proof

abstract.tex

@@ -18,14 +18,14 @@
 forests, including a central limit theorem for the estimated regression
 function and a characterization of the bias. We show how to conduct feasible
 and valid nonparametric inference by constructing confidence intervals, and
-further provide a debiasing procedure which enables minimax-optimal estimation
+further provide a debiasing procedure that enables minimax-optimal estimation
 rates for smooth function classes in arbitrary dimension.
 
 Next, we turn our attention to nonparametric kernel density estimation with
 dependent dyadic network data. We present results for minimax-optimal
 estimation, including a novel lower bound for the dyadic uniform convergence
 rate, and develop methodology for uniform inference via confidence bands and
-counterfactual analysis. Our methods are based on strong approximation and are
+counterfactual analysis. Our methods are based on strong approximations and are
 designed to be adaptive to potential dyadic degeneracy. We give empirical
 results with simulated and real-world economic trade data.
 

acknowledgments.tex

@@ -2,16 +2,16 @@
 
 I am extremely fortunate to have been surrounded by many truly wonderful people
 over the course of my career, and without their support this dissertation would
-not have been possible. While it is infeasible for me to identify every one of
-them individually, I would like to mention a few names in particular, to
+not have been possible. While it is impossible for me to identify every one of
+them individually, I would like to mention a few names in particular to
 recognize those who have been especially important to me during the last few
 years.
 
 Firstly, I would like to express my utmost gratitude to my Ph.D.\ adviser,
 Matias Cattaneo. Working with Matias has been genuinely inspirational for me,
 and I could not have asked for a more rewarding start to my journey as a
-researcher. From the very beginning he has guided me expertly through my
-studies, providing hands-on assistance when required, while also allowing me the
+researcher. From the very beginning, he has guided me expertly through my
+studies, providing hands-on assistance when required while also allowing me the
 independence necessary to develop as an academic. I hope that, during the four
 years we have worked together, I have acquired just a fraction of his formidable
 mathematical intuition, keen attention to detail, boundless creativity, and
@@ -28,14 +28,14 @@
 Amir Ali Ahmadi, Ramon van Handel, Mikl{\'o}s R{\'a}cz, and Mykhaylo Shkolnikov,
 my colleagues Sanjeev Kulkarni and Roc{\'i}o Titiunik,
 and my former supervisor Mihai Cucuringu.
-I am thankful also for the staff members at Princeton who have been
-perpetually helpful, and would like to identify Kim
+I am also thankful for the staff members at Princeton who have been
+perpetually helpful, and I would like to identify Kim
 Lupinacci in particular; her assistance in all things administrative has been
 invaluable.
 
 I am grateful to my fellow graduate students in the ORFE department for their
 technical expertise and generosity with their time, and for making Sherrerd
-Hall such a vibrant and exciting space; especially
+Hall such a vibrant and exciting space, especially
 Jose Avilez,
 Pier Beneventano,
 Ben Budway,
@@ -90,7 +90,7 @@
 and Anita Zhang.
 Thank you to the Princeton Chapel Choir for being such a wonderful
 community of musicians and a source of close friends,
-and to our directors Nicole Aldrich and Penna Rose, and organist Eric Plutz.
+and to our directors, Nicole Aldrich and Penna Rose, and organist Eric Plutz.
 
 Lastly, yet most importantly, I want to thank my family for their unwavering
 support throughout my studies. My visits back home have been a source of joy

introduction.tex

@@ -15,25 +15,25 @@ \chapter{Introduction}
 random forests, neural networks, and many more.
 
 % nonparametric estimation is good
-The benefits of the nonparametric framework are clear; statistical procedures
+The benefits of the nonparametric framework are clear: statistical procedures
 can be formulated in cases where the stringent assumptions of parametric models
 are untestable, demonstrably violated, or simply unreasonable.
-Further, the resulting
-methods often as a consequence inherit desirable robustness properties against
-various forms of misspecification or misuse. The class of problems which can be
+As a consequence,
+the resulting methods often inherit desirable robustness properties against
+various forms of misspecification or misuse. The class of problems that can be
 formulated is correspondingly larger: arbitrary distributions and
 relationships can be characterized and estimated in a principled manner.
 
 % nonparametric estimation is hard
 Nonetheless, these attractive properties do come at a price. In particular, as
 its name suggests, the nonparametric approach forgoes the ability to reduce
-a complex statistical problem to that of estimating a fixed finite number of
-parameters. Rather, nonparametric procedures typically involve making inference
+a complex statistical problem to that of estimating a fixed, finite number of
+parameters. Rather, nonparametric procedures typically involve making inferences
 about a growing number of parameters simultaneously, as witnessed in
 high-dimensional regimes, or even directly handling infinite-dimensional
 objects such as entire regression or density functions. As a consequence,
 nonparametric estimators are usually less efficient than their corresponding
-correctly specified parametric counterparts, when these are available; rates of
+correctly specified parametric counterparts, when they are available; rates of
 convergence tend to be slower, and confidence sets more conservative. Another
 challenge is that theoretical mathematical analyses of nonparametric estimators
 are often significantly more demanding than those required for low-dimensional
@@ -54,7 +54,7 @@ \chapter{Introduction}
 a central concept in classical statistics, and despite the rapid
 recent development of theory for modern nonparametric estimators, their
 applicability to statistical inference is in certain cases rather less well
-studied: theoretically sound and practically implementable inference procedures
+studied; theoretically sound and practically implementable inference procedures
 are sometimes absent in the literature.
 
 % complex data
@@ -64,11 +64,11 @@ \chapter{Introduction}
 framework of independent and identically distributed samples, and instead might
 consist of time series, stochastic processes, networks,
 or high-dimensional or functional data, to name just a few.
-Therefore it is important to understand how nonparametric methods might be
+Therefore, it is important to understand how nonparametric methods might be
 adapted to correctly handle these data types, maintaining fast estimation rates
 and valid techniques for statistical inference. The technical challenges
 associated with such an endeavor are non-trivial; many standard techniques are
-ineffective in the presence of dependent or infinite-dimensional data for
+ineffective in the presence of dependent or infinite-dimensional data, for
 example. As such, the development of new mathematical results in probability
 theory plays an important role in the comprehensive treatment of nonparametric
 statistics with complex data.
@@ -85,7 +85,7 @@ \section*{Overview of the dissertation}
 % what are random forests
 Random forests are popular ensembling-based methods for classification and
 regression, which are well known for their good performance, flexibility,
-robustness and efficiency. The majority of random forest models share the
+robustness, and efficiency. The majority of random forest models share the
 following common framework for producing estimates of a classification or
 regression function using covariates and a response variable. Firstly, the
 covariate space is partitioned in some algorithmic manner, possibly using a
@@ -103,9 +103,9 @@ \section*{Overview of the dissertation}
 % mondrian random forests
 One interesting such example is that of the Mondrian random forest, in which
 the underlying partitions (or trees) are constructed independently of the data.
-Naturally this restriction rules out many classical random forest models which
+Naturally, this restriction rules out many classical random forest models, which
 exhibit a complex and data-dependent partitioning scheme. Instead, trees are
-sampled from a canonical stochastic process, known as the Mondrian process,
+sampled from a canonical stochastic process known as the Mondrian process,
 which endows the resulting tree and forest estimators with various agreeable
 features.
 
@@ -117,7 +117,7 @@ \section*{Overview of the dissertation}
 consistent variance estimator, allows one to perform asymptotically valid
 statistical inference, such as constructing confidence intervals, on the
 unknown regression function. We also provide a debiasing procedure for Mondrian
-random forests which allows them to achieve minimax-optimal estimation rates
+random forests, which allows them to achieve minimax-optimal estimation rates
 with H{\"o}lder smooth regression functions, for any smoothness parameter and
 in arbitrary dimension.
 
@@ -136,7 +136,7 @@ \section*{Overview of the dissertation}
 
 % broad scope
 We focus on nonparametric estimation and inference with dyadic
-data, and in particular we seek methods which are robust in the sense that our
+data, and in particular we seek methods that are robust in the sense that our
 results should hold uniformly across the support of the data. Such uniformity
 guarantees allow for statistical inference in a broader range of settings,
 including specification testing and distributional counterfactual analysis. We
@@ -155,8 +155,8 @@ \section*{Overview of the dissertation}
 causal inference and program evaluation.
 % why it is difficult
 A crucial feature of dyadic distributions is that they may be ``degenerate'' at
-certain points in the support of the data, a property making our analysis
-somewhat delicate. Nonetheless our methods for uniform inference remain robust
+certain points in the support of the data, a property that makes our analysis
+somewhat delicate. Nonetheless, our methods for uniform inference remain robust
 to the potential presence of such points.
 % applications
 For implementation purposes, we discuss inference procedures based on positive