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b.lab-Ch4a - Outline Preliminaries ”Proof” of E S 2 =...

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Unformatted text preview: Outline Preliminaries ”Proof” of E ( S 2 ) = σ 2 ”Proof” of Var ( X + Y ) = Var ( X- Y ) = σ 2 X + σ 2 Y for X , Y indep. Lab Assignment #4. Due date: 10/21. Email to Ms. Kraus. Lab4: Introduction to Statistical Simulations M. George Akritas M. George Akritas Lab4: Introduction to Statistical Simulations Outline Preliminaries ”Proof” of E ( S 2 ) = σ 2 ”Proof” of Var ( X + Y ) = Var ( X- Y ) = σ 2 X + σ 2 Y for X , Y indep. Lab Assignment #4. Due date: 10/21. Email to Ms. Kraus. Preliminaries ”Proof” by Simulation Matrices in R The apply function ”Proof” of E ( S 2 ) = σ 2 ”Proof” of Var( X + Y ) = Var( X- Y ) = σ 2 X + σ 2 Y for X , Y indep. Lab Assignment #4. Due date: 10/21. Email to Ms. Kraus. M. George Akritas Lab4: Introduction to Statistical Simulations Outline Preliminaries ”Proof” of E ( S 2 ) = σ 2 ”Proof” of Var ( X + Y ) = Var ( X- Y ) = σ 2 X + σ 2 Y for X , Y indep. Lab Assignment #4. Due date: 10/21. Email to Ms. Kraus. ”Proof” by Simulation Matrices in R The apply function I A simulation consists of repeated generation of random samples and application of a statistical procedure on each sample. I It is used for providing numerical evidence in support of, or in contradiction to, certain probabilistic/statistical claims, or simply for investigating properties of a statistical procedure. I We will use simulations to verify that E ( S 2 ) = σ 2 , while E ( S ) 6 = σ. I In the process we will also verify that 1. As the sample size n → ∞ ), S 2 → σ 2 , and S → σ. 2. For independent X 1 , X 2 , Var( X 1 + X 2 ) = Var( X 1- X 2 ) = σ 2 1 + σ 2 2 M. George Akritas Lab4: Introduction to Statistical Simulations Outline Preliminaries ”Proof” of E ( S 2 ) = σ 2 ”Proof” of Var ( X + Y ) = Var ( X- Y ) = σ 2 X + σ 2 Y for X , Y indep. Lab Assignment #4. Due date: 10/21. Email to Ms. Kraus. ”Proof” by Simulation Matrices in R The apply function New terminology: Consistent and Unbiased estimators I Because S 2 → σ 2 , as n → ∞ , we say that S 2 is a consistent estimator of σ 2 . S is also a consistent estimator of σ . I Because E ( S 2 ) = σ 2 , we say that S 2 is an unbiased estimator of σ 2 . I Because E ( S ) 6 = σ , S is a biased estimator of σ ....
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b.lab-Ch4a - Outline Preliminaries ”Proof” of E S 2 =...

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