A Kinder Egg Problem

Question

My friend and I recently saw our old passions for Kinder Surprise toys reignited with a new animal toy line which resembled the old toys we were missing. To our dismay, however, this series did not include a "check-list" of toys to be collected. Hence arose the natural problem of estimating the number of distinct toys in the series, and when to stop buying. (We have an embarrassing amount of data.)

As neither of us has much experience with this type of problem, we do not know any standard approaches. That being said, here is what we cooked up: Suppose that there are $N$ distinct toys in the series. We can calculate the probability of observing $v_1$ toys repeated once ("singles"), $v_2$ toys repeated twice ("doubles"), etc. If this were indeed our data, we assume the most likely thing happened, and maximize the calculated probability as a function of $N$. I recognize that there are problems with this approach. Could anyone suggest alternatives?

The data is sourced only from purchases in the supermarket. For fun, here is some data, presented as a quintuple with the number of singles, doubles, etc.: $(5,2,3,2,0)$ was a week ago, $(5,2,3,1,1)$ and $(5,2,2,2,1)$ are more recent.

This question is similar, except here we assume that the value labelled $S$ there is infinite.

Answer 1

You could potentially approach this as an inference from an occupancy problem, depending on the sampling method. For simplicity, let's assume that there are $N$ types of toys and your sampling method gives you an IID sample of $n$ toys that are equiprobable over the different types. Let $K_n$ denote the number of disinct toys in your sample. Given the values $N$ and $n$ this =random variable follows the classical occupancy distribution (see e.g., O'Neill 2019), with probability mass function:

$$\mathbb{P}(K_n = k) = \text{Occ}(k|n,N) = \frac{(N)_k \cdot S(n,k)}{N^n} \quad \quad \quad \text{for all } 1 \leqslant k \leqslant \min(n,N).$$

Observing the occupancy value $K_n=k$ gives you the log-likelihood function:

$$\ell_k(N) = \sum_{i=1}^k \log (N-i+1) - n \log(N) \quad \quad \quad \quad \quad \text{for all } N \geqslant k.$$

You can find the details for computing the MLE and MoM estimators for $N$ in this related question. If you can specify the number of toys you've collected and the number of distinct toys you got I can complete the estimation. (I'm impressed that you have an embarrassing amount of data on Kinder egg toys; if you'd be interested in sharing that data, it would make a nice example for inferences in occupancy problems.)