# Positive-Binomial Distribution Density, distribution function, quantile function and random generation for the positive-binomial distribution. ```r dposbinom(x, size, prob, log = FALSE) pposbinom(q, size, prob) qposbinom(p, size, prob) rposbinom(n, size, prob) ``` ## Arguments - `x, q`: vector of quantiles. - `p`: vector of probabilities. - `n`: number of observations. Fed into `runif`. - `size`: number of trials. It is the $N$ symbol in the formula given in `posbinomial` and should be positive. - `prob`: probability of success on each trial. Should be in $(0,1)$. - `log`: See `dbinom`. ## Details The positive-binomial distribution is a binomial distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is $$ \mu / (1-(1-\mu)^N)%mu / (1-(1-mu)^N) $$ where $mu$ is the argument `prob` above. As $mu$ increases, the positive-binomial and binomial distributions become more similar. Unlike similar functions for the binomial distribution, a zero value of `prob` is not permitted here. ## Returns `dposbinom` gives the density, `pposbinom` gives the distribution function, `qposbinom` gives the quantile function, and `rposbinom` generates random deviates. ## Author(s) T. W. Yee. ## Note These functions are or are likely to be deprecated. Use `Gaitdbinom` instead. For `dposbinom()`, if arguments `size` or `prob` equal 0 then a `NaN` is returned. The family function `posbinomial` estimates the parameters by maximum likelihood estimation. ## See Also `posbinomial`, `dposbern`, `Gaitdbinom`, `zabinomial`, `zibinomial`, `Binomial`. ## Examples ```r prob <- 0.2; size <- 10 table(y <- rposbinom(n = 1000, size, prob)) mean(y) # Sample mean size * prob / (1 - (1 - prob)^size) # Population mean (ii <- dposbinom(0:size, size, prob)) cumsum(ii) - pposbinom(0:size, size, prob) # Should be 0s table(rposbinom(100, size, prob)) table(qposbinom(runif(1000), size, prob)) round(dposbinom(1:10, size, prob) * 1000) # Should be similar ## Not run: barplot(rbind(dposbinom(x = 0:size, size, prob), dbinom(x = 0:size, size, prob)), beside = TRUE, col = c("blue", "green"), main = paste("Positive-binomial(", size, ",", prob, ") (blue) vs", " Binomial(", size, ",", prob, ") (green)", sep = ""), names.arg = as.character(0:size), las = 1) ## End(Not run) # Simulated data example nn <- 1000; sizeval1 <- 10; sizeval2 <- 20 pdata <- data.frame(x2 = seq(0, 1, length = nn)) pdata <- transform(pdata, prob1 = logitlink(-2 + 2 * x2, inv = TRUE), prob2 = logitlink(-1 + 1 * x2, inv = TRUE), sizev1 = rep(sizeval1, len = nn), sizev2 = rep(sizeval2, len = nn)) pdata <- transform(pdata, y1 = rposbinom(nn, sizev1, prob = prob1), y2 = rposbinom(nn, sizev2, prob = prob2)) with(pdata, table(y1)) with(pdata, table(y2)) # Multiple responses fit2 <- vglm(cbind(y1, y2) ~ x2, posbinomial(multip = TRUE), trace = TRUE, pdata, weight = cbind(sizev1, sizev2)) coef(fit2, matrix = TRUE) ```

posbinomUC function