estimating the total species richness within large communities of species , using only samplings limited in sizes , is a common , long standing challenge which has elicited numerous procedures of estimations . for a few decades , a series of so - called nonparametricestimators provide elegant and convenient solutions to this question . these new estimators are ordinarily simple format formulations which , moreover , require no specific assumption regarding the statistical distribution of the abundances of species ( which , thus , makes these formulations  nonparametric  ) . within this category of formulations , anne chao proposed a very popular nonparametric expression , which actually stands among the most commonly used estimators of the total species richness of a sampled community . let  be the number of species that are missed by ( i.e. , unrecorded within ) the limited sampling of a large community of species . then , according to chao 's formulation , 
 ( 1)=f122f2 
 with f1 and f2 as the numbers of species encountered only once and twice , respectively , in the sample . this expression was later generalised   as 
 ( 2)=f1x(x!fx)(1/(x1 ) ) 
 with fx standing for the number of species recorded x times in the sample . these expressions , derived on a statistical basis , provide a lower - bound estimate of the number  of missing species in a sampled community [ 1 , 2 ] . that is , chao 's formulation is expected to provide only a minimal threshold ( in the statistical sense ) for the estimated species richness . yet , several decades of practice ( especially in the field of ecological purposes and biodiversity surveys ) call for placing chao 's expression among the most valuable and reliable estimators [ 37 ] since , in many occasions , this expression may well appear to provide approximatelycentered rather than only lower - bound estimates . in short , although designed conceptually as a lower - bound evaluation , chao 's formulation may , nonetheless , fairly often provide rather centered estimates in the common practice . hereafter , we address this apparent paradox and propose new insights and argumentations , issued from an alternative , algebraic derivation of the originally statistically derived formulation by chao . we demonstrate ( see appendix a for mathematical details ) that the general expression of chao 's estimator , established originally on a statistical basis , may also admit , alternatively , an algebraic derivation , leading to exactly the same expression as the statistically derived formulation  = [ f1/(x!fx ) ] . yet , while the statistical derivation of chao 's formulation requires no particular restriction (  nonparametric  ) , the algebraic derivation , implies a particular shape for the expected decrease of the proportion /s of missing ( i.e. , unrecorded ) species when the sample size n increases . in fact , as demonstrated in appendix a , this asymptotic decrease should conform to a negative exponential : 
 ( 3)s = exp(kn ) 
 with(i)s  as the species richness , that is , the total ( unknown ) number of species of the community,(ii)k  as a constant,(iii)n  as the sample size , that is , the number of individuals recorded in the sample . s  as the species richness , that is , the total ( unknown ) number of species of the community , n  as the sample size , that is , the number of individuals recorded in the sample . in turn , this particular form of the decrease of the number of missing species with enlarging sample sizes constrains the shape of the species abundance distribution ( the  sad ,  that is , the distribution of species abundances when species are conventionally ranked by decreasing order of abundance ) . according to ( 3 ) , the number r  of recorded species in the sample is 
 ( 4)r = s=s1expkn . let k be the number of individuals belonging to the less abundant species among the  r  species recorded in the sample ( i.e. , the species of rank  r  when species are ranked by decreasing order of abundance ) . then , the relative abundance ar of the species of rank r is expected to be inversely proportional to n , as 
 ( 5)ar = kn . 
 according to ( 4 ) and ( 5 ) , 
 ( 6)r = s1expkkar . 
 according to the sample size n , every species may be called to play the role of the less abundant species within the sample ( since , by continuously decreasing the size n of the sample , each species of the community [ including , at last , the most common ] would successively play the role of the least abundant species in the sample ) . this equation thus describes the shape of the species abundance distribution ( ai = f(i ) ) when the proportion of unrecorded species in a sample is exponentially decreasing with the sample size , that is , the shape of the species abundance distribution which conditions the validity of the algebraic derivation of chao estimator . as the algebraic derivation is deterministic by essence , it therefore provides a centered estimate of  and of the resulting total species richness s , instead of being only a lower - bound estimate , as is the case in the nonparametric context . as mentioned above , the algebraic derivation of chao 's formulation requires that the sampled community satisfies , at least approximately , the particular type of sad defined by ( 7 ) and illustrated at figure 1 . this restrictive condition is the  price  to be paid for the more accurate estimate , namely , the loss of the strict  nonparametric  character of the statistically based conception . yet , this condition assigned to the shape of the distribution of species abundances might not be so restrictive in practice , at least as a first approximation . reasons for this may be as follows : ( a)an asymptotic decrease to zero of  with n ( equation ( 3 ) ) seems logical and intuitive , because aiming to estimate the total number of species in a community implicitly requires that this number does exist and might be actually reached progressively with sampling size n increasing continuously ; ( b)among the different types of accumulation curves with such an asymptotic evolution , the negative exponential answer of  to increasing sampling size is , admittedly , one among the most simple , robust , and seemingly common [ 8 , 9 ] ; ( c)the sigmoidal shape of the prescribed sad ( equation ( 7 ) and figure 1 ) is not so far from the most classically referred empirical types , broken - stick and log - normal distributions . yet for example , the equation i = f(ai ) for the sad corresponding to a broken - stick distribution is i  s  exp(s  ai ) , which is formally different from ( 7 ) . an asymptotic decrease to zero of  with n ( equation ( 3 ) ) seems logical and intuitive , because aiming to estimate the total number of species in a community implicitly requires that this number does exist and might be actually reached progressively with sampling size n increasing continuously ; among the different types of accumulation curves with such an asymptotic evolution , the negative exponential answer of  to increasing sampling size is , admittedly , one among the most simple , robust , and seemingly common [ 8 , 9 ] ; the sigmoidal shape of the prescribed sad ( equation ( 7 ) and figure 1 ) is not so far from the most classically referred empirical types , broken - stick and log - normal distributions . for example , the equation i = f(ai ) for the sad corresponding to a broken - stick distribution is i  s  exp(s  ai ) , which is formally different from ( 7 ) . accordingly , it is no real surprise that chao 's formula often approaches a strictly centred estimate , in spite of being only a lower - bound estimate in all generality . this would explain why , in ecological practice in particular , chao 's formulation is yet considered one of the more accurate and reliable estimators of the total species richness within partially sampled communities . as mentioned in particular by gotelli ( personal communication ) , a trend would remain for chao 's estimates to increase somehow when a series of sampling of growing sizes are extracted from the same community instead of remaining ideally stable on average . this , however , is not necessarily contradictory to preceding arguments but should certainly result from residual discrepancy between the real sad and the ideal model described by ( 7 ) and exemplified at figure 1 .