## Nonparametric inference under biased sampling from a finite population.(English)Zbl 0767.62032

Let $${\mathcal U}=\{U_ 1,U_ 2,\dots,U_ N\}$$ denote a finite population of $$N$$ units and let $$Y_ j$$ be a characteristic associated with $$U_ j$$, $$j=1,\dots,N$$. A sample of size $$n$$ is selected successively without replacement and with probability proportional to some measure of size $$\{W_ 1,\dots,W_ N\}$$, where $$W_ j=W(Y_ j)$$, is a positive function of the unknown population characteristics. Specifically, let $$z_ 1,\dots,z_ k$$ denote the distinct values in the finite population with multiplicities $$N_ 1,\dots,N_ k$$. It is of interest to estimate the parameters $$N_ 1,\dots,N_ k$$ from the given sample.
The paper deals with maximum likelihood estimation. Consistency, asymptotic distribution and efficiency of the MLE are shown. The asymptotic and finite-sample behavior is compared through a limited simulation study. The results are applied to the analysis of oil and gas discovery data from the North Sea basin. For the asymptotic results, it is assumed that the number of distinct values in the population is fixed as $$N\to\infty$$. It is also assumed that the sampling proportion $$f_ N=n/N$$ satisfies $$0<\lim_{N\to\infty} f_ N<1$$.
Reviewer: K.Alam (Clemson)

### MSC:

 62G05 Nonparametric estimation 62D05 Sampling theory, sample surveys
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