`mclustBootstrapLRT.Rd`

Perform the likelihood ratio test (LRT) for assessing the number of mixture components in a specific finite mixture model parameterisation. The observed significance is approximated by using the (parametric) bootstrap for the likelihood ratio test statistic (LRTS).

mclustBootstrapLRT(data, modelName = NULL, nboot = 999, level = 0.05, maxG = NULL, verbose = interactive(), ...) # S3 method for mclustBootstrapLRT print(x, ...) # S3 method for mclustBootstrapLRT plot(x, G = 1, hist.col = "grey", hist.border = "lightgrey", breaks = "Scott", col = "forestgreen", lwd = 2, lty = 3, main = NULL, ...)

data | A numeric vector, matrix, or data frame of observations. Categorical variables are not allowed. If a matrix or data frame, rows correspond to observations and columns correspond to variables. |
---|---|

modelName | A character string indicating the mixture model to be fitted.
The help file for |

nboot | The number of bootstrap replications to use (by default 999). |

level | The significance level to be used to terminate the sequential bootstrap procedure. |

maxG | The maximum number of mixture components \(G\) to test. If not provided
the procedure is stopped when a test is not significant at the specified |

verbose | A logical controlling if a text progress bar is displayed during the bootstrap procedure. By default is |

... | Further arguments passed to or from other methods. In particular, see the optional arguments in |

x | An |

G | A value specifying the number of components for which to plot the bootstrap distribution. |

hist.col | The colour to be used to fill the bars of the histogram. |

hist.border | The color of the border around the bars of the histogram. |

breaks | See the argument in function |

col, lwd, lty | The color, line width and line type to be used to represent the observed LRT statistic. |

main | The title for the graph. |

The implemented algorithm for computing the LRT observed significance using the bootstrap is the following.
Let \(G_0\) be the number of mixture components under the null hypothesis versus \(G_1 = G_0+1\) under the alternative. Bootstrap samples are drawn by simulating data under the null hypothesis. Then, the p-value may be approximated using eq. (13) on McLachlan and Rathnayake (2014). Equivalently, using the notation of Davison and Hinkley (1997) it may be computed as
$$\textnormal{p-value} = \frac{1 + \#\{LRT^*_b \ge LRTS_{obs}\}}{B+1}$$
where

\(B\) = number of bootstrap samples

\(LRT_{obs}\) = LRTS computed on the observed data

\(LRT^*_b\) = LRTS computed on the \(b\)th bootstrap sample.

An object of class `'mclustBootstrapLRT'`

with the following components:

A vector of number of components tested under the null hypothesis.

A character string specifying the mixture model as provided in the function call (see above).

The observed values of the LRTS.

A matrix of dimension `nboot`

x the number of components tested
containing the bootstrap values of LRTS.

A vector of p-values.

Davison, A. and Hinkley, D. (1997) *Bootstrap Methods and Their Applications*. Cambridge University Press.

McLachlan G.J. (1987) On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. *Applied Statistics*, 36, 318-324.

McLachlan, G.J. and Peel, D. (2000) *Finite Mixture Models*. Wiley.

McLachlan, G.J. and Rathnayake, S. (2014) On the number of components in a Gaussian mixture model. *Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery*, 4(5), pp. 341-355.