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A coupling with uniform observations supplies valid dominance critical values; ties and discrete ranks require separate calculations.
Read postConditional stochastic dominance can be tested using only a fixed number of observations nearest each target covariate value.
Read postPlug-in estimation of the population sensitivity bounds looks straightforward until an estimated propensity score changes which branch of a minimum or maximum is active.
Read postMissing network links change both the measured number of neighbors and their measured treatment exposure.
Read postConditional independence sets the probability of treatment equal to the propensity score even after conditioning on a potential outcome.
Read postNetwork formation models ask why links are present before using those links to explain outcomes.
Read postTax reforms create variation in net-of-tax rates, but an individual's realized rate also depends on earnings.
Read postRandom draws create a tractable reference fluctuation, while the original sample mean determines how many repetitions inference can tolerate.
Read postA Strict Local Martingale and the Meaning of Boundedness. Notes on stochastic processes.
Read postRandomized subsampling trades some local power for inference without an estimated dependence ordering.
Read postIn a just-identified IV regression, the usual t statistic depends on the strength of the first stage even under the null.
Read postEstimated outcome regressions can improve network-policy evaluation, and estimated exposure probabilities are needed outside a fully known experiment.
Read postNovelty detection starts with labeled examples of ordinary observations and an unlabeled sample containing both ordinary and novel observations.
Read postWhen the ordering of dependence is unknown, a block bootstrap may be impossible to implement: there is no reliable way to choose blocks that preserve the dependence.
Read postThe study of geometry extends to any discrete setting, where we do not necessarily (a priori) expect it.
Read postA randomized saturation design first assigns treatment fractions to clusters and then assigns individuals within clusters.
Read postA necessary inequality becomes a sharp testable restriction only after showing that every data distribution satisfying it can arise under the null.
Read postRandomizing treatment does not randomize a mediator.
Read postMeasurement error in a running variable need not destroy an RD design.
Read postAn exposure mapping summarizes which features of the treatment vector matter for a unit's outcome.
Read postThe proof of post-double selection needs more than a prediction-error bound.
Read postTwo-stage experiments distinguish changing an individual's treatment from changing the treatment environment around that individual.
Read postBecause functions need not be differentiable in general, one construction we can follow for defining derivatives is through the integration-by-parts formula; this is, heuristically, the foundation for defining a weak derivative.
Read postA treatment rule changes both a person's own treatment and the treatments received by their neighbors.
Read postAn observed bunching peak can spread across the kink because earnings are measured with error or choices are imperfectly adjusted.
Read postFurther Results on Stochastic Integrals. Notes on stochastic processes.
Read postA nonconvex risk can have stationary points that are not minimizers.
Read postA regression's control space is unchanged by an invertible change of basis.
Read postA cluster-robust standard error adds squared cluster scores.
Read postTo show the preceding statement, we need the following result, often referred to as the pasting lemma.
Read postThe streak tests become easier to understand after expanding their ratios.
Read postA streak statistic compares success after several successes with an appropriate baseline.
Read postThis result has a lot of consequences, so it is definitely worth proving in full detail.
Read postA single social network can contain many observations but far fewer independent pieces of information.
Read postThe marginal slope of an observed outcome regression combines a causal response with selection.
Read postInference under interference can proceed without estimating every cross-potential-outcome covariance.
Read postA mass point in a continuous treatment can reveal selection at the boundary.
Read postPartitioning a network does not make the resulting groups independent.
Read postThe difficulty with selecting controls is that a variable can be easy to miss in the outcome regression and still matter for the treatment coefficient.
Read postHere is an important and fundamental result for smooth vector fields, which guarantees a lot of the discussions about vector fields will make sense.
Read postBurkholder-Davis-Gundy inequalities. Notes on stochastic processes.
Read postThe constructions above allow us to distinguish between good and bad sets, where bad is understood in the sense of having no large enough subset that can be written as a union of regular sets (i.e.
Read postE-values connect hypothesis tests to betting payoffs; their calibration and combination rules determine which guarantees survive reuse.
Read postA bunching mass is an integral over an unobserved counterfactual interval, as in the kink response calculation.
Read postPrediction-powered inference uses a labeled sample to correct the errors in a much larger set of predictions.
Read postA notch creates a discrete loss in consumption at an earnings threshold.
Read postNetwork TMLE combines a causal intervention formula with a targeted update of an outcome regression.
Read postBefore diving into the statement and proof of the area formula, we take a slight detour to look at linear maps again in a bit more detail.
Read postIt would be ambitious (if not, overly ambitious) to try to squeeze the main ideas of geometric measure theory into this collection of notes.
Read postItō’s formula is extremely useful, and there are a few direct applications.
Read postCombining all the ingredients above, we have a proof of the classical result due to Bernstein.
Read postNonparametric Minimax Risk and the Varshamov–Gilbert Lemma. Notes on nonparametric estimation.
Read postConstruction of the Stochastic Integral. Notes on stochastic processes.
Read postMinimal Submanifolds and First Variation. Notes on differential geometry.
Read postProjection Estimators and Linear Minimax Risk. Notes on nonparametric estimation.
Read postThis concludes the proof of the desired statement on the upper bound of the uniform risk.
Read postThe study of global differential geometry deals with understanding the geometry of entire manifolds, and makes statements about, e.g.
Read postWith a fixed number of clusters, increasing the number of observations inside each cluster does not justify a many-cluster normal approximation.
Read postTo begin the discussion, we first illustrate a remark about the Hessian operator.
Read postQuadratic Variation and the Doob–Meyer Decomposition. Notes on stochastic processes.
Read postWe can also work with manifolds with a lower bound on their sectional curvature.
Read postA tax kink changes the marginal return to earnings while leaving the budget set continuous.
Read postStein’s Identity, Shrinkage, and Unbiased Risk Estimation. Notes on nonparametric efficiency.
Read postThis note goes over the details of asymptotic efficiency and adaptation arguments in analyzing nonparametric estimation procedures.
Read postA result of great importance is the following, often referred to as the isotopy lemma.
Read postAn interesting field of questions arise from the study of connections between geometry and topology.
Read postA major question in the study of stochastic processes is how weak convergence can be used to characterize convergence in law almost surely.
Read postWhether standard errors should be clustered depends on the sampling and treatment-assignment design.
Read postFrom the exposition above, it becomes clear that we are interested in solving a differential equation captured by the following result.
Read postCut times, and therefore cut locus, are generally difficult to determine, save for in special examples.
Read postBrownian Path Oscillations and Continuity Bounds. Notes on stochastic processes.
Read postOne way to circumvent descriptions of best or most optimal is to look at two facts:
Read postCurvature Bounds and Global Geometric Consequences. Notes on differential geometry.
Read postTo formalize long-range independence, I define a few additional objects.
Read postWe now extend the ideas of the index form to the space of vector fields in a natural way.
Read postThis construction leads us to the following desirable result.
Read postApplication: Cartan’s Theorem. Notes on differential geometry.
Read postKernel Density Estimation: Pointwise and Integrated Risk. Notes on nonparametric estimation.
Read postLocal Asymptotic Normality. Notes on local asymptotic normality and efficiency.
Read postWe can extend this discussion much further: uniformly most accurate confidence bounds is subsumed in the theory of unbiased confidence sets.
Read postEvery energy minimizing curve is a geodesic; as we have noted, the first variation of the energy functional must vanish for every proper variation of such curve.
Read postA useful result for proving the theorem to follow is the following inequality.
Read postIt turns out this problem can be avoided, in theory, by allowing for randomization—as formalized in the Neymann-Pearson lemma.
Read postWhy Complex Projective Space Is Hausdorff. Notes on differential geometry.
Read postThe first result we will discuss is a characterization of Jacobi fields: they are variation fields of variations through geodesics.
Read postBrownian Motion: Markov Properties and Path Continuity. Notes on stochastic processes.
Read postThe shock-based argument for shift-share IV allows exposure shares to be correlated with outcome errors.
Read postMartingales: Maximal and Upcrossing Inequalities. Notes on martingale inequalities.
Read postA shift-share instrument combines exposure shares with aggregate shocks.
Read postTo start, we prove the spectral theorem for self-adjoint operators on finite-dimensional Hilbert spaces.
Read postWe recap a number of concepts before studying submanifolds and how to characterize their intrinsic geometric properties.
Read postThe following result gives us an explicit description of the natural connection between localized empirical Gaussian complexity and Gaussian process.
Read postRademacher and Gaussian Complexity. Notes on empirical processes.
Read postUsing these tools, we can develop the following idea, called Gaussian interpolation.
Read postWe revisit the study of empirical processes again, by returning to Boolean functions.
Read postLocalization and the Uniform Law. Notes on empirical processes.
Read postWe need a preliminary step in order to establish the lower bound of Rademacher complexity.
Read postTo continue our discussion of Lie algebras, we briefly return to discussing the algebraic structure underlying Lie groups.
Read postFinite-class Bernstein bounds and precise local bracketing theorems separate variance control from envelope control.
Read postConditional chaining bounds empirical-process suprema; population bracketing adds a scale-dependent truncation argument.
Read postThis argument is the standard local quadratic expansion and stochastic equicontinuity route used in Pollard-style treatments of M-estimation, and matches the general asymptotic normality statement for M-estimators (via an argmax/argmin theorem).
Read postThis section details more about Lie groups, and some of its applications.
Read postThe maximal inequality due to the Chernoff bound immediately gives us the following.
Read postTo facilitate the ensuing analyses, we need the following higher-order asymptotic expansions of the roots we’ve been studying.
Read postSymmetrization arguments also work for bounding tail probabilities.
Read postAs we have seen above, the method of tensorization allows us to derive useful and nontrivial bounds in a lot of scenarios.
Read postThe discussion above can be generalized to procedure known as the Chernoff bound.
Read postThe following statement is one of the most important results in the study of planar geometry.
Read postWe focus on the following four important principles that underpin most statistical analyses.
Read postFirst, we make some general connections (no pun intended!) to the previous section.
Read postThe theory of permutation and and randomization tests, while powerful and has good properties, can only be used in certain problems where the randomization hypothesis holds.
Read postIt will be useful to understand the energy functions and its variations in the sequel.
Read postLet us begin with a statistical example that motivates the study of empirical processes.
Read postMoment Inequalities and Random Series. Notes on probability and statistics.
Read postWe are now ready to state the impossibility result, due to Bahadur-Savage.
Read postThe technical lemma allows us to investigate the asymptotics without needing to verify Lindeberg condition (in)directly.
Read postWhen we sample with replacements, the asymptotic theory is relatively standard.
Read postWe now briefly review some concepts from linear algebra and differential topology.
Read postA particularly insightful exercise is to prove the following result.
Read postWe begin to examine one of the most important consequences of the construction of geodesics.
Read postWe can basically summarize the discussion above in the following theorem.
Read postVerifying qmd is usually not straightforward, but here is a sufficient condition for verifying if a family of parametric models is qmd.
Read postNow we proceed to define geodesics as critical points of the energy functional.
Read postSometimes in practice, the Lindeberg condition is difficult to verify.
Read postWe now build the theory of tangent spaces, from that on Euclidean spaces to manifolds.
Read postProjection Methods and U-Statistics. Notes on projections, u-statistics, and ranks.
Read postHere are some useful facts that we will use throughout the study of Riemannian geometry.
Read postAn average need not concentrate, and a sum of many terms need not have a normal limit.
Read postThis section collects most of the preliminary details of classical differential geometry.
Read postStudent and chi-square limits both start from sums of squared normal variables.
Read postConvergence in probability permits rare large errors; almost-sure convergence concerns entire sample paths.
Read postContiguity becomes useful when the data-generating law changes with the sample size.
Read postWeak convergence controls the limiting law, but it does not determine a coupling or probabilities of arbitrary Borel sets.
Read postWeak Convergence, Tightness, and Metric Spaces. Notes on stochastic convergence.
Read postThere are a few straightforward cases of conditional expectations:
Read postThe Radon–Nikodym theorem constructs a density by maximising its mass; Hahn’s decomposition shows why no absolutely continuous remainder can remain.
Read postThe notation for stochastic order abbreviates probability statements.
Read postA fundamental result that belies all results in the section is the following, due to Skorokhod.
Read postModes of Stochastic Convergence. Notes on stochastic convergence.
Read postProbability theory relies on the following fact about extensions of measures.
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