Abstract
In this paper we consider two nonparametric mixtures of quadratic natural exponential families with unknown mixing densities. We propose a statistic to test the equality of these mixing densities when the two natural exponential families are known. The test is based on moment characterizations of the distributions. The number of moments is retained automatically by a data driven technique. Some examples and simulations of implementation of the procedure are provided.
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References
Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions. Dover Publications.
Akaike, H. (1974). A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 6, 716–723.
Albers, W., Kallenberg, W. and Martini, F. (2001). Data driven rank tests for classes of tail alternatives. J. Am. Stat. Assoc. 96, 685–696.
Asmussen, S. and Albrecher, H. 2010 Ruin Probabilities. Advanced Series on Statistical Science Applied Probability. World Scientific.
Bangert, M., Hennig, P. and Oelfke, U. (2010) Using an infinite von mises-fisher mixture model to cluster treatment beam directions in external radiation therapy. In: International Conference on Machine Learning and Applications.
Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, Chichester.
Carleman, T. (1926). Les fonctions quasi analytiques. Collection de Monographies sur la Théorie des Fonctions, Gauthier-Villars, Paris.
Carroll, R., Ruppert, D., Stefanski, L. and Crainiceanu, C. (2010). Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition. Taylor & Francis.
Chee, C.-S. and Wang, Y. (2013). Minimum quadratic distance density estimation using nonparametric mixtures. Comput. Stat. Data Anal. 57, 1, 1–16.
D’Agostino, R. B. and Stephens, M. A. (1986). Goodness-Of-Fit Techniques. Marcel Dekker, New York.
Denuit, M. (1997). A new distribution of poisson-type for the number of claims. ASTIN Bulletin 27, 229–242.
Doukhan, P., Pommeret, D. and Reboul, L. (2015). Data driven smooth test of comparison for dependent sequences. J. Multivar. Anal. 139, 147–165.
Dubey, A., Hwang, S., Rangel, C., Rasmussen, C., Ghahramani, Z. and Wild, D. L. (2004). Clustering protein sequence and structure space with infinite gaussian mixture models. World Scientific, 399–410.
Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Am. Stat. Assoc. 91, 674–688.
Ghattas, B., Pommeret, D., Reboul, L. and Yao, A. (2011). Data driven smooth test for paired populations. Journal of Statistical Planning and Inference 141, 1, 262–275.
Ghosh, A. (2001). Testing equality of two densities using Neyman’s smooth test. Tech. rep., Department of Economics, University of Illinois.
Griffin, J. E. (2010). Default priors for density estimation with mixture models. Bayesian Analysis 5, 1, 45–64.
Groeneboom, P. 1984 Estimating a Monotone Density. Report MS / Centrum voor Wiskunde en Informatica, Department of Mathematical Statistics. Stichting Math. Centrum.
Hart, J. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York.
Inglot, T., Jurlewicz, T. and Ledwina, T. (1990). On neyman-type smooth tests of fit. Statistics 21, 549–568.
Inglot, T., Kallenberg, W. and Ledwina, T. (1997). Data-driven smooth tests for composite hypotheses. Ann. Stat. 25, 1222–1250.
Inglot, T. and Ledwina, T. (2006). Towards data driven selection of a penalty function for data driven neyman test. Linear Algebra Appl. 417, 124–133.
Janic-Wróblewska, J. and Ledwina, T. (2000). Data driven rank test for two-sample problem. Scand. J. Stat. 27, 281–297.
Kallenberg, W. and Ledwina, T. (1995). On data-driven Neyman’s tests. Probab. Math. Stat. 15, 409–426.
Kallenberg, W. and Ledwina, T. (1995a). Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests. Ann. Stat. 23, 1594–1608.
Ledwina, T. (1994). Data-driven version of Neyman’s smooth test of fit. J. Am. Stat. Assoc. 89, 1000–1005.
Lindsay, B. G. (1989). Moment matrices: Applications in mixtures. Ann. Stat. 17, 2, 722–740.
Morris, C. (1982). Natural exponential families with quadratic variance functions. Ann. Stat. 10, 65–80.
Munk, Axel S.J.-P.V.J.G.G. (2011). Neyman smooth goodness-of-fit tests for the marginal distribution of dependent data. Ann. Inst. Statist. Math. 63, 5, 939–959.
Neyman, J. (1937). Smooth test for goodness of fit. Skandinavisk Aktuarietidskrift 20, 149–199.
Pommeret, D. (2005). Distance of a mixture from its parent distribution. Sankhyā 67, 4, 699–714.
Rayner, J. and Best, D. (1989). Smooth Tests of Goodness of Fit. Oxford University Press, New York.
Rayner, J. and Best, D. (2001). A Contingency Table Approach to Nonparametric Testing. Chapman and Hall/CRC.
Schwarz, G. (1978). Estimating the dimension of a model. Ann. Stat. 6.2, 461–464.
van and de Geer, S. (2003). Asymptotic theory for maximum likelihood in nonparametric mixture models. Comput. Stat. Data Anal. 41, 453–464.
Wang, J. (2007). A linearization procedure and a vdm/ecm algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models. Comput. Stat. Data Anal. 51, 2946–2957.
Wang, X.-F. and Wang, B. (2011). Deconvolution estimation in measurement error models: The r package decon. J. Stat. Softw. 39, 10, 1–24.
Wylupek, G. (2010). Data driven k sample test. Technometrics 52, 107–123.
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Pommeret, D. Comparing Two Mixing Densities in Nonparametric Mixture Models. Sankhya A 78, 133–153 (2016). https://doi.org/10.1007/s13171-015-0067-6
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DOI: https://doi.org/10.1007/s13171-015-0067-6
Keywords and phrases
- Akaike’s rule
- Natural exponential families
- Nonparametric mixture
- Quadratic variance function
- Schwarz’s rule
- Smooth test