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How do I show the consistency of an estimator of a parameter, say $\mu$, that is not based on the sample size $n$ but a function of $n_{i}$'s where $\sum_{i=1}^{K}n_{i}=n$ ? Consider for example the estimator $\hat{\mu}=\frac{1}{K}\sum_{i=1}^{K}\frac{Y_{i}}{n_{i}}$ where $Y_{i}$'s are independent r.v.'s with mean $n_{i}\mu$ and variance $\sigma^{2}n_{i}$.

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  • $\begingroup$ Do $n_i\to\infty$? Or $K\to\infty$? What's the relationship between $n_i$'s? $\endgroup$ – Julius Vainora Dec 12 '18 at 0:39
  • $\begingroup$ $K$ is finite. We know $\sum_{i=1}^{K}n_{i}=n$ where $n_{i}>0$. How do I define a sequence of estimators based on $\hat{\mu}$ and $n$ to check its consistency ? The definition of consistency involves $n$. $\endgroup$ – Sayan Dec 12 '18 at 0:43

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