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diff --git a/jasa-2019-0653-R1.tex b/jasa-2019-0653-R1.tex new file mode 100644 index 0000000..8591e78 --- /dev/null +++ b/jasa-2019-0653-R1.tex @@ -0,0 +1,69 @@ +\documentclass[10pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[hmargin=1.2in, vmargin=1.2in]{geometry} +\usepackage{amsmath,amsfonts} + +\title{\large Review of \emph{Real-time Regression Analysis of Streaming Clustered Data with Possible Abnormal Data Batches}} +\date{} + +\begin{document} + +\maketitle + +This is an update on a previous review of the same paper after reading the +authors' revision. Overall I would like to thank the authors for taking my +comments and questions in serious considerations and improving the paper +accordingly. + +\paragraph{1.} +The main change at the technical level has been a clarification about the +regime with which the number of samples is taken to grow to infinity, with now +two distinct regime: one where the size of each batch is constant and the +number of batches grows to infinity, and one where the first batch's size grows +to infinity (with two sub-regimes depending on whether the subsequent batches +can also grow to infinity). + +The asymptotic analysis of the estimator in the first of these two regime was not +previously covered by the original proof, but in this revision the authors +added a separate analysis for this case while also clarifying the proof in the +other case. + +Thanks to this improvement, I have now reached a reasonable level confidence in +the correctness of the stated results and believe that the paper is technically +sound. + +\paragraph{} A minor suggestion to improve the argument given on line 17, page +43 in the appendix, which lacks rigor as currently written ($n$ hasn't been + defined and it seems to suggest that all batches have the same size, which + is not without loss of generality, it is also not clear in which sense the + approximation $\simeq$ needs to be understood). + + By definition one has $n_j = N_j - N_{j-1}$ hence, defining $N_0=0$: + \begin{align*} + \sum_{j=1}^{b-1} \frac{n_j}{\sqrt{N_j}} = \sum_{j=1}^{b-1} + \frac{N_j-N_{j-1}}{\sqrt{N_j}} + \leq + \sum_{j=1}^{b-1} + \int_{N_{j-1}}^{N_j} + \frac{dt}{\sqrt{t}} = \int_{0}^{N_j}\frac{dt}{\sqrt{t}} = 2\sqrt{N_j}\,, + \end{align*} + where the inequality holds since $t\mapsto 1/\sqrt{t}$ is a decreasing + function. + +\paragraph{2.} The authors also clarified the details of how the Newton-Raphson +method is used, in particular conditions guaranteeing convergence and the +convergence criterion used in the numerical experiments. + +While I agree that the numerical experiments clearly show that convergence of +the NR method does happen quickly in practice, I was not convinced by the +authors' explanation that there is no need to control the residual error in the +theoretical analysis, and in particular make sure that it does not accumulate +over the iterations of the recursive procedure. The authors claim that it is +the ``conventional practice in the statistical literature'', but my impression +is that nested procedure (where a subroutine, like NR here, is used in each +iteration) are becoming increasingly common for online estimation (following +a similar trend in the fields of stochastic optimization and machine learning) +and it is now standard to do and end-to-end analysis of the entire procedure, +including the error terms accrued at each iteration. +\end{document} |