- 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.

### STAT1018 - Intro Business Stat

The STAT 1018 honors section, which fulfills the STAT 1010 requirement offers an introduction to probability and statistics for students who have studied calculus and are seeking a class with mathematical content. The class will assume good command of the material in MATH 1400 as a prerequisite. Although multivariate calculus is not a formal prerequisite for the class concurrent enrollment in MATH 1410 or MATH 1610 may be a good choice for first or second year students who are interested in developing firm foundations for the more mathematical statistics classes. Stat 1018 is particularly recommended for those considering a statistics and data science concentration or minor.

STAT1018001 ( Syllabus )

### STAT4330 - 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.

STAT4330001 ( Syllabus )

STAT4330002 ( Syllabus )

### STAT1018 - Intro Business Stat

The STAT 1018 honors section, which fulfills the STAT 1010 requirement offers an introduction to probability and statistics for students who have studied calculus and are seeking a class with mathematical content. The class will assume good command of the material in MATH 1400 as a prerequisite. Although multivariate calculus is not a formal prerequisite for the class concurrent enrollment in MATH 1410 or MATH 1610 may be a good choice for first or second year students who are interested in developing firm foundations for the more mathematical statistics classes. Stat 1018 is particularly recommended for those considering a statistics and data science concentration or minor.

### STAT3990 - Independent Study

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

### STAT4300 - Probability

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

### STAT4330 - 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.

### STAT5100 - 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.

### STAT5160 - Adv Stat Inference II

STAT 5160 is a natural continuation of STAT 5150, and the main focus is on asymptotic evaluations and regression models. Time permitting, it also discusses some basic nonparametric statistical methods.

### STAT5330 - 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.

### STAT8990 - Independent Study

Written permission of instructor, the department MBA advisor and 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