# test the order of a mixture of normals with unequal variance

### 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.

### See Also

plotmix.norm, pmle.norm, rmix.norm

### Examples

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