# Suppose x and y have such a limit distribution then x

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: percent about 365*(0.5) times, and to decrease by one percent about 365*(0.1) times. That leads to a year end value of (1.01)182.5 (0.99)36.5 = 4.26.) 162 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES 4.10 Minimum mean square error estimation 4.10.1 Constant estimators Let Y be a random variable with some known distribution. Suppose Y is not observed but that we wish to estimate Y . If we use a constant δ to estimate Y , the estimation error will be Y − δ . The mean square error (MSE) for estimating Y by δ is deﬁned by E [(Y − δ )2 ]. By LOTUS, if Y is a continuous-type random variable, ∞ (y − δ )2 fY (y )dy. MSE (for estimation of Y by a constant δ ) = (4.25) −∞ We seek to ﬁnd δ to minimize the MSE. Since (Y − δ )2 = Y 2 − 2δY + δ 2 , we can use linearity of expectation to get E [(Y − δ )2 ] = E [Y 2 ] − 2δE [Y ] + δ 2 . This is quadratic in δ , and the derivative with resect to δ is −2E [Y ] + 2δ. Therefore the minimum occurs at δ ∗ = E [Y ]. For this value of δ , the MSE is E [(Y − δ ∗ )2 ] = Var(Y ). In summary,...
View Full Document

## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

Ask a homework question - tutors are online