# cnm: Maximum Likelihood Estimation of a Nonparametric Mixture... In nspmix: Nonparametric and Semiparametric Mixture Estimation

## 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.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```cnm( x, init = NULL, model = c("npmle", "proportions"), maxit = 100, tol = 1e-06, grid = 100, plot = c("null", "gradient", "probability"), verbose = 0 ) ```

## 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).

## Details

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 >= 0 and sum_{j=1}^k pi_j =1sum_{j=1}^k pi_j =1.

A nonparametric mixture model has a density of the form

f(x; G) = Integral 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)`

`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`.

## 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.

## Author(s)

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.

`nnls`, `npnorm`, `nppois`, `cnmms`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## Simulated data x = rnppois(1000, disc(c(1,4), c(0.7,0.3))) # Poisson mixture (r = cnm(x)) plot(r, x) x = rnpnorm(1000, 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 mix0 = disc(seq(min(x\$v),max(x\$v), len=100)) # over a finite grid plot(cnm(x, init=list(beta=0.5, mix=mix0), model="p"), x, add=TRUE, col="blue") # An approximate NPMLE ## Real-world data data(thai) plot(cnm(x <- nppois(thai)), x) # Poisson mixture data(brca) plot(cnm(x <- npnorm(brca)), x) # Normal mixture ```