Introduction
An inescapable problem in microbial ecology is that a sample from an environment typically does not observe all species present in that environment. In [14], this problem has 1been recently linked to the concepts of coverage probability (i.e. the probability that a member from the environment is represented in the sample) and the closely related discovery or unobserved probability (i.e. the probability that a previously unobserved species is seen with another random observation from that environment). The mathematical treatment of coverage is not limited, however, to microbial ecology and has found applications in varied contexts, including gene expression, microbial ecology, optimization, and even numismatics.
The point estimation of coverage and discovery probability seem to have been first addressed by Turing and Good [2] to help decipher the Enigma Code, and subsequent work has provided point predictors and prediction intervals for these quantities under various assumptions [1], [3]–[5].
Following Robbins [6] and in more generality Starr [7], an unbiased estimator of the expected discovery probability of a sample of size 
is
(1)where 
is the number of species observed exactly 
-times in a sample with replacement of size 
. Using the theory of U-statistics developed by Halmos [8], Clayton and Frees [9] show that the above estimator is the uniformly minimum variance unbiased estimator (UMVUE) of the expected discovery probability of a sample of size 
based on an enlarged sample of size 
.
A quantity analogous to the discovery probability of a sample from a single environment but in the context of two environments is dissimilarity, which we broadly define as the probability that a draw in one environment is not represented in a random sample (of a given size) from a possibly different environment. Estimating the dissimilarity of two microbial environments is therefore closely related to the problem of assessing the species that are unique to each environment, and the concept of dissimilarity may find applications to measure sample quality and allocate additional sampling resources, for example, for a more robust and reliable estimation of the UniFrac distance [10], [11] between pairs of environments. Dissimilarity may find applications in other and very different contexts. For instance, in SELEX experiments [12]–a laboratory technique in which an initial pool of synthesized random RNA sequences is repeatedly screened to yield a pool containing only sequences with given biological functions–the dissimilarity of two RNA pools corresponds to the probability of finding binding site solutions in one pool that are absent in the other.
In this manuscript, we study an estimator of dissimilarity probability similar to Robbins' and Starr's statistic for discovery probability. Our estimator is optimal among the appropriate class of unbiased statistics, while being approximately normally distributed in a general case. The variance of this statistic is estimated using a consistent jackknife. As proof of concept, we analyze samples of processed V35 16S rRNA data from the Human Microbiome Project [13].
Probabilistic Formulation and Inference Problem
To study dissimilarity probability, we use the mathematical model of a pair of urns, where each urn has an unknown composition of balls of different colors, and where there is no a priori knowledge of the contents of either urn. Information concerning the urn composition is inferred from repeated draws with replacement from that urn.
In what follows, 
and 
are independent sequences of independent and identically distributed (i.i.d.) discrete random variables with probability mass functions 
and 
, respectively. Without loss of generality we assume that 
and 
are supported over possibly infinite subsets of 
, and think of outcomes from these distributions as “colors”: i.e. we speak of color-
, color-
, etc. Let 
denote the set of colors 
such that 
, and similarly define 
. Under this perspective, 
denotes the color of the 
-th ball drawn with replacement from urn-
. Similarly, 
is the color of the 
-th ball drawn with replacement from urn-
. Note that based on our formulation, distinct draws are always independent.
The mathematical analysis that follows was motivated by the problem of estimating the fraction of balls in urn-
with a color that is absent in urn-
. We can write this parameter as
(2)where
(3)The parameter 
measures the proportion of urn-
which is unique from urn-
. On the other hand, 
is a measure of the effectiveness of 
-samples from urn-
to determine uniqueness in urn-
. This motivates us to refer to the quantity in (2) as the dissimilarity of urn-
from urn-
, and to the quantity in (3) as the average dissimilarity of urn-
relative to 
-draws from urn-
. Note that these parameters are in general asymmetric in the roles of the urns. In what follows, urns-
and -
are assumed fixed, which motivates us to remove subscripts and write 
instead of 
.
Unfortunately, one cannot estimate unbiasedly the dissimilarity of one urn from another based on finite samples, as stated in the following result. (See the Materials and Methods section for the proofs of all of our results).
Theorem 1
(No unbiased estimator of dissimilarity.) There is no unbiased estimator of 
based on finite samples from two arbitrary urns-
and -
.
Furthermore, estimating 
accurately without further assumptions on the compositions of urns-
and -
seems a difficult if not impossible task. For instance, arbitrarily small perturbations of urn-
are likely to be unnoticed in a sample of a given size from this urn but may drastically affect the dissimilarity of other urns from urn-
. To demonstrate this idea, consider a parameter 
and let 
, 
and 
. If 
then 
while, for each 
, 
.
In contrast with the above, for fixed 
, 
depends continuously on 
e.g. under the metric
where 
denotes the total variation of a signed measure 
over 
such that 
. This is the case because
The above implies that 
is continuous with respect to any metric equivalent to 
. Many such metrics can be conceived. For instance, if 
denotes the probability measure associated with 
samples with replacement from urn-
that are independent of 
samples with replacement from urn-
then 
is also continuous with respect to any of the metrics 
, with 
, because
Because of the above considerations, we discourage the direct estimation of 
and focus on the problem of estimating 
accurately.
Results
Consider a finite number of draws with replacement 
and 
, from urn-
and urn-
, respectively, where 
are assumed fixed. Using this data we can estimate 
, for 
, via the estimator:
(4)where
(5)We refer to 
as the 
-statistics summarizing the data from both urns. Due to the well-known relation: 
, at most 
of these estimators are non-zero. This sparsity may be exploited in the calculation of the right-hand side of (4) over a large range of 
's.
Our statistic in 
is the U-statistic associated with the kernel 
, where 
is used to denote the indicator function of the event within the brackets (Iverson's bracket notation). Following the approach by Halmos in [8], we can show that this U-statistic is optimal amongst the unbiased estimators of 
for 
. We note that no additional samples from either urn are necessary to estimate 
unbiasedly over this range when 
. This contrasts with the estimator in equation (1), which requires sample enlargement for unbiased estimation of discovery probability of a sample of size 
.
Theorem 2 (Minimum variance unbiased estimator.) If 
and 
then 
is the unique uniformly minimum variance unbiased estimator of 
. Further, no unbiased estimator of 
exists for 
or 
.
Our next result shows that 
converges uniformly in probability to 
over the largest possible range where unbiased estimation of the later parameter is possible, despite the non-Markovian nature of 
when applied sequentially over 
. The result asserts that 
is likely to be a good approximation of 
, uniformly for 
, when 
and 
are large. The method of proof uses an approach by Hoeffding [14] for the exact calculation of the variance of a 
-statistic.
Theorem 3 (Uniform convergence in probability.) Independently of how 
and 
tend to infinity, it follows for each 
that
(6)We may estimate the variance of 
for 
via a leave-one-out or also called delete-
jackknife estimator, using an approach studied by Efron and Stein [15] and Shao and Wu [16].
To account for variability in the 
-data through a leave-one-out jackknife estimate, we require that 
and let
(7)
On the other hand, to account for variability in the 
-data, consider for 
and 
the statistics
(8)
Clearly, 
; in particular, the 
-statistics are a refinement of the 
-statistics. Define 
and, for 
, define
(9)where
(10)
(11)
Our estimator of the variance of 
is obtained by summing the variance attributable to the 
-data and the 
-data and is given by
(12)for 
; in particular, 
is our jackknife estimate of the standard deviation of 
.
To assess the quality of 
as an estimate of the variance of 
and the asymptotic distribution of the later statistic, we require a few assumptions that rule out degenerate cases. The following conditions are used in the remaining theorems in this section:
.- there are at least two colors in
that occur in different proportions in urn-
; in particular, the conditional probability 
is not a uniform distribution.
- urn-
contains at least one color that is absent in urn-
; in particular, 
.
and 
grow to infinity at a comparable rate i.e. 
, which means that there exist finite constants 
such that 
, as 
tend to infinity.
Conditions (a–c) imply that 
has a strictly positive variance and that a projection random variable, intermediate between 
and 
, has also a strictly positive variance. The idea of projection is motivated by the analysis of Grams and Serfling in [17].
Condition (d) is technical and only used to show that the result in Theorem 5 holds for the largest possible range of values of 
namely, for 
. See [18] for results with uniformity related to Theorem 4, as well as uniformity results when condition (d) is not assumed.
Because the variance of 
, from now on denoted 
, and its estimate 
tend to zero as 
and 
increase, the unnormalized consistency result is unsatisfactory. As an alternative, we can show that 
is a consistent estimator relative to 
, as stated next.
Theorem 4 (Asymptotic consistency of variance estimation.) If conditions (a)–(c) are satisfied then, for each 
and 
, it applies that
(13)Finally, under conditions (a)–(d), we show that 
is asymptotically normally distributed for all 
, as 
and 
increase at a comparable rate.
Theorem 5 (Asymptotic normality.) Let 
i.e. 
has a standard normal distribution. If conditions (a)–(d) are satisfied then
(14)for all real number 
.
The non-trivial aspect of the above result is the asymptotic normality of 
when 
, e.g. 
, as the results we have found in the literature [14],[19],[20] only guarantee the asymptotic normality of our estimator of 
for fixed 
. We note that, due to Slutsky's theorem [21], it follows from (13) and (14) that the ratio
has, for fixed 
, approximately a standard normal distribution when 
and 
are large and of a comparable order of magnitude.
Discussion
As proof of concept, we use our estimators to analyze data from the Human Microbiome Project (HMP) [13]. In particular, our samples are V35 16S rRNA data, processed by Qiime into an operational taxonomic unit (OTU) count table format (see File S1). Each of the 
samples analyzed have more than 
successfully identified bacteria (see File S2). We sort these samples by the body location metadata describing the origin of the sample. This sorting yields the assignments displayed in Table 1.
We present our estimates of 
for all 
possible sample comparisons in Figure 1, i.e., we estimate the average dissimilarity of sample-
relative to the full sample-
. Due to (4), observe that 
. At the given sample sizes, we can differentiate four broad groups of environments: stool, vagina, oral/throat and skin/nostril. We differentiate a larger proportion of oral/throat bacteria found in stool than stool bacteria found in the oral/throat environments. We may also differentiate the throat, gingival and saliva samples, but cannot reliably differentiate between tongue and throat samples or between the subgingival and supragingival plaques. On the other hand, the stool samples have larger proportions of unique bacteria relative to other stool samples of the same type, and vaginal samples also have this property. In contrast the skin/nostril samples have relatively few bacteria that are not identified in other skin/nostril samples.
The above effects may be a property of the environments from which samples are taken, or an effect of noise from inaccurate estimates due to sampling. To rule out the later interpretation, we show estimates of the standard deviation of 
based on the jackknife estimator 
from (12) in Figure 2. As 
is zero, the error estimate is given by 
. We see from (7), with 
, that
Assuming a normal distribution and an accurate jackknife estimate of variance, 
will be in the interval 
with at least approximately 95% confidence, for any choice of sample comparisons in our data; in particular, on a linear scale, we expect at least 95% of the estimates in Figure 1 to be accurate in at least the first two digits.
As we mentioned earlier, estimating 
accurately is a difficult problem. We end this section with two heuristics to assess how representative 
is of 
, when urn-
has at least two colors and at least one color in common with urn-
. First, observe that:
(15)
In particular, 
is a strictly concave-up and monotonically decreasing function of the real-variable 
. Hence, if 
is close to the asymptotic value 
, then 
should be of small magnitude. We call the later quantity the discrete derivative of 
at 
. Since we may estimate the discrete derivative from our data, the following heuristic arises: relatively large values of 
are evidence that 
is not a good approximation of 
.
Figure 3 shows the heat map of 
for each pair of samples. These estimates are of order 
for the majority of the comparisons, and spike to 
for several sample-
of varied environment types, when sample-
is associated with a skin or vaginal sample. In particular, further sampling effort from environments associated with certain vaginal, oral or stool samples are likely to reveal bacteria associated with broadly defined skin or vaginal environments.
Another heuristic may be more useful to assess how close 
is to 
, particularly when the previous heuristic is inconclusive. As motivation, observe that 
, because of the identity in (15), where
Furthermore, 
, where 
is certain finite constant. We can justify this approximation only when 
is well approximated by a linear function of 
, in which case we let 
denote the estimated value for 
obtained from the linear regression. Since 
, the following more precise heuristic comes to light: 
is a good approximation of 
if the linear regression of 
for 
near 
gives a good fit, 
is small relative to 
, and 
is also small.
To fix ideas we have applied the above heuristic to three pairs of samples: 
, 
and 
, with each ordered pair denoting urn-
and urn-
, respectively. As seen in Table 2 for these three cases, 
is at least 14-times larger than 
; in particular, due to the asymptotic normality of the later statistic, an appropriate use of the heuristic is reduced to a good linear fit and a small 
value. In all three cases, 
was computed from the estimates 
, with 
.
For the 
-pair, 
and the regression error, measured as the largest absolute residual associated with the best linear fit, are zero to machine precision, suggesting that 
is a good approximation of 
. This is reinforced by the blue plot in Figure 4. On the other hand, for the 
-pair, the regression error is small, suggesting that the linear approximation 
is good for 
. However, because 
, we cannot guarantee that 
is a good approximation of 
. In fact, as seen in the red-plot in Figure 4, 
, with 
, exposes a steady and almost linear decay that suggests that 
may be much smaller than 
. Finally, for the 
-pair, the regression error is large and the heuristic is therefore inconclusive. Due to the green-plot in Figure 4, the lack of fit indicates that the exponential rate of decay of 
to 
has not yet been captured by the data from these urns. Note that the heuristic based on the discrete derivative shows no evidence that 
is far from 
.
Materials and Methods
Here we prove the theorems given in the Results section. The key idea to prove each theorem may be summarized as follows.
To show Theorem 1, we identify pairs of urns for which unbiased estimation of 
is impossible for any statistic. To show Theorem 2, we exploit the diversity of possible urn distributions to show that there are relatively few unbiased estimators of 
and, in fact, there is a single unbiased estimator 
that is symmetric on the data. The uniqueness of the symmetric estimator is obtained via a completeness argument: a symmetric statistic having expected value zero is shown to correspond to a polynomial with identically zero coefficients, which themselves correspond to values returned by the statistic when presented with specific data. The symmetric estimator is a U-statistic in that it corresponds to an average of unbiased estimates of 
, based on all possible sub-samples of size 
and 
from the samples of urn-
and -
, respectively. As any asymmetric estimator has higher variance than a corresponding symmetric estimator, the symmetric estimator must be the UMVUE.
To show Theorem 3 we use bounds on the variance of the U-statistic and show that, uniformly for relatively small 
, 
converges to 
in the 
-norm. In contrast, for relatively large values of 
, we exploit the monotonicity of 
and 
to show uniform convergence.
Finally, theorems 4 and 5 are shown using an approximation of 
by sums i.i.d. random variables, as well as results concerning the variance of both 
and its approximation. In particular, the approximation satisfies the hypotheses the Central Limit Theorem and Law of Large Numbers, which we use to transfer these results to 
.
In what follows, 
denotes the set of all probability distributions that are finitely supported over 
.
Proof of Theorem 1
Consider in 
probability distributions of the form 
, 
and 
, where 
is a given parameter. Any statistic 
which takes as input 
draws from urn-
and 
draws from urn-
has that 
is a polynomial of degree at most 
in the variable 
; in particular, it is a continuous function of 
over the interval 
. Since 
has a discontinuity at 
over this interval, there exists no estimator of 
that is unbiased over pairs of distributions in 
.
We use lemmas 6–11 to first show Theorem 2. The method of proof of this theorem follows an approach similar to the one used by Halmos [8] for single distributions, which we extend here naturally to the setting of two distributions.
Our next result implies that no uniformly unbiased estimator of 
is possible when using less than one sample from urn-
and 
samples from urn-
.
Lemma 6 If 
is unbiased for 
for all 
, then 
and 
.
Proof. Consider in 
probability distributions of the form 
, 
, 
and 
, where 
are arbitrary real numbers. Clearly, 
is a linear combination of polynomials of degree 
in 
and 
in 
and, as a result, it is a polynomial of degree at most 
in 
and 
in 
. Since 
has degree 
in 
and 
in 
, and 
is unbiased for 
, we conclude that 
and 
.
The form of 
given in equation (4) is convenient for computation but, for mathematical analysis, we prefer its 
-statistic form associated with the kernel function 
.
In what follows, 
denotes the set of all functions 
that are one-to-one.
Lemma 7
(16)where 
.
Proof. Fix 
and suppose that color 
occurs 
-times in 
. If 
then any sublist of size 
of 
contains 
, hence 
, for all 
. On the other hand, if 
then 
. Since the rightmost sum only depends on the number of times that color 
was observed in 
, we may use the 
-statistics defined in equation (5) to rewrite:
The right-hand side above now corresponds to the definition of 
given in equation (4).
In what follows, we say that a function 
is 
-symmetric when
for all 
and permutations 
and 
of 
and 
, respectively. Alternatively, 
is 
-symmetric if and only if it may be regarded a function of 
, where 
and 
correspond to the order statistics 
and 
, respectively. Accordingly, a statistic of 
is called 
-symmetric when it may be represented in the form 
, for some 
-symmetric function 
. It is immediate from Lemma 7 that 
is 
-symmetric.
The next result asserts that the variance of any non-symmetric unbiased estimator of 
may be reduced by a corresponding symmetric unbiased estimator. The proof is based on the well-known fact that conditioning preserves the mean of a statistic and cannot increase its variance.
Lemma 8 An asymmetric unbiased estimator of 
that is square-integrable has a strictly larger variance than a corresponding 
-symmetric unbiased estimator.
Proof. Let 
denote the sigma-field generated by the random vector 
and suppose that the statistic 
is unbiased for 
and square-integrable. In particular, 
is a well-defined statistic and there is an 
-symmetric function 
such that 
. Clearly, 
is unbiased for 
and 
-symmetric. Since 
, Jensen's inequality for conditional expectations [22] implies that 
, with equality if and only if 
is 
-symmetric.
Since 
is 
-symmetric and bounded, the above lemma implies that if an UMVUE for 
exists then it must be 
-symmetric. Next, we show that there is a unique symmetric and unbiased estimator of 
, which immediately implies that 
is the UMVUE.
In what follows, 
denote integers. We say that a polynomial 
is 
-homogeneous when it is a linear combination of polynomials of the form 
, with 
and 
. Furthermore, we say that 
satisfies the partial vanishing condition if 
whenever 
, 
and 
.
The next lemma is an intermediate step to show that a 
-homogeneous polynomial which satisfies the partial vanishing condition is the zero polynomial, which is shown in Lemma 10.
Lemma 9 If 
is a 
-homogeneous polynomial in the real variables 
, with 
, that satisfies the partial vanishing condition, then 
whenever 
, 
and 
.
Proof. Fix 
such that 
and 
and observe that
because 
is a 
-homogeneous polynomial. Notice now that the right hand-side above is zero because 
satisfies the partial vanishing condition.
Lemma 10 Let 
be a 
-homogeneous polynomial in the real variables 
, with 
. If 
satisfies the partial vanishing condition then 
identically.
Proof. We prove the lemma using structural induction on 
for all 
.
If 
then a 
-homogeneous polynomial 
must be of the form 
, for an appropriate constant 
. As such a polynomial satisfies the partial-vanishing condition only when 
, the base case for induction is established.
Next, consider a 
-homogeneous polynomial 
, with 
, that satisfies the partial vanishing condition, and let 
denote its degree with respect to the variable 
. In particular, there are polynomials 
in the variables 
such that
Now fix 
such that 
and 
. Because 
satisfies the partial vanishing condition, Lemma 9 implies that 
for all 
. In particular, for each 
, 
whenever 
, 
and 
. Thus each 
satisfies the partial vanishing condition. Since 
is a 
-homogeneous polynomial, the inductive hypothesis implies that 
identically and hence 
identically. The same argument shows that if 
, with 
, is a 
-homogeneous polynomial that satisfies the partial vanishing condition then 
identically, completing the inductive proof of the lemma.
Our final resultbefore proving Theorem 2 implies that 
cannot admit more than one symmetric and unbiased estimator. Its proof depends on the variety of distributions in 
, and uses the requirement that our estimator must be unbiased for any pair of distributions chosen from 
.
Lemma 11 If 
is an 
-symmetric function such that 
, for all 
, then 
identically.
Proof. Consider a point 
and define 
and 
as the cardinalities of the sets 
and 
, respectively. Furthermore, let 
denote the distinct elements in the set 
and define 
to be the number of times that 
appears in this set. Furthermore, let 
be a probability distribution such that 
and define 
. In a completely analogous manner define 
, 
, 
and 
.
Notice that 
is a polynomial in the real variables 
that satisfies the hypothesis of Lemma 10; in particular, this polynomial is identically zero. However, because 
is 
-symmetric, the coefficient of 
in 
is
implying that 
.
Proof of Theorem 2
From Lemma 8, as we mentioned already, if the UMVUE for 
exists then it must be 
-symmetric. Suppose there are two 
-symmetric functions such that 
and 
are unbiased for 
. Applying Lemma 11 to 
shows that 
, and 
admits therefore a unique symmetric and unbiased estimator. From Lemma 7, 
is 
-symmetric and unbiased for 
hence it is the UMVUE for 
. From Lemma 6, it follows that no unbiased estimator of 
exists for 
or 
.
Our next goal is to show Theorem 3, for which we prove first lemmas 12–13. We note that the later lemma applies in a much more general context than our treatment of dissimilarity.
Lemma 12 If, for each 
, 
is an integer such that 
then
(17)
(18)uniformly for 
as 
.
Proof. First observe that for all 
sufficiently large and 
, it applies that
Note that 
, for all 
. As a result, we may bound the exponential factor on the right-hand side above as follows:
Since 
and 
, uniformly for all 
as 
, (17) follows.
To show (18), first note the combinatorial identity
(19)
Proceeding in an analogous manner as we did to show (17), we see now that the term associated with the index 
in the above summation satisfies that
for all 
sufficiently large and 
. Since 
and 
, the above inequalities together with (17) and (19) establish (18).
Lemma 13 Define 
, where 
is a bounded 
-symmetric function, and let
be the U-statistic of 
associated with 
draws from urn-
and 
draws from urn-
; in particular, 
. Furthermore, assume that
,- there is a function
such that 
,
; in particular, 
.
Under the above assumptions, it follows that
Proof. Define 
; in particular, 
and 
, 
and 
, for any 
, as 
. The proof of the theorem is reduced to show that
(20)
(21)
Next, we compute the variance of 
following an approach similar to Hoeffding [14]. Because 
is 
-symmetric, a tedious yet standard calculation shows that
(22)where
(23)
(24)
Clearly, 
. On the other hand, if 
is any random variable with finite expectation and 
are sigma-fields then 
, due to well-known properties of conditional expectations [22]. In particular, for each 
, we have that
(25)
Consequently, (22) implies that
(26)
We claim that
(27)
Indeed, using an argument similar as above, we find that
Due to assumptions (i)-(ii) and the Bounded Convergence Theorem, the right-hand side above tends to 
, and the claim follows.
It follows from (26) and (27) that
Finally, because of assumption (iii),
Since each term on the right-hand side above tends to zero as 
, (20) follows.
We now show (21). As 
and 
, it follows by (22) and Lemma 12 that
uniformly for 
as 
. In particular,
Due to the definition of the coefficients 
, the right-hand side above tends to zero, and (21) follows.
Proof of Theorem 3
Note that 
, with 
. We show that the kernel function 
and the U-statistics 
satisfy the hypotheses of Lemma 13. From this the theorem is immediate because 
-convergence implies convergence in probability.
Clearly 
is 
-symmetric and 
, which shows assumption (i) in Lemma 13. On the other hand, due to the Law of Large Numbers, 
almost surely, from which assumption (ii) in the lemma also follows.
Finally, to show assumption (iii), recall that 
is the set of one-to-one functions from 
into 
; in particular, 
. Now note that for each indicator of the form 
, with 
, there are 
choices of 
outside the set 
. Because 
, it follows that 
for all 
. This shows condition (iii) in Lemma 13, and Theorem 3 follows.
Similarly, 
corresponds to the jackknife summed over each possible deletion of a single 
-data, which is more precisely given by
(29)where 
is the set of one-to-one functions from 
into 
.
Recall that 
is the number of colors seen 
times in draws from urn-
and 
times in draws from urn-
, giving that 
.
Fix 
and suppose that 
is of a color that contributes to 
, for some 
Removing 
from the data decrements 
and increments 
by one unit. Proceeding similarly as in the case for 
, if 
is used to denote the 
-statistics when observation 
is removed from sample-
, then
where 
and 
are as defined in (10) and (11). Noting that for each 
there are 
draws from urn-
that contribute to 
, the form in (9) follows.
In what follows, we specialize the coefficients in (23) and (24) to the kernel function of dissimilarity, 
. From now on, for each 
and 
, define
(30)
(31)
Above it is understood that the sigma-field generated by 
when 
is 
; in particular, 
, for all 
.
The following asymptotic properties of 
are useful in the remaining proofs.
Lemma 14 Assume that conditions (a)-(c) are satisfied and define 
. It follows that 
and 
. Furthermore
(32)
(33)
(34)
Proof. Observe that conditions (a)–(b) imply that 
. In addition, condition (b) implies that 
, whereas condition (c) implies that 
.
Next, consider the set
i.e. 
is the set of rarest colors in urn-
which are also in urn-
. Also note that
(35)As an intermediate step before showing (32), we prove that
(36)
For this, first observe that
Hence
from which (36) now easily follows.
To show (32) note that (36) implies
which establishes (32).
Now note that
which establishes (33).
Next we show (34), which we note gives more precise information than (27). Consider the random variable 
defined as the smallest 
such that 
. We may bound the probability of 
being large by 
, where 
is finite because of condition (a). On the other hand, note that
Define 
and observe that, over the event 
, 
. Since 
, we obtain that
The identity in equation (34) is now a direct consequence of (35).
Our next goal is to show Theorems 4 and 5. To do so we rely on the method of projection by Grams and Serfling [17]. This approach approximates 
by the random variable
The projection is the best approximation in terms of mean squared error to 
that is a linear combination of individual functions of each datapoint.
Under the stated conditions, 
is the sum of two independent sums of non-degenerate i.i.d. random variables and therefore satisfies the hypotheses of the classical central limit theorem. The variance of the projection is easier to analyze and estimate than the 
-statistic directly, which is relevant in establishing consistency for the jackknife estimation of variance.
Let
be the remainder of 
that is not accounted for by its projection. When 
is small relative to 
, 
is mostly explained by 
in relative terms.
The next lemma summarizes results about the asymptotic properties of 
, particularly with relation to the scale of 
as given by its variance.
Lemma 15 We have that
(37)
(38)
Under assumptions (a)-(c), for a fixed 
, we have that
(39)
Furthermore, under assumptions (a)–(d) we have that
(40)
(41)for all 
.
Proof. A direct calculation from the form given in (16) gives that
(42)
(43)
As 
, (37) follows.
To show (38), first observe that
(44)
Next, using the definition of the projection, we obtain that
from which (38) follows, due to the identity in (44). Note that the last identity implies that 
and 
are uncorrelated.
Before continuing, we note that (41) is a direct consequence of (38), (40) and Chebyshev's inequality [22]. To complete the proof of the lemma all reduces therefore to show (41) under conditions (a)–(d). Indeed, if 
and we let 
then due to the identities in (22) and (37) and Lemma 12, we obtain under (a)–(c) that
uniformly for all 
, as 
. Since 
for all 
, we have thus shown (39). Furthermore, note that 
for all 
; in particular, due to (33) and conditions (a)–(d), we can assert that 
. Since 
, the above identity together with the one in (37) let us conclude that
as 
. Because of condition (d), the big-O term above tends to 
. As a result:
(45)On the other hand, (38) implies that 
. Hence, using (19) and (25) to bound from above the variance of the U-statistic, we obtain:
as 
, where for the last identity we have used (32) and (34). Since 
, it follows from the above identity that
In particular, if the base-
in the logarithm is selected to satisfy that 
, then
(46)
The identities in equation (45) and (46) show (41), which completes the proof of the lemma.
Proof of Theorem 5
For a fixed 
, note that 
is the sum of two independent sums of non-degenerate i.i.d. random variables and thus,
is asymptotically a standard Normal random variable as 
by the classical Central Limit Theorem. We would like to show however that this convergence also applies if we let 
vary with 
and 
. We do so using the Berry-Esseen inequality [23]. Motivated by this we define the random variables
Note that 
, and that
We need to show that
(47)uniformly for 
, as 
.
Note that from (42) and (43),
Let
It follows from (37) that
But note that 
. Since, according to Lemma 14, 
decreases exponentially fast, we obtain
uniformly for all 
, as 
. On the other hand, 
. Furthermore, (33) implies that 
. Since 
, for some finite constant 
we find that
which shows (47).
The above establishes convergence in distribution of 
to a standard normal random variable uniformly for 
, as 
. The end of the proof is an adaptation of the proof of Slutsky's Theorem [21]. Indeed, note that
(48)
From this identity, it follows for any fixed 
that
The first term on the right-hand side of the above inequality can be made as close to 
as wanted, uniformly for 
, as 
, because of (40). On the other hand, the second term tends to 
uniformly for 
because of (41). Letting , shows that
Similarly, using (48), we have:
and a similar argument as before shows now that
which completes the proof of the theorem.
We finally show Theorem 4, for which we first show the following result.
Lemma 16 Let 
be the set of one-to-one functions from 
into 
. Consider the kernel 
, and define
(49)
(50)
(51)
(52)Then, for each 
and 
,
(53)
(54)Proof. Fix 
. We first use a result by Sen [24] to show that, for each 
:
(55)
(56)
(57)
(58)
in an almost sure sense. Indeed, assume without loss of generality that 
. As the kernel functions found in (49) and (51) are bounded, the hypotheses of Theorem 1 in [24] are satisfied, from which (55) and (57) are immediate. Similarly, because 
and 
are discrete random variables, (56) and (58) also follow from [24].
Define
(59)
(60)and observe that
Furthermore, due to (55)–(58), we have that
(61)
(62)
But note that, for 
, 
and 
are independent and hence uncorrelated. Similarly, the random variables 
and 
are independent. Since 
and 
, it follows from (61) and (62), and the Bounded Convergence Theorem [22] that
(63)
(64)as 
.
Finally, by (30) and (31) it follows that
In particular, again by the Bounded Convergence Theorem, we have that 
and 
. Since
the lemma is now a direct consequence of (63) and (64), and Theorem 1.5.4 of Durrett [22].
Proof of Theorem 4
Fix 
. Using (16) we have that
It follows by (28) and (29) that
(65)
(66)where 
and 
are as in (59) and (60), respectively. Furthermore, observe that
In particular, due to (37), we obtain that
By Lemma 16, 
converges in probability to 
, while similarly 
converges in probability to 
; in particular, the first two terms on the right-hand side of the inequality converge to 
in probability. Since 
and 
, the same can be said about the last two terms of the inequality. Consequently, 
converges to 
in probability, as 
. As stated in (39), however, conditions (a)–(c) imply that 
and 
are asymptotically equivalent as 
, from which the theorem follows.
