# Positive-Negative Binomial Distribution Density, distribution function, quantile function and random generation for the positive-negative binomial distribution. ```r dposnegbin(x, size, prob = NULL, munb = NULL, log = FALSE) pposnegbin(q, size, prob = NULL, munb = NULL, lower.tail = TRUE, log.p = FALSE) qposnegbin(p, size, prob = NULL, munb = NULL) rposnegbin(n, size, prob = NULL, munb = NULL) ``` ## Arguments - `x, q`: vector of quantiles. - `p`: vector of probabilities. - `n`: number of observations. Fed into `runif`. - `size, prob, munb, log`: Same arguments as that of an ordinary negative binomial distribution (see `dnbinom`). Some arguments have been renamed slightly. Short vectors are recycled. The parameter `1/size` is known as a dispersion parameter; as `size` approaches infinity, the negative binomial distribution approaches a Poisson distribution. Note that `prob` must lie in $(0,1)$, otherwise a `NaN` is returned. - `log.p, lower.tail`: Same arguments as that of an ordinary negative binomial distribution (see `pnbinom`). ## Details The positive-negative binomial distribution is a negative binomial distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is $$ \mu / (1-p(0))%munb / (1-p(0)) $$ where $munb$ the mean of an ordinary negative binomial distribution. ## Returns `dposnegbin` gives the density, `pposnegbin` gives the distribution function, `qposnegbin` gives the quantile function, and `rposnegbin` generates $n$ random deviates. ## References Welsh, A. H., Cunningham, R. B., Donnelly, C. F. and Lindenmayer, D. B. (1996). Modelling the abundances of rare species: statistical models for counts with extra zeros. **Ecological Modelling**, 88 , 297--308. ## Author(s) T. W. Yee ## Note These functions are or are likely to be deprecated. Use `Gaitdnbinom` instead. ## See Also `Gaitdnbinom`, `posnegbinomial`, `zanegbinomial`, `zinegbinomial`, `rnbinom`. ## Examples ```r munb <- 5; size <- 4; n <- 1000 table(y <- rposnegbin(n, munb = munb, size = size)) mean(y) # Sample mean munb / (1 - (size / (size + munb))^size) # Population mean munb / pnbinom(0, mu = munb, size, lower.tail = FALSE) # Same x <- (-1):17 (ii <- dposnegbin(x, munb = munb, size = size)) max(abs(cumsum(ii) - pposnegbin(x, munb = munb, size))) # 0? ## Not run: x <- 0:10 barplot(rbind(dposnegbin(x, munb = munb, size = size), dnbinom(x, mu = munb, size = size)), beside = TRUE, col = c("blue","green"), main = paste0("dposnegbin(munb = ", munb, ", size = ", size, ") (blue) vs dnbinom(mu = ", munb, ", size = ", size, ") (green)"), names.arg = as.character(x)) ## End(Not run) # Another test for pposnegbin() nn <- 5000 mytab <- cumsum(table(rposnegbin(nn, munb = munb, size))) / nn myans <- pposnegbin(sort(as.numeric(names(mytab))), munb = munb, size) max(abs(mytab - myans)) # Should be 0 ```

posnegbinUC function