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Related Concept Videos

Multiple Regression01:25

Multiple Regression

Multiple regression assesses a linear relationship between one response or dependent variable and two or more independent variables. It has many practical applications.
Farmers can use multiple regression to determine the crop yield based on more than one factor, such as water availability, fertilizer, soil properties, etc. Here, the crop yield is the response or dependent variable as it depends on the other independent variables. The analysis requires the construction of a scatter plot...
Regression Analysis01:11

Regression Analysis

Regression analysis is a statistical tool that describes a mathematical relationship between a dependent variable and one or more independent variables.
In regression analysis, a regression equation is determined based on the line of best fit– a line that best fits the data points plotted in a graph. This line is also called the regression line. The algebraic equation for the regression line is called the regression equation. It is represented as:
Regression Toward the Mean01:52

Regression Toward the Mean

Regression toward the mean (“RTM”) is a phenomenon in which extremely high or low values—for example, and individual’s blood pressure at a particular moment—appear closer to a group’s average upon remeasuring. Although this statistical peculiarity is the result of random error and chance, it has been problematic across various medical, scientific, financial and psychological applications. In particular, RTM, if not taken into account, can interfere when researchers try to extrapolate results...
Correlation and Regression00:53

Correlation and Regression

In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a negative...
Prediction Intervals01:03

Prediction Intervals

The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y. 
The...
Truncation in Survival Analysis01:09

Truncation in Survival Analysis

Truncation in survival analysis refers to the exclusion of individuals or events from the dataset based on specific criteria related to the time of the event. This exclusion can happen in two primary forms: left truncation and right truncation.
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Related Experiment Videos

On robust regression with high-dimensional predictors.

Noureddine El Karoui1, Derek Bean, Peter J Bickel

  • 1Department of Statistics, University of California, Berkeley, CA 94720, USA. nkaroui@stat.berkeley.edu

Proceedings of the National Academy of Sciences of the United States of America
|August 20, 2013
PubMed
Summary
This summary is machine-generated.

We developed a new method to analyze large-scale regression M-estimates, finding a key variable that simplifies complex models. This reveals when least squares outperform other methods for specific error types.

Keywords:
concentration of measurehigh-dimensional statisticsprox function

Related Experiment Videos

Area of Science:

  • Statistics
  • Machine Learning
  • Econometrics

Background:

  • Regression M-estimates are crucial for robust statistical modeling.
  • Analyzing these estimates with large numbers of covariates (p) and observations (n) presents significant computational and theoretical challenges.
  • Existing methods often struggle with the high-dimensional regime where p approaches n.

Purpose of the Study:

  • To derive an exact stochastic representation for regression M-estimates in the large-p, large-n setting (p ≤ n).
  • To identify a central scalar random variable and its deterministic limit, rρ(κ), as the ratio p/n approaches κ > 0.
  • To characterize rρ(κ) using a novel nonlinear system of deterministic equations.

Main Methods:

  • Derivation of an exact stochastic representation for the M-estimate distribution.
  • Analysis of the limit of a scalar random variable as p/n → κ.
  • Development of a nonlinear system of two deterministic equations to characterize the limit rρ(κ).

Main Results:

  • An exact stochastic representation for regression M-estimates is established.
  • A scalar random variable and its deterministic limit, rρ(κ), are identified as central to the representation.
  • A nonlinear system of two deterministic equations is discovered, characterizing rρ(κ) and its dependence on the objective function (ρ) and statistical model.
  • It is shown that for large p/n ratios, least squares can be superior to least absolute deviations for double-exponential errors, a surprising finding.

Conclusions:

  • The study provides a powerful new framework for understanding regression M-estimates in high-dimensional settings.
  • The derived system of equations offers deeper insights into the behavior of M-estimates and their dependence on model specifics.
  • The findings challenge conventional wisdom, demonstrating that simpler methods like least squares can be optimal under certain conditions, particularly with specific error distributions.