# Mixture of Two Poisson Distributions Estimates the three parameters of a mixture of two Poisson distributions by maximum likelihood estimation. ```r mix2poisson(lphi = "logitlink", llambda = "loglink", iphi = 0.5, il1 = NULL, il2 = NULL, qmu = c(0.2, 0.8), nsimEIM = 100, zero = "phi") ``` ## Arguments - `lphi, llambda`: Link functions for the parameter $phi$ and $lambda$. See `Links` for more choices. - `iphi`: Initial value for $phi$, whose value must lie between 0 and 1. - `il1, il2`: Optional initial value for $lambda1$ and $lambda2$. These values must be positive. The default is to compute initial values internally using the argument `qmu`. - `qmu`: Vector with two values giving the probabilities relating to the sample quantiles for obtaining initial values for $lambda1$ and $lambda2$. The two values are fed in as the `probs` argument into `quantile`. - `nsimEIM, zero`: See `CommonVGAMffArguments`. ## Details The probability function can be loosely written as $$ P(Y=y) = \phi \, Poisson(\lambda_1) +(1-\phi) \, Poisson(\lambda_2)%P(Y=y) = phi * Poisson(lambda1) + (1-phi) * Poisson(lambda2) $$ where $phi$ is the probability an observation belongs to the first group, and $y=0,1,2,...$. The parameter $phi$ satisfies $0 < phi < 1$. The mean of $Y$ is $phi*lambda1 + (1-phi)*lambda2$ and this is returned as the fitted values. By default, the three linear/additive predictors are c("$(logit(phi),\n$", "$ log(lambda1), log(lambda2))^T$"). ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`. ## Warning This `VGAM` family function requires care for a successful application. In particular, good initial values are required because of the presence of local solutions. Therefore running this function with several different combinations of arguments such as `iphi`, `il1`, `il2`, `qmu` is highly recommended. Graphical methods such as `hist` can be used as an aid. With grouped data (i.e., using the `weights` argument) one has to use a large value of `nsimEIM`; see the example below. This `VGAM` family function is experimental and should be used with care. ## Author(s) T. W. Yee ## Note The response must be integer-valued since `dpois` is invoked. Fitting this model successfully to data can be difficult due to local solutions and ill-conditioned data. It pays to fit the model several times with different initial values, and check that the best fit looks reasonable. Plotting the results is recommended. This function works better as $lambda1$ and $lambda2$ become more different. The default control argument `trace = TRUE` is to encourage monitoring convergence. ## See Also `rpois`, `poissonff`, `mix2normal`. ## Examples ```r ## Not run: # Example 1: simulated data nn <- 1000 mu1 <- exp(2.5) # Also known as lambda1 mu2 <- exp(3) (phi <- logitlink(-0.5, inverse = TRUE)) mdata <- data.frame(y = rpois(nn, ifelse(runif(nn) < phi, mu1, mu2))) mfit <- vglm(y ~ 1, mix2poisson, data = mdata) coef(mfit, matrix = TRUE) # Compare the results with the truth round(rbind('Estimated' = Coef(mfit), 'Truth' = c(phi, mu1, mu2)), 2) ty <- with(mdata, table(y)) plot(names(ty), ty, type = "h", main = "Orange=estimate, blue=truth", ylab = "Frequency", xlab = "y") abline(v = Coef(mfit)[-1], lty = 2, col = "orange", lwd = 2) abline(v = c(mu1, mu2), lty = 2, col = "blue", lwd = 2) # Example 2: London Times data (Lange, 1997, p.31) ltdata1 <- data.frame(deaths = 0:9, freq = c(162,267,271, 185,111,61,27,8,3,1)) ltdata2 <- data.frame(y = with(ltdata1, rep(deaths, freq))) # Usually this does not work well unless nsimEIM is large Mfit <- vglm(deaths ~ 1, weight = freq, data = ltdata1, mix2poisson(iphi=0.3, il1=1, il2=2.5, nsimEIM=5000)) # This works better in general Mfit = vglm(y ~ 1, mix2poisson(iphi=0.3, il1=1, il2=2.5), ltdata2) coef(Mfit, matrix = TRUE) Coef(Mfit) ## End(Not run) ```

mix2poisson function