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Mathematics of Operational Research Contents Table of Contents i ...
www.statslab.cam.ac.uk/~rrw1/mor/morweber.pdf15 Mar 2016: Mathematics of Operational Research. Contents. Table of Contents i. Schedules v. 1 Lagrangian Methods 11.1 Lagrangian methods. 11.2 The Lagrange dual. 31.3 Supporting hyperplanes. 3. 2 Convex and Linear Optimization 62.1 Convexity and strong duality. -
Mathematics of Operational Research Example Sheet 2 R. Weber ...
www.statslab.cam.ac.uk/~rrw1/mor/examples2.pdf2 Dec 2015: maximizesi,tj. { i. si j. tj : αiβj si tj 0, i, j}. -
8 8 1/2 1/2 1/2 1/2 1 1/2 1/4 ...
www.statslab.cam.ac.uk/~rrw1/markov/slides.pdf14 Nov 2011: 25. . Theorem 5.8. Suppose P is irreducible and recurrent.Then for all j I we have P(Tj < ) = 1. -
21 Paper 4, Section I 9H Markov ChainsLet X0, ...
www.statslab.cam.ac.uk/~rrw1/markov/MarkovChainTriposQuestions.pdf17 Sep 2015: Assume p > q. Let Tj = inf{n > 1 : Xn = j} if this is finite, and Tj = otherwise. ... Let Ti = inf{n > 1 : Xn = i}. For each i 6= j letqij = P(Tj < Ti | X0 = i) and mij = E(Tj | X0 = i). -
Markov Chains These notes contain material prepared by colleagues ...
www.statslab.cam.ac.uk/~rrw1/markov/M.pdf22 May 2013: Pj(Tj < ) Pj(Hi < )Pi(Hj < ) = 1. 5.5 Relation with closed classes. ... P(X0 = i)Pi(Tj < ). so it suffices to show that Pi(Tj < ) = 1 for all i I. -
Asymptotics and optimal bandwidth selection for highest density…
www.statslab.cam.ac.uk/~rjs57/Samworth10.pdf21 Apr 2010: tn, tj tn], where D4 = 12 μ2(K)f ′′(xj ) D1;. • ... tn = o(n1/6).Then. P(f̂h(x. tj ) < f̂h,τ. ) ( tf ′(xj ) D4n1/2h5/2{R(K)fτ 2D3,j D2}1/2). -
CHOICE OF NEIGHBOUR ORDER FOR NEAREST-NEIGHBOUR CLASSIFICATION RULE…
www.statslab.cam.ac.uk/~rjs57/HPSLV.pdf28 Aug 2007: 2/d , then, with v defined by. (4.8), and tj = νj1 {ν/k2(ν)}. -
Maximum likelihood estimation of a multidimensional log-concave…
www.statslab.cam.ac.uk/~rjs57/CSSFinalLV.pdf24 Mar 2010: Maximum likelihood estimation of a multidimensional log-concave density. Madeleine Cule and Richard Samworth†University of Cambridge, UK. and Michael StewartUniversity of Sydney, Australia. Summary. Let X1,. , Xn be independent and identically -
Statistical modellingRajen D. Shah r.shah@statslab.cam.ac.uk Course…
www.statslab.cam.ac.uk/~rds37/teaching/statistical_modelling/notes.pdf14 Feb 2019: and (Xj )TXj = X. Tj (I Pj)Xj = ‖(I Pj)Xj‖2. -
Modern Statistical MethodsRajen D. Shah r.shah@statslab.cam.ac.uk…
www.statslab.cam.ac.uk/~rds37/teaching/modern_stat_methods/notes2.pdf15 Jan 2023: Modern Statistical MethodsRajen D. Shah r.shah@statslab.cam.ac.uk. Course webpage:http://www.statslab.cam.ac.uk/rds37/modern_stat_methods.html. In this course we will study a selection of important modern statistical methods. Thisselection is -
MATHEMATICS OF MACHINE LEARNING Part IIExample Sheet 2 (of ...
www.statslab.cam.ac.uk/~rds37/teaching/machine_learning/Qu2.pdf20 Feb 2024: F̂(t1,. , tp) =1. n. ni=1. 1A(Wi). where A =pj=1(, tj]. ... n. [Hint: Consider H := {1A : A =pj=1(, tj], tj R}, and H := {h : h H}, and. -
Journal of Machine Learning Research ? (????) ?-?? Submitted ...
www.statslab.cam.ac.uk/~rds37/papers/shah16.pdf9 Jun 2016: Journal of Machine Learning Research? (?)? -? Submitted 11/13; Revised 3/16; Published? /? Modelling Interactions in High-dimensional Data withBacktracking. Rajen D. Shah r.shah@statslab.cam.ac.ukStatistical Laboratory. University of Cambridge. -
Statistical modellingLecturer: Alberto J. Coca…
www.statslab.cam.ac.uk/~qz280/teaching/modelling-2022/notes_3Dec19.pdf29 Apr 2024: and (Xj )TXj = X. Tj (I Pj)Xj = ‖(I Pj)Xj‖2. -
Mendelian Randomization: Old and New Insights Qingyuan Zhao…
www.statslab.cam.ac.uk/~qz280/talk/penn-biostat-2021/slides.pdf29 Apr 2024: 4): Q-Q plot of standardized residual:. tj (β̂) =Γ̂j β̂γ̂j. 1 β̂2. ... Robust adjusted profile score (RAPS). I Define standardized residual: tj (β,τ2) =. Γ̂j βγ̂j1 β2 τ2. I For some robust loss ρ (let ψ = ρ′), the RAPS equations -
Leverage Mendelian Randomization to Learn MeaningfulRepresentations…
www.statslab.cam.ac.uk/~qz280/talk/lmrl-2021/slides.pdf29 Apr 2024: ψ(ρ)1 (β,τ. 2) =. pj=1. ( β. tj)ψ(tj ),. ψ(ρ)2 (β,τ. ... 2) =. pj=1. tj ψ(tj ) E[Tψ(T )], for T N(0, 1). -
The Randomization Principle in Causal Inference: A Modern Look ...
www.statslab.cam.ac.uk/~qz280/talk/imperial-2022/slides.pdf29 Apr 2024: Consider any function g : Z [M] anda collection of test statistics: Tj : Z W R, j [M].The p-value of the CRT is given by. -
Mendelian Randomization: Old and New Insights Qingyuan Zhao…
www.statslab.cam.ac.uk/~qz280/talk/epfl-2021/slides.pdf29 Apr 2024: 4): Q-Q plot of standardized residual:. tj (β̂) =Γ̂j β̂γ̂j. 1 β̂2. ... Robust adjusted profile score (RAPS). I Define standardized residual: tj (β,τ2) =. Γ̂j βγ̂j1 β2 τ2. I For some robust loss ρ (let ψ = ρ′), the RAPS equations -
Simultaneous Hypothesis Testing using Internal Negative Controls…
www.statslab.cam.ac.uk/~qz280/publication/ranc/slides.pdf29 Apr 2024: 2 (Ti)iI (Tj)jInc;3 (Tj)jInc is mutually independent;. 4 (Ti)iI is PRDS on (Ti)iI0;. ... R(t) 1, 0 t 1. Further define. Vnc(t) :=jInc. 1{Tj t}, V̄nc(t) :=n (Vnc(t) 2). -
Mendelian randomization: From genetic association to epidemiological…
www.statslab.cam.ac.uk/~qz280/publication/mr-partially-bayes/slides.pdf29 Apr 2024: Robust adjusted profile score (RAPS). I Define standardized residual: tj (β,τ2) =. Γ̂j βγ̂j(σ2Yj τ. 2) β2σ2Xj. I For some robust loss function ρ, the RAPS are. ... ψ(ρ)1 (β,τ. 2) =. pj=1. ρ′(tj ). βtj,. ψ(ρ)2 (β,τ. 2) =. pj=1. -
Confounder Adjustment in Multiple Hypothesis Testing
www.statslab.cam.ac.uk/~qz280/publication/cate/slides.pdf29 Apr 2024: p. tj =. nβ̂j. σ̂j. 1 ‖α̂‖2, Pj = 2(1 Φ(|tj|)).
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