Pengfei Li

1. EM-test in finite mixture model: MixtureInf

The EM-test is a likelihood based method for testing the order of a finite mixture. Some ideas of the EM-test can be traced back to the modified likelihood ratio test (Chen et al., 2001 and 2004), but it has its own advantages. (a) The EM-test has simple limiting distributions under less regularity conditions and therefore is more widely applicable. (b) Its theoretical analysis is less challenging. More details can be found in the reference. So far the R package MixtureInf provides the R functions for

(a) Testing the order of a finite mixture with univariate mixing parameter

The EM-test can be used to test

H_0: order of a mixture=m0 versus H_A: order of a mixture>m0,

where m0 is an arbitrary positive integer. The description and the properties of the EM-test can be found in Li and Chen (2010) and Li, Chen and Marriott (2009).

(b) Testing the order of a normal mixtures with unknown variances

Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. (a) The log-likelihood function is unbounded. (b) The Fisher information with respect to the mixing proportion maybe infinite. (c)The normal mixture model is not strongly identifiable if the variance parameter is unknown. Our EM-test can easily handle the above three challenges. The description and properties can be found in Chen and Li (2009) and Chen, Li and Fu (2012).

Reference:

1. Chen, H., Chen, J. and Kalbﬂeisch, J. D. (2001) A modiﬁed likelihood ratio test for homogeneity in ﬁnite mixture models. JRSSB, 63, 19-29.

2. Chen, H., Chen, J. and Kalbﬂeisch, J. D. (2004) Testing for a ﬁnite mixture model with two components. JRSSB, 66, 95-115.

3. Chen, J. and Li, P. (2009) Hypothesis test for normal mixture models: the EM approach. The Annals of Statistics, 37, 2523-2542.

4. Li, P. and Chen, J. (2010) Testing the order of a finite mixture model. JASA, 105, 1084-1092.

5. Li, P., Chen, J. and Marriott, P. (2009) Non-finite Fisher information and homogeneity: the EM approach. Biometrika, 96, 411-42.

6. Chen, J., Li, P. and Fu, Y. (2012) Inference on the order of a normal mixture. JASA, 107, 1096-1105.

2. Testing homogeneity in a scale mixture of normal distributions: EMScale.R

The EM-test is extended to test the homogeneity in the scale mixture of normal distributions in Niu, Li and Zhang (2016). The R function in EMScale.R can be used to calculate the EM-test statistics.

Reference:

1. Niu, X.*, Li, P. and Zhang, P. (2016). Testing homogeneity in a scale mixture of normal distributions. Statistical papers, 57, 499-516.

3. Abundance estimation in discrete-time capture-recapture experiments: abun.zip

Liu, Li and Qin (2017) proposed the empirical likelihood based methods for estimating and constructing the confidence interval for the abundance in discrete-time capture-recapture experiments. In abun.R, the gabun function implements the empirical likelihood and conditional likelihood methods for the general case, and the sabun function implements these methods for the special case. See the accompanying example.R for the use of these functions.

Reference:

1. Liu, Y., Li, P. and Qin, J. (2017). Maximum empirical likelihood estimation for abundance in a closed population from capture-recapture data. Biometrika, 104, 527-543.