Asymptotics for minimisers of convex processes

By Nils Lid Hjort, D. Pollard. 2011

244 citations recorded by the source corpus.

By means of two simple convexity arguments we are able to develop a gen- eral method for proving consistency and asymptotic normality of estimators that are deflned by minimisation of convex criterion functions. This method is then applied to a fair range of difierent statistical estimation problems, including Cox regression, logistic and Poisson regression, least absolute deviation regression outside model conditions, and pseudo-likelihood estimation for Markov chains. Our paper has two aims. The flrst is to exposit the method itself, which in many cases, under reasonable regularity conditions, leads to new proofs that are simpler than the traditional proofs. Our second aim is to exploit the method to its limits for logistic regression and Cox regression, where we seek asymptotic results under as weak regularity conditions as possible. For Cox regression in particular we are able to weaken previously published regularity conditions substantially.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3b98e8c2b30822db7f0d72eb3d2f33c7cf88f427

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In Statistics and Probability

Citations in the corpus

Unconditional Quantile Regressions (2007, methodology): “The quantile regression estimates reported in the corresponding columns show, as in 26We use the same June 1997 Detailed Natality Data published by the National Center for Health Statistics as used by Koenker and Hallock (2001). 27See Card, Lemieux, and Riddell (2004) for a detailed discussion and s”
WEIGHTED LEAST ABSOLUTE DEVIATIONS ESTIMATION FOR ARMA MODELS WITH INFINITE VARIANCE (2007, background): “therefore the asymptotic normality of LADE may be readily derived from the convex lemma ( Hjort and Pollard 1993 ).”
Quantile and quantile-function estimations under density ratio model (2013, background): “See Hjort and Pollard (1993) for this justification.”
Goodness of Fit via Non‐parametric Likelihood Ratios (2004, background): “that the maximiser of K(b) is b = a. The maximiser ba of Kn goes in probability to the maximiser a of K, see convexity lemmas in Andersen & Gill (1982, Appendix) and Hjort & Pollard (1994) .”
Cluster-Robust Bootstrap Inference in Quantile Regression Models (2014, unknown): “qn,1−α(Ω̂ ∗ n,T ) = arg minq∈R E ∗ hn(q) and Eh has a unique minimum, Lemma 2 of Hjort and Pollard (1993) gives qn,1−α(Ω̂ ∗ n,T )→ q1−α(Ω∞,T ).”
Unconditional Quantile Treatment Effects Under Endogeneity (2007, unknown): “D Additional lemmas used for the proofs Here we restate the rst two lemmas of Hjort and Pollard (1993) for the convenience of the reader. Lemma 1 of Hjort and Pollard (1993): From pointwise to uniform.”
Quantile-adaptive model-free variable screening for high-dimensional heterogeneous data (2013, methodology): “This is a direct application of Lemma 2 of Hjort and Pollard (1993) making use of the convexity of the objective function.”
Frequentist Model Average Estimators (2003, methodology): “be given for log-concave models using convexity arguments of Hjort and Pollard (1994) .”
UNIFORM BIAS STUDY AND BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATORS OF THE CONDITIONAL QUANTILE FUNCTION (2011, background): “Techniques to study En(α;h, x) for a fixed argument α, h and x are given in Hjort and Pollard (1993).”
Focused Information Criteria and Model Averaging for the Cox Hazard Regression Model (2006, methodology): “See also Longford (2005). When a model selection scheme like the AIC or FIC is followed to produce an estimator, it is important to realize that the real variance involved is larger than it would be had the selected model been given in advance.”
Asymptotic analysis of high-dimensional LAD regression with Lasso (2016, methodology): “From the convex minimization theorem in Hjort and Pollard (1993) we get”
Laplace Periodogram for Time Series Analysis (2008, background): “This result can also be justified by citing the Basic Corollary of Hjort and Pollard (1993). Note that Zn(δ ) can be reparameterized as a function of β :=β 0 +δ n−1/2 such that Zn(δ ) = 2 ∑{|yt −xT t β |−|ut |}.”
Generalized bootstrap for estimators of minimizers of convex functions (2003, methodology): “A diHerent approach to treating stochastic convex functions is illustrated in Knight (1998a, b), Geyer (1996), Hjort and Pollard (1993).”
Quantile Regression Estimator for GARCH Models (2013, background): “Hence, (9) is validated by the results in Geyer (1996) and Hjort & Pollard (1993).”
Optimal subsampling for quantile regression in big data (2020, unknown): “30) Since Z̆∗ n(λ) is convex, from the corollary in page 2 of Hjort and Pollard (2011), its minimizer, √ n(β̆ − βt), satisfies that √ n(β̆opt − βt) = −D−1 N W̆ ∗ n + oP (1), where β̆opt = β̆Lopt for π Lopt i and β̆opt = β̆Aopt for π Aopt i .”