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Asymptotic Expansion of Risk for a Regression Model with respect to $\alpha$-Divergence with an Application to the Sample Size Problem -- Complete Version

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Yo Sheena

Statistics Theory · Submitted 12 Mar 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Especially comparing the value of the (approximated) risk with that of a binomial distribution, we can give a certain standard for the sample size required to estimate the regression model.

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arXiv ID1703.10107 (version 2)
Pith IDK5E23627, Pith's permanent record of this paper and its review
CitationYo Sheena. "Asymptotic Expansion of Risk for a Regression Model with respect to $\alpha$-Divergence with an Application to the Sample Size Problem -- Complete Version." arXiv:1703.10107 (2017). https://pith.science/paper/K5E23627
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,064 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.19 · human participants 0.03 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.56 · compares with prior results 0.61 · mentions code or data availability 0.06.

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arXiv listing abstract v2. Results and the main claim are underlined; .

For a regression model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and $\chi^2$ divergence. The asymptotic expansion of the risk with respect to the sample size $n$ is given up to the order $n^{-2}$. We are interested in how the risk convergence speed (to zero) is affected by the error term distributions of the regression model and the magnitude of the joint moments of the standardized explanatory variables. Besides the general result (which is given by Mathematica program), we consider three concrete error term distributions; a normal distribution, a t-distribution and a skew-normal distribution. We use the (approximated) risk of m.l.e. as a measure of the difficulty of estimation for the regression model. Especially comparing the value of the (approximated) risk with that of a binomial distribution, we can give a certain standard for the sample size required to estimate the regression model.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 For a regression model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and $\chi^2$ divergence. contribution 0.56 0.62 0.06 asserted 0.97
s1 The asymptotic expansion of the risk with respect to the sample size $n$ is given up to the order $n^{-2}$. result 0.73 0.93 0.26 asserted 1.00
s2 We are interested in how the risk convergence speed (to zero) is affected by the error term distributions of the regression model and the magnitude of the joint moments of the standardized explanatory variables. contribution 0.78 0.37 0.13 unclear 0.53
s3 Besides the general result (which is given by Mathematica program), we consider three concrete error term distributions; a normal distribution, a t-distribution and a skew-normal distribution. method 0.74 0.90 0.05 asserted 0.98
s4 We use the (approximated) risk of m.l.e. as a measure of the difficulty of estimation for the regression model. method 0.61 0.81 0.08 asserted 1.00
s5 Especially comparing the value of the (approximated) risk with that of a binomial distribution, we can give a certain standard for the sample size required to estimate the regression model. interpretation 0.49 0.87 0.37 asserted 0.53

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

For a regression model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback-Leibler divergence, Hellinger distance and $\chi^2$ divergence. The asymptotic expansion of the risk with respect to the sample size $n$ is given up to the order $n^{-2}$. We are interested in how the risk convergence speed (to zero) is affected by the error term distributions of the regression model and the magnitude of the joint moments of the standardized explanatory variables. Besides the general result (which is given by Mathematica program), we consider three concrete error term distributions; a normal distribution, a t-distribution and a skew-normal distribution. We use the (approximated) risk of m.l.e. as a measure of the difficulty of estimation for the regression model. Especially comparing the value of the (approximated) risk with that of a binomial distribution, we can give a certain standard for the sample size required to estimate the regression model.

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