Suppose $\mathcal B=\{1,2,\ldots,b\}$ is the set of all possible coupons, with $\mathbf p = ( p_1,p_2,\ldots,p_b)$ assigning the probability of occurrence for all coupons in $\mathcal B$.
The "traditional coupon collecting problem" can be summarized as follows: Assume we have a variable numbers $n$ of draws from a non-depleting set of coupons (urn with coupons distributed as $\mathbf p$). We can count the drawn numbers of coupons according the different kinds of coupons in $\mathcal B$ with a occupancy vector $(X_1,X_2,..X_b)$, with $X_i$ is the number of coupons from type $i$, and $\sum_i X_i=n$. A prominent problem investigate in $\mathbb E[n]$, i.e. expected number of needed to fulfill a constraint on minimal quotas $X_i \geq q_i$, with predefined quotas $\mathbf q = ( q_1,q_2,...,q_b)$. Investigated, e.g. in [1,2].
A kind twisted problem I am interested in is the following: Assume we have a fixed number of $N$ draws from the urn of coupons with non-uniform distribution $\mathbf p$, can estimate or bound the number of coupon-types, which we will not observe in our $N$ draws? To be more formal, let $Y_i$ be an indicator variable with $Y_i=1$ for $X_i=0$ otherwise $Y_i=0$, and we are interested in $\mathbb E[G]$, where $G=\sum_i Y_i$.Evidently, for uniform distribution the expected number of vacancies in the occupancy vector will be minimal.
In fact I am interested in bounding $\mathbb E[G]$ incorporating an information theoretic measure on $\mathbf p$, e.g. the entropy of the distribution $\mathbf p$, the total variation distance of $\mathbf p$ to uniform distribution or maybe some useful f-divergence. At least some bound dependend on maybe $\max_i p_i$ and $\min_i p_i$ would be helpful.
If you have any idea how to tackle my problem, I am glad to hear your advice or answer. Thanks in advance.
It is easy to calculate $\mathbb{E}[G]$ using linearity of expectation. We have $\mathbb{E}[Y_i] = \Pr[X_i=0] = (1-p_i)^N$. Therefore $$\mathbb{E}[G] = \sum_{i=1}^b (1-p_i)^N.$$ Whether you can bound it in terms of any other functions of $\mathbf{p}$ is up to you.