Nonparametric estimation of species richness using discrete k-monotone distributions

Nonparametric mixture models are commonly used to estimate the number of unobserved species for their robustness of modelling heterogeneity. In particular, Poisson mixtures are popular in this regard, but they are also known to have the boundary problem that causes unstable estimation. A family of shape-restricted distributions known as discrete k-monotone distributions is proposed for species richness estimation. These distributions are in fact also nonparametric mixtures and can thus be fitted rapidly via some algorithms that are made available recently. As shown by empirical studies, as compared with Poisson mixtures, their use avoids the boundary problem and gives more stable and more accurate estimates.

[1]  B. Lindsay.  The Geometry of Mixture Likelihoods: A General Theory , 1983 .

[2]  B. Lindsay,et al.  Estimating the number of classes , 2007, 0708.2153.

[3]  M. E. Muldoon,et al.  Higher Monotonicity Properties of Certain Sturm-Liouville Functions. III , 1970, Canadian Journal of Mathematics.

[4]  J. Norris,et al.  Non-parametric MLE for Poisson species abundance models allowing for heterogeneity between species , 1998, Environmental and Ecological Statistics.

[5]  Dankmar Böhning,et al.  A Bagging‐Based Correction for the Mixture Model Estimator of Population Size , 2008, Biometrical journal. Biometrische Zeitschrift.

[6]  Bruce G. Lindsay,et al.  An exponential partial prior for improving nonparametric maximum likelihood estimation in mixture models , 2008 .

[7]  Note on Completely Monotone Densities , 1969 .

[8]  D. Bertram.  Studies on the transmission of cotton rat filariasis; the variability of the intensities of infection in the individuals of the vector, Liponyssus bacoti, its causation and its bearing on the problem of quantitative transmission. , 1949, Annals of tropical medicine and parasitology.

[9]  J. Bunge,et al.  Parametric Models for Estimating the Number of Classes , 2008, Biometrical journal. Biometrische Zeitschrift.

[10]  J. Bunge,et al.  Objective Bayesian estimation for the number of species , 2010 .

[11]  Yong Wang,et al.  Density estimation using non-parametric and semi-parametric mixtures , 2012 .

[12]  Lalitha Sanathanan,et al.  ESTIMATING THE SIZE OF A MULTINOMIAL POPULATION , 1972 .

[13]  Yong Wang,et al.  Maximum likelihood computation for fitting semiparametric mixture models , 2010, Stat. Comput..

[14]  Yong Wang.  On fast computation of the non‐parametric maximum likelihood estimate of a mixing distribution , 2007 .

[15]  Fadoua Balabdaoui,et al.  Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds , 2010, Statistica Neerlandica.

[16]  D. Böhning,et al.  Nonparametric maximum likelihood estimation of population size based on the counting distribution , 2005 .

[17]  M. Spevack,et al.  A complete and systematic concordance to the works of Shakespeare , 1968 .

[18]  A. Chao.  Species Estimation and Applications , 2006 .

[19]  Ji-ping Wang.  SPECIES: An R Package for Species Richness Estimation , 2011 .

[20]  Antonio Punzo,et al.  Discrete approximations of continuous and mixed measures on a compact interval , 2012 .

[21]  Ji-ping Wang,et al.  Estimating species richness by a Poisson-compound gamma model. , 2010, Biometrika.

[22]  B. Efron,et al.  Estimating the number of unseen species: How many words did Shakespeare know? Biometrika 63 , 1976 .

[23]  S. Huet,et al.  Nonparametric species richness estimation under convexity constraint , 2014, 1404.4830.

[24]  Robert K. Colwell,et al.  ESTIMATION OF SPECIES RICHNESS: MIXTURE MODELS, THE ROLE OF RARE SPECIES, AND INFERENTIAL CHALLENGES , 2005 .

[25]  B. Lindsay,et al.  A Penalized Nonparametric Maximum Likelihood Approach to Species Richness Estimation , 2005 .

[26]  Lalitha Sanathanan.  Estimating the Size of a Truncated Sample , 1977 .

[27]  R. Beaver,et al.  Estimation of the Number of Classes in a Population , 2007 .

[28]  Dankmar Böhning,et al.  Equivalence of Truncated Count Mixture Distributions and Mixtures of Truncated Count Distributions , 2006, Biometrics.

[29]  J. Bunge,et al.  Estimating the Number of Species: A Review , 1993 .