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Understanding Empirical Distribution Function in Statistics

Learn about the empirical distribution function, its theorems, and statistical functions. Explore topics such as mean, variance, median, and plug-in estimator. Discover examples and insights on statistical functionals and correlation.

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Understanding Empirical Distribution Function in Statistics

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  1. All of Statistics:Chapter 7 Toby Xu UW-Madison 07/02/07

  2. The Empirical Distribution Function • Def: The Empirical distribution function is the CDF that puts mass 1/n at each data point Xi. Formally, • Where

  3. Theorems: The supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. • MSE= • DKW inequality:

  4. DKW: confidence level • L(x)=max{ ,0} • U(x)=min{ ,1} • Where • For an F

  5. Statistical Functions • A statistical function T(F) is any function of F. • Mean: • Variance : • Median : m=F-1(1/2) • Plug-in estimator of is defined by • If for some function r(x) then T is called a linear function

  6. Statistical Functionals continued • The plug-in estimator for linear functional • Assume we can find se, then for many cases: • Normal-based interval for 95% CL

  7. Examples: • The Mean: let , the plug-in estimator is . • The Variance:

  8. Examples Continued • The Skewness: • Correlation:

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