cnm: Maximum Likelihood Estimation of a Nonparametric Mixture Model

Description

Function cnm can be used to compute the maximum likelihood estimate of a nonparametric mixing distribution (NPMLE) that has a one-dimensional mixing parameter, or simply the mixing proportions with support points held fixed.

A finite mixture model has a density of the form

$$f(x; \pi, \theta, \beta) = \sum_{j=1}^k \pi_j f(x; \theta_j, \beta).$$

where $\pi_j \ge 0$ and $\sum_{j=1}^k \pi_j $$ =1$.

A nonparametric mixture model has a density of the form

$$f(x; G) = \int f(x; \theta) d G(\theta),$$ where $G$ is a mixing distribution that is completely unspecified. The maximum likelihood estimate of the nonparametric $G$, or the NPMLE of $G$, is known to be a discrete distribution function.

Function cnm implements the CNM algorithm that is proposed in Wang (2007) and the hierarchical CNM algorithm of Wang and Taylor (2013). The implementation is generic using S3 object-oriented programming, in the sense that it works for an arbitrary family of mixture models defined by the user. The user, however, needs to supply the implementations of the following functions for their self-defined family of mixture models, as they are needed internally by function cnm:

initial(x, beta, mix, kmax)

valid(x, beta, theta)

logd(x, beta, pt, which)

gridpoints(x, beta, grid)

suppspace(x, beta)

length(x)

print(x, ...)

weight(x, ...)

While not needed by the algorithm for finding the solution, one may also implement

plot(x, mix, beta, ...)

so that the fitted model can be shown graphically in a user-defined way. Inside cnm, it is used when plot="probability" so that the convergence of the algorithm can be graphically monitored.

For creating a new class, the user may consult the implementations of these functions for the families of mixture models included in the package, e.g., npnorm and nppois.

Usage

cnm(
  x,
  init = NULL,
  model = c("npmle", "proportions"),
  maxit = 100,
  tol = 1e-06,
  grid = 100,
  plot = c("null", "gradient", "probability"),
  verbose = 0
)

Value

family: the name of the mixture family that is used to fit to the data.
num.iterations: number of iterations required by the algorithm
max.gradient: maximum value of the gradient function, evaluated at the beginning of the final iteration
convergence: convergence code. =0 means a success, and =1 reaching the maximum number of iterations
ll: log-likelihood value at convergence
mix: MLE of the mixing distribution, being an object of the class disc for discrete distributions.
beta: value of the structural parameter, that is held fixed throughout the computation.

Arguments

x: a data object of some class that is fully defined by the user. The user needs to supply certain functions as described below.
init: list of user-provided initial values for the mixing distribution mix and the structural parameter beta.
model: the type of model that is to estimated: the non-parametric MLE (if npmle), or mixing proportions only (if proportions).
maxit: maximum number of iterations.
tol: a tolerance value needed to terminate an algorithm. Specifically, the algorithm is terminated, if the increase of the log-likelihood value after an iteration is less than tol.
grid: number of grid points that are used by the algorithm to locate all the local maxima of the gradient function. A larger number increases the chance of locating all local maxima, at the expense of an increased computational cost. The locations of the grid points are determined by the function gridpoints provided by each individual mixture family, and they do not have to be equally spaced. If needed, a gridpoints function may choose to return a different number of grid points than specified by grid.
plot: whether a plot is produced at each iteration. Useful for monitoring the convergence of the algorithm. If ="null", no plot is produced. If ="gradient", it plots the gradient curves and if ="probability", the plot function defined by the user for the class is used.
verbose: verbosity level for printing intermediate results in each iteration, including none (= 0), the log-likelihood value (= 1), the maximum gradient (= 2), the support points of the mixing distribution (= 3), the mixing proportions (= 4), and if available, the value of the structural parameter beta (= 5).

Author

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. (2007). On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Ser. B, 69, 185-198.

Wang, Y. (2010). Maximum likelihood computation for fitting semiparametric mixture models. Statistics and Computing, 20, 75-86

Wang, Y. and Taylor, S. M. (2013). Efficient computation of nonparametric survival functions via a hierarchical mixture formulation. Statistics and Computing, 23, 713-725.

Examples

Run this code


## Simulated data
x = rnppois(200, disc(c(1,4), c(0.7,0.3))) # Poisson mixture
(r = cnm(x))
plot(r, x)

x = rnpnorm(200, disc(c(0,4), c(0.3,0.7)), sd=1) # Normal mixture
plot(cnm(x), x)                        # sd = 1
plot(cnm(x, init=list(beta=0.5)), x)   # sd = 0.5

## Real-world data
data(thai)
plot(cnm(x <- nppois(thai)), x)     # Poisson mixture

data(brca)
plot(cnm(x <- npnorm(brca)), x)     # Normal mixture

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