Abstract
Let us have a sample satisfying d-dimensional Gaussian mixture model (d is supposed to be large). The problem of classification of the sample is considered. Because of large dimension it is natural to project the sample to k-dimensional (k = 1, 2, . . .) linear subspaces using projection pursuit method which gives the best selection of these subspaces. Having an estimate of the discriminant subspace we can perform classification using projected sample thus avoiding ’curse of dimensionality’. An essential step in this method is testing goodness-of-fit of the estimated d-dimensional model assuming that distribution on the complement space is standard Gaussian. We present a simple, data-driven and computationally efficient procedure for testing goodness-of-fit. The procedure is based on well-known interpretation of testing goodness-of-fit as the classification problem, a special sequential data partition procedure, randomization and resampling, elements of sequentialtesting.Monte-Carlosimulations are used to assess the performance of the procedure.
Highlights
Let X = XN be a sample of size N satisfying d-dimensional Gaussian mixture model with distribution function (d.f.) F
The step-by-step procedure applied to the standardized sample is the following: 1. Finding the best linear subspace of dimension k using the projection pursuit method
If we use common methods in the Step 3 the problem is the comparison of some non-parametric density estimate with some parametric density estimate in high dimensional space
Summary
Because of the high dimension of the considered space it is natural to project the sample X to linear subspaces of dimension k If the distribution of the standardized projected sample on the complementary space is standard Gaussian this linear subspace H is called discriminant subspace. Having the estimate of the discriminant subspace it is easier to perform the classification using the projected sample. D, until hypothesis of standard Gaussian distribution on the complementary space holds for some k): 1. Finding the best linear subspace of dimension k using the projection pursuit method (see, e.g., [4]). 2. Estimation of the parameters of Gaussian mixture (see, e.g., [3]) from the sample projected to the linear subspace of dimension k
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