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Let's say I am drawing a random sample of n values from a population with N values, and in the population, U of them are unique.

If I observe u unique values in the sample, why can't I use that to estimate U? Is it a biased estimator, and if so, why?

I read this question methods for estimating unique no of values in a population but fundamentally I want to understand why it's an issue at all.

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It's very plainly biased because:

(a) it's quite easy for the number of distinct values observed in the sample to be lower than the corresponding number in the population (it has positive probability), but

(b) it's impossible for the number of distinct values in the sample to exceed the corresponding number in the population

Consequently the expected number in the sample is below the value in the population; there's nothing to compensate for the proportion of times it's too low.

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