In this issue, we look at the consequences of having only incomplete data. For example, suppose a random variable $X$ has a normal distribution with mean $\mu $ and variance $\sigma ^2$, and both parameters need to be estimated. With usual data, which we call complete data, namely iid copies $X_1, X_2, \cdots , X_n$ of $X$, we can estimate $\mu $ and $\sigma ^2$ easily; $\bar{X}$ and $s^2$ are consistent and fully efficient. But, if we only have the maximum and the minimum of the $X_i$’s, then although we can still estimate $\mu $ and $\sigma ^2$ consistently, we can no longer estimate them efficiently. This is the price we pay for only having incomplete data. In some cases, data may be so incomplete that even consistent estimation of a parameter may be impossible.

Here is the exact problem of this issue:

For each of the following cases, either prove that consistent estimation of the indicated parameter is not possible, or demonstrate a concrete consistent estimate of the indicated parameter:

(a) $X_i \stackrel {{\it iid }} {\sim } N(\mu , 1), i = 1, 2, \cdots , n, -\infty < \mu < \infty $, and we get to observe only $Y_i, i = 1, 2, \cdots , n$, where $Y_i = |X_i|$; we want to estimate $\mu $; (b) $X_i \stackrel {{\it iid }} {\sim } \mbox{Poisson}(\lambda ), i = 1, 2, \cdots , n, 0 < \lambda < \infty $, and we get to observe only whether each$X_i$ is larger than $k$ or $\leq k$ for some fixed specified positive integer $k$; we want to estimate $\lambda $; (c) $X_i, i = 1, 2, \cdots , n$ are iid exponential with mean $\lambda , 0 < \lambda < \infty $, and we get to observe only the fractional parts of the $X_i$’s; we want to estimate $\lambda $. The solution to the previous Puzzle is here.

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