- Walter C. Bladstrom Professor
- Professor of Statistics and Data Science

**Primary Email:**

lowm@wharton.upenn.edu

**office Address:**425 Academic Research Building

265 South 37th Street

Philadelphia, PA 19104

**Research Interests: **decision theory, nonparametric function estimation, statistical inference

PhD, Cornell University, 1989

ScB, Brown University, 1983

Wharton: 1991-present (named Walter C. Bladstrom Professor, 2007; Anheuser-Busch Term Assistant Professor of Statistics 1991-96)

Visiting appointments: University of California, Berkeley; University of Illinois

For more information go to my Personal page.

Tony Cai and Mark G. Low (2015),

**A Framework for Estimation of Convex Functions**,*Statistica Sinica*, 25, pp. 423-456.Mark G. Low and Zongming Ma (2015),

**Discussion: Frequentist coverage of adaptive nonparametric Bayesian credible sets**,*The Annals of Statistics*, 43, pp. 1448-1454.Tony Cai, Mark G. Low, Zongming Ma (2014),

**Adaptive confidence bands for nonparametric regression functions**,*Journal of the American Statistical Association*, 109 (507), pp. 1054-1070.**Abstract:**A new formulation for the construction of adaptive condence bands in nonparametric function estimation problems is proposed. Condence bands are constructed which have size that adapts to the smoothness of the function while guaranteeing that the relative excess mass of the function lying outside the band and the measure of the set of points where the function lies outside the band is small. It is shown that the bands adapt over a maximum range of Lipschitz classes. The procedure can be easily modied and used for other nonparametric function estimation models.Tony Cai, Mark G. Low, Yin Xia (2013),

**Adaptive Confidence Intervals for Regression Functions Under Shape Constraints**,*The Annals of Statistics*, 41, 722-750.Tony Cai and Mark G. Low (2011),

**Testing Composite Hypotheses, Hermite Polynomials, and Optimal Estimation of a Nonsmooth Functional**,*Annals of Statistics*, Vol. 39, 1012-1041.Mark G. Low (2010),

**Chi-square lower bounds**,*IMS Collections*, 6, 22-31.Tony Cai, Mark G. Low, Linda Zhao (2009),

**Sharp Adaptive Estimation by a Blockwise Method**,*J. Nonparametric Statistics*, Vol. 21, 839-850.Mark G. Low and Zhou, H. (2007),

**A complement to Le Cam’s Theorem**,*Annals of Statistics*, 35, 1146-1165.Tony Cai, Jiashun Jin, Mark G. Low (2007),

**Estimation and Confidence Sets for Sparse Normal Mixtures**,*Annals of Statistics*, 2421-2449.**Abstract:**For high dimensional statistical models, researchers have begun to focus on situations which can be described as having relatively few moderately large coefficients. Such situations lead to some very subtle statistical problems. In particular, Ingster and Donoho and Jin have considered a sparse normal means testing problem, in which they described the precise demarcation or detection boundary. Meinshausen and Rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region, but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible. In the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large.We show that the detection region described by Ingster and Donoho and Jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates. This theory is developed further and minimax rates of convergence are derived. A procedure is constructed which attains the optimal rate of convergence in this setting. Furthermore, the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval. Simulations are used to enable comparison with the work of Meinshausen and Rice, where a procedure is given but where rates of convergence have not been discussed. Extensions to more general Gaussian mixture models are also given.Tony Cai, Mark G. Low, Linda Zhao (2007),

**Trade-offs Between Global and Local Risks in Nonparametric Function Estimation**,*Bernoulli*, Vol. 13, 1-19.**Abstract:**The problem of loss adaptation is investigated: given a fixed parameter, the goal is to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. Given the class of estimator sequences that achieve the minimax rate, over a fixed Besov space, for estimating the entire function a lower bound is given on the performance for estimating the function at each point. This bound is larger by a logarithmic factor than the usual minimax rate for estimation at a point when the global and local minimax rates of convergence differ. A lower bound for the maximum global risk is given for estimators that achieve optimal minimax rates of convergence at every point. An inequality concerning estimation in a two-parameter statistical problem plays a key role in the proof. It can be considered as a generalization of an inequality due to Brown and Low. This may be of independent interest. A particular wavelet estimator is constructed which is globally optimal and which attains the lower bound for the local risk provided by our inequality.

### STAT433 - Stochastic Processes

An introduction to Stochastic Processes. The primary focus is on Markov Chains, Martingales and Gaussian Processes. We will discuss many interesting applications from physics to economics. Topics may include: simulations of path functions, game theory and linear programming, stochastic optimization, Brownian Motion and Black-Scholes. This course may be taken concurrently with the prerequisite with instructor permission.

STAT433001 ( Syllabus )

STAT433002 ( Syllabus )

### BSTA621 - STATISTICAL INFERENCE I

This class will cover the fundamental concepts of statistical inference. Topics include sufficiency, consistency, finding and evaluating point estimators, finding and evaluating interval estimators, hypothesis testing, and asymptotic evaluations for point and interval estimation. Prerequisite: If course requirements not met, permission of instructor.

### STAT101 - INTRO BUSINESS STAT

Data summaries and descriptive statistics; introduction to a statistical computer package; Probability: distributions, expectation, variance, covariance, portfolios, central limit theorem; statistical inference of univariate data; Statistical inference for bivariate data: inference for intrinsically linear simple regression models. This course will have a business focus, but is not inappropriate for students in the college. This course may be taken concurrently with the prerequisite with instructor permission.

### STAT102 - INTRO BUSINESS STAT

Continuation of STAT 101. A thorough treatment of multiple regression, model selection, analysis of variance, linear logistic regression; introduction to time series. Business applications. This course may be taken concurrently with the prerequisite with instructor permission.

### STAT111 - INTRODUCTORY STATISTICS

Introduction to concepts in probability. Basic statistical inference procedures of estimation, confidence intervals and hypothesis testing directed towards applications in science and medicine. The use of the JMP statistical package. Knowledge of high school algebra is required for this course.

### STAT430 - PROBABILITY

Discrete and continuous sample spaces and probability; random variables, distributions, independence; expectation and generating functions; Markov chains and recurrence theory.

### STAT433 - STOCHASTIC PROCESSES

An introduction to Stochastic Processes. The primary focus is on Markov Chains, Martingales and Gaussian Processes. We will discuss many interesting applications from physics to economics. Topics may include: simulations of path functions, game theory and linear programming, stochastic optimization, Brownian Motion and Black-Scholes. This course may be taken concurrently with the prerequisite with instructor permission.

### STAT510 - PROBABILITY

Elements of matrix algebra. Discrete and continuous random variables and their distributions. Moments and moment generating functions. Joint distributions. Functions and transformations of random variables. Law of large numbers and the central limit theorem. Point estimation: sufficiency, maximum likelihood, minimum variance. Confidence intervals. A one-year course in calculus is recommended.

### STAT512 - MATHEMATICAL STATISTICS

An introduction to the mathematical theory of statistics. Estimation, with a focus on properties of sufficient statistics and maximum likelihood estimators. Hypothesis testing, with a focus on likelihood ratio tests and the consequent development of "t" tests and hypothesis tests in regression and ANOVA. Nonparametric procedures.

### STAT516 - ADV STAT INFERENCE II

STAT 516 is a natural continuation of STAT 515, and the main focus is on asymptotic evaluations and regression models. Time permitting, it also discusses some basic nonparametric statistical methods.

### STAT533 - STOCHASTIC PROCESSES

An introduction to Stochastic Processes. The primary focus is on Markov Chains, Martingales and Gaussian Processes. We will discuss many interesting applications from physics to economics. Topics may include: simulations of path functions, game theory and linear programming, stochastic optimization, Brownian Motion and Black-Scholes.

### STAT621 - ACC REGRESSION ANALYSIS

STAT 621 is intended for students with recent, practical knowledge of the use of regression analysis in the context of business applications. This course covers the material of STAT 613, but omits the foundations to focus on regression modeling. The course reviews statistical hypothesis testing and confidence intervals for the sake of standardizing terminology and introducing software, and then moves into regression modeling. The pace presumes recent exposure to both the theory and practice of regression and will not be accommodating to students who have not seen or used these methods previously. The interpretation of regression models within the context of applications will be stressed, presuming knowledge of the underlying assumptions and derivations. The scope of regression modeling that is covered includes multiple regression analysis with categorical effects, regression diagnostic procedures, interactions, and time series structure. The presentation of the course relies on computer software that will be introduced in the initial lectures. Recent exposure to the theory and practice of regression modeling is recommended.

### STAT899 - INDEPENDENT STUDY

Written permission of instructor, the department MBA advisor and course coordinator required to enroll.

### STAT991 - SEM IN ADV APPL OF STAT

This seminar will be taken by doctoral candidates after the completion of most of their coursework. Topics vary from year to year and are chosen from advance probability, statistical inference, robust methods, and decision theory with principal emphasis on applications.

### STAT995 - DISSERTATION

### STAT999 - INDEPENDENT STUDY

Written permission of instructor and the department course coordinator required to enroll.

- Wharton Teaching Excellence Award, 2020
- Wharton Undergraduate Teaching Award, 2013 Description
2011, 2013

- Medallion Lecturer, Institute of Mathematical Statistics, 2008
- Fellow, Institute of Mathematical Statistics, 2004
- Mathematical Sciences Post-Doctoral Fellow, NSF, 1991

Tony Cai and Mark G. Low (2015), **A Framework for Estimation of Convex Functions**, *Statistica Sinica*, 25, pp. 423-456.

All Research