Convergences and Limit Theorems Motivation for Convergence Concepts Motivation for Convergence Concepts Question:How can one measure the closeness of Xn to u and the closeness of S2 to o2? [Randomness of Estimators]Since Xn and S2 are random variables,the convergence concepts for nonstochas- tic sequences do not apply. Both Xn and S2 are mappings from S to the real line; that is,Xn:S→R and S2:S→R+,where S is the sample space of the underlying random experiment. Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16,2020 26/186
Convergences and Limit Theorems Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16, 2020 26/186 Motivation for Convergence Concepts Motivation for Convergence Concepts
Convergences and Limit Theorems Motivation for Convergence Concepts Motivation for Convergence Concepts Suppose a random experiment is conducted,a basic outcome s∈S occurs..Then we observe a data set x"=(xri,···,rn), where xi=Xi(s).This is a realization of the random sam- ple X".From the data set x",we can compute an estimate n=Xn(s)for u and an estimate s2=S2(s)for o2.Different outcomes s will yield different estimates for u and o2 respec- tively for any given n. Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16,2020 27/186
Convergences and Limit Theorems Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16, 2020 27/186 Motivation for Convergence Concepts Motivation for Convergence Concepts
Convergences and Limit Theorems Motivation for Convergence Concepts Motivation for Convergence Concepts .[Sample Path]In fact,each basic outcome seS will generate a sequence of real numbers {n=Xn(s),n =1,2,...}and a se- quence of real numbers {s2=S2(s),n =1,2,...}respectively. These nonstochastic sequences are called a sample path for the sample mean Xn and a sample path for the sample vari- ance 52 respectively,when a basic outcome s occurs.There are many such sample paths for both Xn and 52 respectively, corresponding to different basic outcomes s e S. Stimulation of standard brownian motion sample path 1 sample path 2 sample path 3 02 0.5 0.g Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16,2020 28/186
Convergences and Limit Theorems Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16, 2020 28/186 Motivation for Convergence Concepts Motivation for Convergence Concepts
Convergences and Limit Theorems Motivation for Convergence Concepts Motivation for Convergence Concepts .We will develop various suitable convergence concepts and distance measures between Xn and u,or between S2 and o2.A common feature of these different con- vergence concepts is that they define that most non- stochastic sequences {Xn(s),n 1,2,...}and {2 (s),n l,2,···}converge to u and o2 respectively. Question:What is meant by "most nonstochastic sequences" for Xn or 52? Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16,2020 29/186
Convergences and Limit Theorems Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16, 2020 29/186 Motivation for Convergence Concepts Motivation for Convergence Concepts
CONTENTS 7.1 Limits and Orders of Magnitude:A Review 7.2 Motivation for Convergence Concepts 7.3 Convergence in Quadratic Mean and Lp-Convergence 7.4 Convergence in Probability 7.5 Almost Sure Convergence 7.6 Convergence in Distribution 7.7 Central Limit Theorems 7.8 Conclusion Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16,2020 30/186
Convergences and Limit Theorems Introduction to Statistics and Econometrics April 16, 2020 30/186 7.1 Limits and Orders of Magnitude: A Review 7.2 Motivation for Convergence Concepts 7.3 Convergence in Quadratic Mean and 𝑳𝑳𝒑𝒑-Convergence 7.4 Convergence in Probability 7.5 Almost Sure Convergence 7.6 Convergence in Distribution 7.7 Central Limit Theorems 7.8 Conclusion CONTENTS