Mixture of Two Exponential Distributions

Estimates the three parameters of a mixture of two exponential distributions by maximum likelihood estimation.

Usage

mix2exp(lphi = "logitlink", llambda = "loglink", iphi = 0.5,
    il1 = NULL, il2 = NULL, qmu = c(0.8, 0.2), nsimEIM = 100,
    zero = "phi")

Arguments

lphi, llambda

Link functions for the parameters $\phi$ and $\lambda$. The latter is the rate parameter and note that the mean of an ordinary exponential distribution is $1 / \lambda$. See Links for more choices.

iphi, il1, il2

Initial value for $\phi$, and optional initial value for $\lambda_1$ and $\lambda_2$. The last two have values that 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 $\lambda_1$ and $\lambda_2$. The two values are fed in as the probs argument into quantile.

nsimEIM, zero

See CommonVGAMffArguments.

Details

The probability density function can be loosely written as $$f(y) = \phi\,Exponential(\lambda_1) + (1-\phi)\,Exponential(\lambda_2)$$ where $\phi$ is the probability an observation belongs to the first group, and $y>0$. The parameter $\phi$ satisfies $0 < \phi < 1$. The mean of $Y$ is $\phi / \lambda_1 + (1-\phi) / \lambda_2$ and this is returned as the fitted values. By default, the three linear/additive predictors are $(logit(\phi), \log(\lambda_1), \log(\lambda_2))^T$.

Value

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.

This VGAM family function is experimental and should be used with care.

Author

T. W. Yee

Note

Fitting this model successfully to data can be difficult due to local solutions, uniqueness problems 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 $\lambda_1$ and $\lambda_2$ become more different. The default control argument trace = TRUE is to encourage monitoring convergence.

Examples

if (FALSE)  lambda1 <- exp(1); lambda2 <- exp(3)
(phi <- logitlink(-1, inverse = TRUE))
#> [1] 0.2689414
mdata <- data.frame(y1 = rexp(nn <- 1000, lambda1))
#> Error: object 'lambda1' not found
mdata <- transform(mdata, y2 = rexp(nn, lambda2))
#> Error: object 'mdata' not found
mdata <- transform(mdata, Y  = ifelse(runif(nn) < phi, y1, y2))
#> Error: object 'mdata' not found
fit <- vglm(Y ~ 1, mix2exp, data = mdata, trace = TRUE)
#> Error in eval(mf, parent.frame()): object 'mdata' not found
coef(fit, matrix = TRUE)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'coef': object 'fit' not found

# Compare the results with the truth
round(rbind('Estimated' = Coef(fit),
            'Truth' = c(phi, lambda1, lambda2)), digits = 2)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'Coef': object 'fit' not found

with(mdata, hist(Y, prob = TRUE, main = "Orange=estimate, blue=truth"))
#> Error: object 'mdata' not found
abline(v = 1 / Coef(fit)[c(2, 3)],  lty = 2, col = "orange", lwd = 2)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'Coef': object 'fit' not found
abline(v = 1 / c(lambda1, lambda2), lty = 2, col = "blue", lwd = 2)
#> Error: object 'lambda1' not found
 # \dontrun{}