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CPS1255 Tsung-Jen Shen et al.
community ecology (Kunin & Gaston 1993). In this context, the development
of robust statistical methods for accurately predicting the occurrence of new
rare species in additional samples is urgent and necessary.

2. Methodology
Assume that an initial sample of n individuals is collected from a
metacommunity of S species in which the species relative abundance
distribution is given by p , p ,… , p . Let the binary random variable Z =1,
1 2 S i, j
signifying that the jth selected individual is identified as species i; otherwise
Z = 0, where j = 1, 2, …, n. ( X , X ,… ,X ) represents species counts with
i, j
2
1
S

= ∑
, in the sample and follows a multinomial distribution with a

=1
grand total of n and occurrence probabilities of p , p ,… , p . Let F (m) be
1 2 S k
the expected number of newly found species absent in the first sample but
have exactly k individuals detected in an additional sample of size m. Similarly,
the binary variable Z (j = n+1, n+2, …, n+m) can be used to describe the
i, j
sampling outcome for the additional sample. Mathematically, we can express
F (m) by
k
é S æ ö ù
F (m) = Eê å I(X = 0)ç m ÷ p (1- p ) m-k ú
k
÷
k ê i=1 i ç k ø i i ú
è
ë
û
æ ö S æ n+m ö æ n+m ö , (1)
ç
=ç m ÷ å P Z ,Z ,… ,Z i,n+m å Z = k ÷ Pç å Z = k÷
÷
ç
÷ ç
÷
ç
è k ø i=1 è i,1 i,2 j=1 i, j ø è j=1 i, j ø
where I(A) is an indicator function such that I(A) = 1 if statement A is true and
k
I(A) = 0 if untrue. Note that the term p (1- p ) n+m-k in F (m) can be
i i k
equivalently interpreted as exactly k individuals of species i coming out of
n+m individuals in the combined original and additional samples. The
derivation from the second equality to the last equality in Eq. 1 is based on the
fact that Z entities can be regarded as independent random Bernoulli
i, j
variates with success rates p while the sampling outcome is that there are k
i
successful outcomes out of n+m trials.
Incorporating Bayesian weights, we proposed an estimator of F (m) as
k
(Shen & Chen 2018)

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