We consider a mixture with varying concentrations in which each component is described by a nonlinear regression model. A modified least squares estimator is used to estimate the regressions parameters. Asymptotic normality of the derived estimators is demonstrated. This result is applied to confidence sets construction. Performance of the confidence sets is assessed by simulations.

Nonlinear regression models are widely used in analysis of statistical data [

In this paper we adopt a semiparametric approach with the use of modified least squares (mLS) technique. The consistency of mLS estimators in regression MVC models was demonstrated in [

The rest of the paper is organized as follows. In Section

In this paper we consider regression technique application to data, which are described by the model of mixture with varying concentrations. It means that each observed subject belongs to one of

For each subject

We will assume that

In what follows we will frequently use expectations and probabilities connected with different mixture components. To present them in a compact form, we introduce formal random vectors

We will also denote by

Similar notation is used for the weights matrix

We are interested in estimating the parameters

In what follows we

Note that so defined mLS estimator can be a point of local minimum of

In this section, we consider asymptotic behavior of

In this paper we make no assumptions on connections between

Note that if a significant fraction of

Then the weights

We will assume that the limits

We will also denote

Conditions of the

See [

Now consider the asymptotic normality of

Assume that

The

This result is just the statement of the theorem. □

Return to the regression mixture model (

Assume that

Apply the results of Section

It is obvious that if Theorem

To accomplish the confidence set construction, we need convenient conditions for the

Since

For all

Observe that

From

In what follows ≥ means the Loewner order for matrices, i.e.,

Observe that

To do this, observe that by (

There are at least two ways to estimate

The second approach to estimation of

In the simulation study, the performance of confidence ellipsoids constructed in Section

The data were generated from a mixture of two components (i.e.,

Each observation contains two variables

True parameters values for the regression model

1 | 2 | |

0.0 | 1.0 | |

2.0 | 2.0 | |

0.5 | 0.5 | |

2 | −1/3 |

In each experiment, we calculated

(i)

(ii)

(iii)

The ellipsoids were constructed by 1000 simulated samples and covering frequencies were calculated. These frequencies are presented in the tables, for each experiment.

Here the error terms were zero mean normal with the variance

Covering frequencies for normal regression errors

first component | second component | |||||

n | oracle | plug-in | jk | oracle | plug-in | jk |

100 | 0.668 | 0.942 | 0.955 | 0.729 | 0.939 | 0.953 |

500 | 0.929 | 0.931 | 0.956 | 0.948 | 0.934 | 0.939 |

1 000 | 0.954 | 0.951 | 0.95 | 0.944 | 0.937 | 0.939 |

5 000 | 0.959 | 0.951 | 0.943 | 0.952 | 0.940 | 0.931 |

7 500 | 0.961 | 0.942 | 0.933 | 0.951 | 0.938 | 0.957 |

1 000 | 0.954 | 0.949 | 0.944 | 0.944 | 0.947 | 0.954 |

Here we consider bounded regression errors, namely

Covering frequencies for uniform regression errors

first component | second component | |||||

n | oracle | plug-in | jk | oracle | plug-in | jk |

100 | 0.593 | 0.952 | 0.909 | 0.684 | 0.964 | 0.939 |

500 | 0.907 | 0.929 | 0.938 | 0.924 | 0.929 | 0.934 |

1 000 | 0.917 | 0.959 | 0.946 | 0.951 | 0.944 | 0.939 |

5 000 | 0.938 | 0.941 | 0.934 | 0.947 | 0.959 | 0.933 |

7 500 | 0.934 | 0.948 | 0.948 | 0.958 | 0.956 | 0.944 |

1 000 | 0.937 | 0.947 | 0.943 | 0.955 | 0.945 | 0.950 |

Here we compare the ellipsoids accuracy on the regression with heavy-tailed errors. The errors are taken with distribution of

Covering frequencies for uniform regression errors

first component | second component | |||||

n | oracle | plug-in | jk | oracle | plug-in | jk |

100 | 0.568 | 0.942 | 0.903 | 0.701 | 0.932 | 0.931 |

500 | 0.900 | 0.938 | 0.944 | 0.917 | 0.929 | 0.956 |

1 000 | 0.936 | 0.932 | 0.926 | 0.944 | 0.931 | 0.919 |

5 000 | 0.930 | 0.949 | 0.945 | 0.942 | 0.936 | 0.948 |

7 500 | 0.946 | 0.939 | 0.953 | 0.941 | 0.953 | 0.926 |

1 000 | 0.955 | 0.947 | 0.935 | 0.925 | 0.944 | 0.936 |

We presented theoretical results on the asymptotic normality of the modified least squares estimators for mixtures of nonlinear regressions. These results were applied to construction of confidence ellipsoids for the regression coefficients. Simulation results show that the proposed ellipsoids can be used for large enough samples.