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Goodness-of-Fit Test Using Local Maximum Likelihood Polynomial Estimator for Sparse Multinomial Data
Jangsun Baek 한국통계학회 2004 Journal of the Korean Statistical Society Vol.33 No.3
We consider the problem of testing cell probabilities in sparse multino- mial data. Aerts et al. (2000) presented T = Pk i=1fp i က E(p i )g2 as a test statistic with the local least square polynomial estimator p i , and derived its asymptotic distribution. The local least square estimator may produce negative estimates for cell probabilities. The local maximum likelihood poly- nomial estimator ^pi, however, guarantees positive estimates for cell proba- bilities and has the same asymptotic performance as the local least square estimator (Baek and Park, 2003). When there are cell probabilities with rel- atively much dierent sizes, the same contribution of the dierence between the estimator and the hypothetical probability at each cell in their test statis- tic would not be proper to measure the total goodness-of-t. We consider a Pearson type of goodness-of-t test statistic, T1 = Pk i=1f^pi က E(^pi)g2=pi instead, and show it follows an asymptotic normal distribution. Also we investigate the asymptotic normality of T2 = Pk i=1f^pi က E(^pi)g2 where the minimum expected cell frequency is very small.
Jangsun Baek,McLachlan, Geoffrey J,Flack, Lloyd K IEEE 2010 IEEE transactions on pattern analysis and machine Vol.32 No.7
<P>Mixtures of factor analyzers enable model-based density estimation to be undertaken for high-dimensional data, where the number of observations n is not very large relative to their dimension p. In practice, there is often the need to further reduce the number of parameters in the specification of the component-covariance matrices. To this end, we propose the use of common component-factor loadings, which considerably reduces further the number of parameters. Moreover, it allows the data to be displayed in low--dimensional plots.</P>