The Poisson Compound Decision Problem Revisited

The compound decision problem for a vector of independent Poisson random variables with possibly different means has a half-century-old solution. However, it appears that the classical solution needs smoothing adjustment. We discuss three such adjustments. We also present another approach that first transforms the problem into the normal compound decision problem. A simulation study shows the effectiveness of the procedures in improving the performance over that of the classical procedure. A real data example is also provided. The procedures depend on a smoothness parameter that can be selected using a nonstandard cross-validation step, which is of independent interest. Finally, we mention some asymptotic results.

[1]  Wenhua Jiang,et al.  Empirical Bayes in-season prediction of baseball batting averages , 2010 .

[2]  Junyong Park.  Nonparametric empirical Bayes estimator in simultaneous estimation of Poisson means with application to mass spectrometry data , 2012 .

[3]  Nicolas W. Hengartner,et al.  Adaptive demixing in Poisson mixture models , 1997 .

[4]  J. Zidek,et al.  Simultaneous Estimation of the Means of Independent Poisson Laws , 1975 .

[5]  Lawrence D. Brown,et al.  NONPARAMETRIC EMPIRICAL BAYES AND COMPOUND DECISION APPROACHES TO ESTIMATION OF A HIGH-DIMENSIONAL VECTOR OF NORMAL MEANS , 2009, 0908.1712.

[6]  J. B. Copas,et al.  Compound Decisions and Empirical Bayes , 1969 .

[7]  I. Johnstone,et al.  On independent statistical decision problems and products of diffusions , 1984 .

[8]  F. T. Wright,et al.  Order restricted statistical inference , 1988 .

[9]  J. Kiefer,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS , 1956 .

[10]  J. S. Maritz,et al.  Empirical Bayes estimation for the Poisson distribution , 1969 .

[11]  Harrison H. Zhou,et al.  The root–unroot algorithm for density estimation as implemented via wavelet block thresholding , 2010 .

[12]  Cun-Hui Zhang,et al.  Compound decision theory and empirical bayes methods , 2003 .

[13]  Intrinsic losses for empirical Bayes estimation: A note on normal and Poisson cases , 1995 .

[14]  Avishai Mandelbaum,et al.  Statistical Analysis of a Telephone Call Center , 2005 .

[15]  Ya'acov Ritov,et al.  Asymptotic e-ciency of simple decisions for the compound decision problem ⁄ , 2008, 0802.1319.

[16]  Eitan Greenshtein,et al.  Comparison of sequential experiments , 1996 .

[17]  B. Efron.  Robbins, Empirical Bayes, And Microarrays , 2001 .

[18]  I. Johnstone.  Admissibility, Difference Equations and Recurrence in Estimating a Poisson Mean , 1984 .

[19]  H. Robbins.  An Empirical Bayes Approach to Statistics , 1956 .

[20]  H. Robbins.  The Empirical Bayes Approach to Statistical Decision Problems , 1964 .

[21]  Eitan Greenshtein,et al.  EMPIRICAL BAYES IN THE PRESENCE OF EXPLANATORY VARIABLES , 2012 .

[22]  Enno Mammen,et al.  Estimating a Smooth Monotone Regression Function , 1991 .

[23]  Wenhua Jiang,et al.  General maximum likelihood empirical Bayes estimation of normal means , 2009, 0908.1709.