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I first want to thank Professors Moulinath Banerjee (MB hereafter) and Jon Wellner (JW hereafter) for their very illuminating and helpful discussions on my review paper. I am particularly appreciative of the very up-to-date material provided on research topics that I did not discuss in the review as well as their intriguing comments on graduate education. I will now comment briefly on these two items and then conclude with some final words.
Both discussants described many of the very interesting recent developments in shape-constrained inference, and MB drew a connection between inferential challenges in shape-constrained inference and recent developments in non-standard pivotal limiting distributions of likelihood ratio tests (LRTs). Both discussants also provided some insights into open research questions regarding mixed-rate asymptotics and some important recent developments in this area, including Radchenko’s paper.
MB also described some interesting possible extensions of the pivotal LRTs to richer classes of semiparametric survival models and other complex censoring settings. He also raised the interesting challenge of conducting inference and assessing optimality for parameters that are not root-n consistent. The boundary estimation problems also discussed by BM are particularly intriguing and are of personal interest to me. This area is an extension of the very difficult change-point estimation problem and can involve complicated mixed-rate asymptotics and other challenging analytical and inferential issues. The area also has the potential to extend machine learning and related techniques to address important biomedical classification problems.
JW also points out the very important St. Flour Lecture of Aad van der Vaart’s and the accompanying description of open problems in semiparametrics. The work JW mentions on the interplay between classical sampling theory (including finite sampling), empirical process theory, and semi-parametric models with two-phase designs is also important practically and involves interesting, non-standard inferential theory. I also appreciate the additional references on results for empirical processes involving dependent data. The development of efficient computational algorithms for semiparametric estimation JW mentions is also important and would require a non-traditional, multidisciplinary approach.
It is difficult to keep track of all of the important developments in semiparametric inference, and I appreciate the additional insights in this direction provided by MB and JW.
Both discussants comment on the challenges of including a modern treatment of empirical processes and semiparametric inference in graduate education. The approach taken at the “other UW” by JW has been very successful in generating a number of very strong empirical process researchers over the years, but a drawback is that a relatively small percentage of graduate students in statistics and biostatistics at UW are exposed to the area. I personally feel, and I am fairly certain that both MB and JW would agree, that a larger percentage of graduate students should be exposed to general concepts in empirical processes and semiparametric inference but perhaps without technical details, while a smaller subset of students should be exposed to the more technical details required for focused research. The challenge, of course, is striking the right balance. It may be valuable for representatives from several departments and universities with common interests in this issue to meet, compare notes, and perhaps publish recommendations.
I also agree with MB’s worries about the apparent decline in emphasis on theory in statistics, biostatistics and econometrics education. The truth is that we need both theory and applications, and both must thrive in statistics education and research, or we will become limited as a discipline in our potential to impact modern scientific developments. One answer is to strengthen both theory and applications training so that a larger number of our graduate students can work on the interface and feel stronger connections to both of these crucially important facets of our discipline.
In conclusion, semiparametric inference is an intriguing and dynamic area of statistical research that has achieved a healthy level of maturity. As statistical scientists, we have a responsibility to do more in terms of ensuring that new developments in estimation and inference become available and usable by practitioners and that we train statisticians who are properly prepared to have a meaningful impact on future developments in science.