Bulletin Editor Anirban DasGupta writes:

The problem asked was to settle the possibility of consistent estimation with incomplete data in three examples, and to provide a concrete one when consistent estimation is possible.

Intuitively, in case (a), you can only infer about $|\mu |$, but not the sign of $\mu$. Denote $Y_i = |X_i|$; the distribution of each $Y_i$ and hence the joint distribution of $(Y_1, \ldots , Y_n)$ depends only on $|\mu |$. Suppose $c(Y_1,\ldots , Y_n)$ is a consistent estimate (tor) of $\mu$. Fix $\mu > 0$. Then, $P_{\mu }(c(Y_1,\ldots , Y_n) < \mu /2) = P_{-\mu }(c(Y_1,\ldots , Y_n) < \mu /2) \to 1$, as $n \to \infty$, which would contradict $c(Y_1,\ldots , Y_n) \stackrel {P} {\Longrightarrow } \mu$ under every $\mu$.

In case (b), consistent estimation of $\lambda$ is possible. This is because the family of Poisson distributions is strictly MLR in $X$ and hence, for any $k$, $\bar{F}_k(\lambda ) = P_\lambda (X > k)$ is strictly increasing in $\lambda$; it is also continuous. Denote $Y_i = I_{X_i > k}$. Then, $\bar{Y}$ is a consistent estimate of $\bar{F}_k(\lambda )$, and by the continuous mapping theorem, $\bar{F}_k^{-1}(\bar{Y})$ is a consistent estimate of $\lambda$; the estimate may be defined arbitrarily (for example, as 1), if $\bar{Y} = 0$ or $n$.

Finally in case (c), perhaps a little surprisingly, consistent estimation of the exponential mean $\lambda$ is possible as well. By a direct calculation, the fractional part $Y = \{X\}$ has the density $g(y|\lambda ) = \frac{1}{\lambda (1-e^{-1/\lambda })}\, e^{-y/\lambda }, 0 < y < 1$. This is a regular one parameter Exponential family and therefore the natural parameter $-\frac{1}{\lambda }$ is consistently estimable, and so, $\lambda$ is also consistently estimable. For example, a concrete consistent estimate can be constructed by estimating $E_\lambda (Y)$, necessarily a strictly increasing function of the natural parameter, by $\bar{Y}$, and then by using the inverse function.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.997795581817627, "perplexity": 154.44131036583832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00633.warc.gz"}