TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 12

CHAPTER 23 The Mean Squared Error as an Initial Criterion of Precision. The question how “close” two random variables are to each other is a central concern in statistics. The goal of statistics is to find observed random variables which are “close” to the unobserved parameters or random outcomes of interest. | CHAPTER 23 The Mean Squared Error as an Initial Criterion of Precision The question how close two random variables are to each other is a central concern in statistics. The goal of statistics is to find observed random variables which are close to the unobserved parameters or random outcomes of interest. These observed random variables are usually called estimators if the unobserved magnitude is nonrandom and predictors if it is random. For scalar random variables we will use the mean squared error as a criterion for closeness. Its definition is MSE read it mean squared error of as an estimator or predictor whatever the case may be of MSE fl E - 2 629 630 23. THE MEAN SQUARED ERROR AS AN INITIAL CRITERION OF PRECISION For our purposes therefore the estimator or predictor of the unknown parameter or unobserved random variable is no worse than the alternative if MSE MSE . This is a criterion which can be applied before any observations are collected and actual estimations are made it is an initial criterion regarding the expected average performance in a series of future trials even though in economics usually only one trial is made . . Comparison of Two Vector Estimators If one wants to compare two vector estimators say and it is often impossible to say which of two estimators is better. It may be the case that fi1 is better than f1 in terms of MSE or some other criterion but 2 is worse than 2. And even if every component is estimated better by than by certain linear combinations tT of the components of may be estimated better by tT than by tT . PROBLEM 294. 2 points Construct an example of two vector estimators and of the same random vector 1 2 so that MSE MSE U for i 1 2 but MSE 1 2 U 2 MSE 1 2 f1 2 - Hint it is easiest to use an example in which all random variables are constants. Another hint the geometric analog would be to find two vectors in a plane and . In each component . . COMPARISON OF TWO VECTOR ESTIMATORS 631 projection on the axes is

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