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

## Description

Functions `cnmms`, `cnmpl` and `cnmap` can be used to compute the maximum likelihood estimate of a semiparametric mixture model that has a one-dimensional mixing parameter. The types of mixture models that can be computed include finite, nonparametric and semiparametric ones.

Function `cnmms` can also be used to compute the maximum likelihood estimate of a finite or nonparametric mixture model.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```cnmms(x, init=NULL, maxit=1000, model=c("spmle","npmle"), tol=1e-6, grid=100, kmax=Inf, plot=c("null", "gradient", "probability"), verbose=0) cnmpl(x, init=NULL, tol=1e-6, tol.npmle=tol*1e-4, grid=100, maxit=1000, plot=c("null", "gradient", "probability"), verbose=0) cnmap(x, init=NULL, maxit=1000, tol=1e-6, grid=100, plot=c("null", "gradient"), verbose=0) ```

## Arguments

 `x` a data object of some class that can be defined fully by the user `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: non-parametric MLE (`npmle`) or semi-parametric MLE (`spmle`). `maxit` maximum number of iterations `tol` a tolerance value that is used to terminate an algorithm. Specifically, the algorithm is terminated, if the relative increase of the log-likelihood value after an iteration is less than `tol`. If an algorithm converges rapidly enough, then `-log10(tol)` is roughly the number of accurate digits in log-likelihood. `tol.npmle` a tolerance value that is used to terminate the computing of the NPMLE internally. `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, an individual `gridpoints` function may return a different number of grid points than specified by `grid`. `kmax` upper bound on the number of support points. This is useful for fitting a finite mixture model. `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 of 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 =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.

A semiparametric mixture model has a density of the form

f(x; G, beta) = Int f(x; theta, beta) d G(theta),

where G is a mixing distribution that is completely unspecified and beta is the structural parameter.

Of the three functions, `cnmms` is recommended for most problems; see Wang (2010).

Functions `cnmms`, `cnmpl` and `cnmap` implement the algorithms CNM-MS, CNM-PL and CNM-AP that are described in Wang (2010). Their implementations are generic using S3 object-oriented programming, in the sense that they can work for an arbitrary family of mixture models that is 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 the functions above:

`initial(x, beta, mix, kmax)`

`valid(x, beta)`

`logd(x, beta, pt, which)`

`gridpoints(x, beta, grid)`

`suppspace(x, beta)`

`length(x)`

`print(x, ...)`

`weights(x, ...)`

While not needed by the algorithms, one may also implement

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

so that the fitted model can be shown graphically in a way that the user desires.

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., `cvp` and `mlogit`.

## Value

 `family` the class of the mixture family that is used to fit to the data. `num.iterations` Number of iterations required by the algorithm `grad` For `cnmms`, it contains the values of the gradient function at the support points and the first derivatives of the log-likelihood with respect to theta and beta. For `cnmpl`, it contains only the first derivatives of the log-likelihood with respect to beta. For `cnmap`, it contains only the gradient of `beta`. `max.gradient` Maximum value of the gradient function, evaluated at the beginning of the final iteration. It is only given by function `cnmap`. `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` MLE of the structural parameter

## Author(s)

Yong Wang <[email protected]>

## 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

`nnls`, `cnm`, `cvp`, `cvps`, `mlogit`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```## Compute the MLE of a finite mixture x = rnpnorm(200, mu=c(0,4), pr=c(0.7,0.3), sd=1) for(k in 1:6) plot(cnmms(x, kmax=k), x, add=(k>1), comp="null", col=k+1, main="Finite Normal Mixtures") legend("topright", 0.3, leg=paste0("k = ",1:6), lty=1, lwd=2, col=2:7) ## Compute a semiparametric MLE # Common variance problem x = rcvps(k=100, ni=5:10, mu=c(0,4), pr=c(0.7,0.3), sd=3) cnmms(x) # CNM-MS algorithm cnmpl(x) # CNM-PL algorithm cnmap(x) # CNM-AP algorithm # Logistic regression with a random intercept x = rmlogit(k=100, gi=3:5, ni=6:10, pt=c(0,4), pr=c(0.7,0.3), beta=c(0,3)) cnmms(x) ### Real-world data # Random intercept logistic model data(toxo) cnmms(mlogit(toxo)) data(betablockers) cnmms(mlogit(betablockers)) data(lungcancer) cnmms(mlogit(lungcancer)) ```