# emtest.norm: test the order of a mixture of normals with unequal variance In MixtureInf: Inference for Finite Mixture Models

## Description

EM-test for the order of a finite mixture of normals with unequal variance.

## Usage

 ```1 2``` ```emtest.norm(x, m0 = 1, pens = NULL, inival = NULL, len = 10, niter = 50, tol = 1e-6, k = 3, rformat = FALSE) ```

## Arguments

 `x` data, can be either a vector or a matrix with the 1st column being the the observed values and the 2nd column being the corresponding frequencies. `m0` order of the finite mixture model under the null hypothesis. default value: m0 = 1. `pens` a 2-dimensions vector with the level of penalty functions for mixing proportion and variance, default value: pens = NULL. (if not provided, it will be determined by the formulas described in Chen, Li and Fu, 2012) `inival` initial values for the EM-algorithm to compute the MLE under the null model, a 3m0-dimensions vector including m0 mixing proportions, m0 component means and m0 component variances, or a matrix with 3m0 columns, default value: inival = NULL. (if not provided, random initial values are used.) `len` number of random initial values for the EM-algorithm, default value: len = 10. `niter` number of iterations for all initial values in the EM-algorithm. The algorithm runs EM-iteration niter times from each initial value. The iteration will restart from the parameter value with the highest likelihood value at the point and run until convergence. default value: niter = 50. `tol` tolerance level for the convergence of the EM-algorithm, default value: tol = 1e-6. `k` number of EM iterations: default value: k = 3. `rformat` form of the digital output: default of R package is used when rformat = T; If rformat = T, the digital output is rounded to the 3rd dicimal place if it is larger than 0.001, keeps 3 significant digits otherwise. The default value of rformat is F.

## Value

Return an object of class EM-test with the following elements:

MLE of the parameters under the null hypothesis (order = m0)

Parameter estimates under the specific alternative whose order is 2m0

EM-test statistic

P-value

Level of penalty

## Author(s)

Shaoting Li, Jiahua Chen and Pengfei Li

## References

Chen, J. and Li, P. (2009). Hypothesis test for normal mixture models: The EM approach. The Annals of Statistics. 37, 2523–2542.

Chen, J., Li, P. and Fu, Y. (2012). Inference on the order of a normal mixture. JASA. 107, 1096–1105.

 ```1 2 3 4``` ```#load the grains data set, #conduct homogeneity test by the EM-test. data(grains) emtest.norm(grains) ```