-p}{p} \,\gamma \,w_k.

The parameters \gamma and p \:(\code{prob}) are estimated by Maximum Likelihood using the likelihood concentrated with respect to the prob parameter.

Value

A list with the results

estimate	Parameter estimates.
sd	Standard deviation for the estimate.
score	Score vector at the estimated parameter vector.
info	Observed information matrix.
cov	Covariance matrix (approx.).

Note

The Negative Binomial Lévy process is an alternative to the Homogeneous Poisson Process when counts are subject to overdispersion. In the NB process, all counts share the same index of dispersion (variance/expectation ratio), namely 1/prob. When prob is close to 1, the counts are nearly Poisson-distributed.

Author(s)

Yves Deville

References

Kozubowski T.J. and Podgórsky K. (2009) "Distributional properties of the negative binomial Lévy process". Probability and Mathematical Statistics 29, pp. 43-71. Lund University Publications.

See Also

NegBinomial for the negative binomial distribution, glm.nb from the MASS package for fitting Generalised Linear Model of the negative binomial family.

Examples

## known parameters
nint <- 100
gam <- 6; prob <- 0.20

## draw random w, then the counts N
w <- rgamma(nint, shape = 3, scale = 1/5)
N <- rnbinom(nint, size = w * gam, prob = prob)
mu <- w * gam * (1 - prob) / prob
Res <- NBlevy(N = N, w = w)

## Use example data 'Brest'
## compute the number of event and duration of the non-skipped periods
gof1 <- gof.date(date = Brest$OTdata$date,
                 skip = Brest$OTmissing,
                 start = Brest$OTinfo$start,
                 end = Brest$OTinfo$end,
                 plot.type = "omit")
ns1 <- gof1$noskip
## fit the NBlevy
fit1 <- NBlevy(N = ns1$nevt, w = ns1$duration)

## use a higher threshold
OT2 <- subset(Brest$OTdata, Surge > 50)
gof2 <- gof.date(date = OT2$date,
                 skip = Brest$OTmissing,
                 start = Brest$OTinfo$start,
                 end = Brest$OTinfo$end,
                 plot.type = "omit")
ns2 <- gof2$noskip
## the NBlevy prob is now closer to 1
fit2 <- NBlevy(N = ns2$nevt, w = ns2$duration)

c(fit1$prob, fit2$prob)

Renext documentation built on Aug. 30, 2023, 1:06 a.m.