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Statistical and Causal Inference for Complex High Dimensional Data

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Statistical Methods, Experimental Design, and Scientific Inference: A Re-issue of Statistical Methods for Research Workers, the Design of Experiments and Statistical Methods and Scientific Inference

Statistical Methods, Experimental Design, and Scientific Inference: A Re-issue of Statistical Methods for Research Workers, the Design of Experiments and Statistical Methods and Scientific Inference

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R.A. Fisher has had more influence on the development of statistical theory and practice than any other twentieth-century statistician. His writings (both in paper and book form) have proved to be as relevant to present-day statisticians as they were when first published. This book brings together as a single volume three of Fisher's most influential textbooks: Statistical Methods for Research Workers, The Design of Experiments , and Statistical Methods and Scientific Inference . In this new edition Frank Yates has provided a foreword which sheds fresh light on Fisher's thinking and on the writing and reception of each of the books. He discusses some of the key issues tackled in the three books and reflects on how the ideas expressed have come to permeate modern statistical practice.R.A. Fisher has had more influence on the development of statistical theory and practice than any other twentieth-century statistician. His writings (both in paper and book form) have proved to be as relevant to present-day statisticians as they were when first published. This book brings together as a single volume three of Fisher's most influential textbooks: Statistical Methods for Research Workers, The Design of Experiments , and Statistical Methods and Scientific Inference . In this new edition Frank Yates has provided a foreword which sheds fresh light on Fisher's thinking and on the writing and reception of each of the books. He discusses some of the key issues tackled in the three books and reflects on how the ideas expressed have come to permeate modern statistical practice.

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2012 Theses Doctoral

Statistical inference in two non-standard regression problems

Seijo, Emilio Francisco

This thesis analyzes two regression models in which their respective least squares estimators have nonstandard asymptotics. It is divided in an introduction and two parts. The introduction motivates the study of nonstandard problems and presents an outline of the contents of the remaining chapters. In part I, the least squares estimator of a multivariate convex regression function is studied in great detail. The main contribution here is a proof of the consistency of the aforementioned estimator in a completely nonparametric setting. Model misspecification, local rates of convergence and multidimensional regression models mixing convexity and componentwise monotonicity constraints will also be considered. Part II deals with change-point regression models and the issues that might arise when applying the bootstrap to these problems. The classical bootstrap is shown to be inconsistent on a simple change-point regression model, and an alternative (smoothed) bootstrap procedure is proposed and proved to be consistent. The superiority of the alternative method is also illustrated through a simulation study. In addition, a version of the continuous mapping theorem specially suited for change-point estimators is proved and used to derive the results concerning the bootstrap.

  • Mathematics

thumnail for SeijoSolis_columbia_0054D_10886.pdf

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Causal and selective inference in complex statistical models

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Carnegie Mellon University

Statistical Inference for Optimal Transport

 Optimal transport is a flexible framework for comparing probability distributions, which has received a recent surge of interest as a methodological tool in statistics. The aim of this thesis is to develop procedures for performing valid and efficient statistical inference for various objects arising from the optimal transport framework. On the one hand, we derive a semiparametric efficient estimator of the quadratic Wasserstein distance between probability measures of arbitrary fixed dimension. On the other hand, we develop a pointwise central limit theorem for the quadratic optimal transport map between multivariate periodic distributions. We also develop nonasymptotic and sequential inferential procedures for various optimal transport divergence functionals. These results provide a step toward the longstanding problem of performing practical inference for optimal transport in arbitrary dimension. Along the way, this thesis studies the related question of performing minimax estimation for optimal transport maps and costs, leading to new minimax upper or lower bounds for these problems in various settings. We close with an application of these ideas to a problem arising in experimental high energy physics, where we show that optimal transport can be used to address the problem of data-driven background estimation, arising in the search for new physical phenomena at the Large Hadron Collider. 

Degree Type

  • Dissertation
  • Statistics and Data Science

Degree Name

  • Doctor of Philosophy (PhD)

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  • Statistical Theory

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Exploring the History of Statistical Inference in Economics : Introduction

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Jeff Biddle , Marcel Boumans; Exploring the History of Statistical Inference in Economics : Introduction . History of Political Economy 1 December 2021; 53 (S1): 1–24. doi: https://doi.org/10.1215/00182702-9414747

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Basic Principles of Statistical Inference

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statistical inference thesis

  • Wanzhu Tu PhD 2  

Part of the book series: Methods in Molecular Biology™ ((MIMB,volume 404))

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In this chapter, we discuss the fundamental principles behind two of the most frequently used statistical inference procedures: confidence interval estimation and hypothesis testing. both procedures are constructed on the sampling distributions that we have learned in previous chapters. To better understand these inference procedures, we focus on the logic of statistical decision making and the role that experimental data play in the decision process. Numerical examples are used to illustrate the implementation of the discussed procedures. This chapter also introduces some of the most important concepts associated with confidence interval estimation and hypothesis testing, including P values, significance level, power, sample size, and two types of errors. We conclude the chapter with a brief discussion on statistical and practical significance of test results.

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So it is a statistic and a random variable!

Mood, A. M., Graybill, F. A., and Boes, D. C. (1974) Introduction to the Theory of Statistics , 3rd ed. New York, McGraw-Hill.

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Arnold, S. F. (1990) Mathematical Statistics . Englewood Cliffs, Prentice-Hall.

Maisel, A. S., Krishnaswamy, P., Nowak, R. M., McCord, J., Hollander, J. E., Duc, P., Omland, T., Storrow, A. B., Abraham, W. T., and Wu, A. H. B. (2002). Rapid measurement of B-type natriuretic reptide in the emergency diagnosis of heart failure. N. Engl. J. Med. 347 (3), 161–167.

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Division of Biostatistics, Indiana University School of Medicine, Indianopolis, IN

Wanzhu Tu PhD

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Department of Biostatistical Sciences, Wake Forest University Health Sciences, Winston-Salem, NC

Walter T. Ambrosius

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© 2007 Humana Press Inc., Totowa, NJ

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Tu, W. (2007). Basic Principles of Statistical Inference. In: Ambrosius, W.T. (eds) Topics in Biostatistics. Methods in Molecular Biology™, vol 404. Humana Press. https://doi.org/10.1007/978-1-59745-530-5_4

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GENERAL-PURPOSE STATISTICAL INFERENCE WITH DIFFERENTIAL PRIVACY GUARANTEES

Differential privacy (DP) uses a probabilistic framework to measure the level of privacy protection of a mechanism that releases data analysis results to the public. Although DP is widely used by both government and industry, there is still a lack of research on statistical inference under DP guarantees. On the one hand, existing DP mechanisms mainly aim to extract dataset-level information instead of population-level information. On the other hand, DP mechanisms introduce calibrated noises into the released statistics, which often results in sampling distributions more complex and intractable than the non-private ones. This dissertation aims to provide general-purpose methods for statistical inference, such as confidence intervals (CIs) and hypothesis tests (HTs), that satisfy the DP guarantees.

In the first part of the dissertation, we examine a DP bootstrap procedure that releases multiple private bootstrap estimates to construct DP CIs. We present new DP guarantees for this procedure and propose to use deconvolution with DP bootstrap estimates to derive CIs for inference tasks such as population mean, logistic regression, and quantile regression. Our method achieves the nominal coverage level in both simulations and real-world experiments and offers the first approach to private inference for quantile regression.

In the second part of the dissertation, we propose to use the simulation-based ``repro sample'' approach to produce CIs and HTs based on DP statistics. Our methodology has finite-sample guarantees and can be applied to a wide variety of private inference problems. It appropriately accounts for biases introduced by DP mechanisms (such as by clamping) and improves over other state-of-the-art inference methods in terms of the coverage and type I error of the private inference.

In the third part of the dissertation, we design a debiased parametric bootstrap framework for DP statistical inference. We propose the adaptive indirect estimator, a novel simulation-based estimator that is consistent and corrects the clamping bias in the DP mechanisms. We also prove that our estimator has the optimal asymptotic variance among all well-behaved consistent estimators, and the parametric bootstrap results based on our estimator are consistent. Simulation studies show that our framework produces valid DP CIs and HTs in finite sample settings, and it is more efficient than other state-of-the-art methods.

Simulation-Based Inference for Differential Privacy

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  • Doctor of Philosophy

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Title: statistical inference for high-dimensional convoluted rank regression.

Abstract: High-dimensional penalized rank regression is a powerful tool for modeling high-dimensional data due to its robustness and estimation efficiency. However, the non-smoothness of the rank loss brings great challenges to the computation. To solve this critical issue, high-dimensional convoluted rank regression is recently proposed, and penalized convoluted rank regression estimators are introduced. However, these developed estimators cannot be directly used to make inference. In this paper, we investigate the inference problem of high-dimensional convoluted rank regression. We first establish estimation error bounds of penalized convoluted rank regression estimators under weaker conditions on the predictors. Based on the penalized convoluted rank regression estimators, we further introduce a debiased estimator. We then provide Bahadur representation for our proposed estimator. We further develop simultaneous inference procedures. A novel bootstrap procedure is proposed and its theoretical validity is also established. Finally, simulation and real data analysis are conducted to illustrate the merits of our proposed methods.

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Stochastic SIS epidemic models and corresponding statistical inference

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statistical inference thesis

  • Pan, Jiafeng.
  • https://stax.strath.ac.uk/thesis_copyright_statement/
  • University of Strathclyde
  • Doctoral (Postgraduate)
  • Doctor of Philosophy (PhD)
  • Department of Mathematics and Statistics.
  • This thesis considers the deterministic SIS epidemic model, which has applications to transmission of real-life diseases, such as pneumococcus, gonorrhea and tuberculosis. Environmental noise can a ect the deterministic system signi cantly. There are various types of noise which can be incorporated into the deterministic dynamics according to di erent situations. The e ect of three types of noise on the deterministic SIS epidemic model have been examined in this thesis, which has not been discussed in previous literature. Firstly, assuming that there exists environmental noise in the disease transmission coe cient, we extend the classical SIS epidemic model from a deterministic framework to a stochastic one by incorporating white noise using the parameter perturbation technique, and formulate it as a stochastic di erential equation (SDE) for the number of infectious individuals I(t). For the model to make sense, we then prove that this SDE has a unique global positive solution I(t) and establish conditions for extinction and persistence of I(t) and compare these with the corresponding conditions for the deterministic SIS epidemic model. We also discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. Secondly, assuming that the parameters in the SIS epidemic model experience an abrupt change around the point of threshold value, we incorporate telegraph noise in the deterministic model. We then establish the explicit solution of the stochastic SIS epidemic model, which is useful in performing computer simulations. We also obtain the conditions for extinction and persistence for this model. Afterwards, we take a further step of incorporating both types of the aforementioned noise in the SIS epidemic model. We not only show the existence of a unique global positive solution but also examine the asymptotic properties, including extinction and persistence. The results are illustrated by computer simulations, including examples based on real life diseases for the rst and second stochastic models. Computer simulations based on the explicit solution and the Euler{Maruyama scheme are compared for the SIS model with telegraph noise. Furthermore, statistical inference is always essential in disease analysis. That is the motivation for us to conduct parameter estimation for the SDE SIS model with white noise introduced. Three estimation methods, least squares estimation, the pseudo-Maximum Likelihood Estimation (pseudo-MLE) method and the Bayesian approach are applied to the SDE SIS model. Our main contribution in least squares estimation and pseudo-MLE is variance estimation. We obtain not only the point estimators but also the interval estimators and the joint con dence regions for both estimation techniques. Additionally we investigate the factors which in uence variance in estimation. As for the Bayesian approach, although strong results have been obtained by using the MCMC technique, we use a di erent method where analytic results are obtained without the need to deal with the signi cant computational cost. Computer simulations are performed to illustrate our theory. The three estimation methods are compared both analytically and in the simulation examples.
  • Doctoral thesis
  • Strathclyde theses - ask staff. Thesis no. : T13458
  • 10.48730/2ptm-6a85

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