# The Negative Binomial Mixture Distribution Density function for a mixture of two negative binomial distributions. ```r dmixnegbinom(x, theta, E, log.p = FALSE) ``` ## Arguments - `x`: vector of (non-negative integer) quantiles. - `theta`: vector of parameters for the negative binomial distribution mixture. - `E`: vector of (non-negative integer) expected counts. - `log.p`: logical; if TRUE, probabilities p are given as log(p). ## Details The mixture of two negative binomial distributions has density $$ P(N = x) = theta[5] f(x;theta[1],theta[2],E) + (1 - theta[5]) f(x;theta[3],theta[4],E), $$ where $$ f(x;\alpha,\beta,E) = \frac{\Gamma(\alpha + x)}{\Gamma(\alpha) x!}\frac{1}{(1 + \beta/E)^x}\frac{1}{(1 + E/\beta)^\alpha} $$ for $x = 0,1,\ldots,\alpha$, $\alpha, \beta, E > 0$ and $0 < theta[5] \leq 1$. The mixture of two negative binomial distributions represents the marginal distribution of the counts $N$ coming from Poisson data with parameter $\lambda$ and a mixture of two gamma distributions as its prior. For details see the paper by Dumouchel (1999). ## Returns `dmixnegbinom` gives the density corresponding to the `E` and `theta` values provided. ## References DuMouchel W. (1999), "Bayesian Data Mining in Large Frequency Tables, with an Application to the FDA Spontaneous Reporting System". The American Statistician, 53, 177-190. Myers, J. A., Venturini, S., Dominici, F. and Morlock, L. (2011), "Random Effects Models for Identifying the Most Harmful Medication Errors in a Large, Voluntary Reporting Database". Technical Report. ## Author(s) Sergio Venturini sergio.venturini@unicatt.it , Jessica A. Myers jmyers6@partners.org ## See Also `dnbinom`, `rmixnegbinom`. ## Examples ```r ## Not run: data("simdata", package = "mederrRank") ni <- simdata@numi theta0 <- c(10, 6, 100, 100, .1) ans <- mixnegbinom.em(simdata, theta0, 50000, 0.01, se = FALSE, stratified = TRUE) theta <- ans$theta.hat N.E <- cbind(ans$N[1:ni], ans$E[1:ni])[sort(ans$N[1:ni], index.return = TRUE)$ix, ] N.ix <- match(unique(N.E[, 1]), N.E[, 1]) N <- N.E[N.ix, 1] E <- N.E[N.ix, 2] dens <- dmixnegbinom(N, theta, E) hist(N.E[, 1], breaks = 40, freq = FALSE) points(N, dens) ## End(Not run) ```

dmixnegbinom function