Suppose x and y have such a limit distribution then x

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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,...
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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.

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