Jul 15, 2019

# Student Puzzle 25: deadline September 5, 2019

Here’s Anirban DasGupta’s latest puzzle. He says:

All of us were told as undergraduates, or perhaps Masters students, that an essential property of a point estimator is that it be consistent. And indeed, we usually or even always select estimators that are consistent. We are going to ask a provocative question in this month’s puzzle: is there consistency in the real world? The problem posed asks you to show that, in fact, what we believe to be consistent, when computed, is not.

Where does the unavoidable inconsistency in the real world come from? Although the human body is an amazing machine, it is not a perfect one. The practical inconsistency comes from human limitations in the precision of a measurement. This inconsistency is incurable and a large sample won’t fix it. Here is the exact problem of this month.

Suppose we have an iid sequence of exponential random variables $X_1, X_2, \cdots$ with mean $\lambda$. Suppose the values are rounded off using an often-used rule: an observation $X$ is written down as zero if $X \leq 0.005$, as $0.01$ if $X$ is between $0.005$ and $0.015$, as $2.00$ if it is between $1.995$ and $2.005$, and so on.
Call this recorded value $Y$ and consider the mean $\bar{Y}$ for a sample of size $n$.

(a) Prove that $\bar{Y}$ is not a consistent estimator of $\lambda$.

(b) Find in closed form a parametric function $h(\lambda )$ such that $\bar{Y}$ is a consistent estimator of $h(\lambda )$.

(c) Derive an asymptotic expansion to one, or if you can, two terms, for $h(\lambda ) – \lambda$ as $\lambda \to \infty$.

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