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Suppose I have N smarties, each of which is one of C distinct colours.

Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that colour. Whilst the distribution of $c_i$ is unknown, we have the vague assumption that no $c_i$ will be very large or small, eg no one colour will make up more more than 99 or less than 1 percent of the total number of smarties.

If I take a sample of size n and determine that it contains $x_i$ smarties of each colour, how can I produce a good estimate the total number of classes?

Note -> I'm familiar with http://arxiv.org/pdf/0708.2153.pdf and http://www.jstor.org/stable/2290471 but was wondering if anyone had anything else to contibute. Also I'm concerned with the case where N is large but not infinite, which is fairly distinct from the first paper.

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So by reading

Haas, Peter J., and Lynne Stokes. "Estimating the number of classes in a finite population." 
 Journal of the American Statistical Association 93.444 (1998): 1475-1487.

I came up with

def random_sample(n,data):
    indices = np.random.choice(np.arange(0,len(data)),size=n,replace=False)
    # That space efficiency wins no awards.
    return data[indices]

def estimate_n_classes(sample,population_size):
    n = float(len(sample))
    dn = float(len(np.unique(sample))) 
    f = Counter([count for label, count in Counter(sample).iteritems()])
    q = n / population_size
    result = ((1 - ((1 - q) * f[1] / n))) ** (-1) * dn
    return result

This is an implementation of $D_{uj1}$ from the paper which was sufficient for my needs.

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