STAT 400 Lecture AL1 4 Spring 2015 Dalpiaz Answers for 6 4 Part 2 Let X 1 X 2 X n be a random sample of size n from the distribution with probability density function 1 1 x f x 0 0 x 1 otherwise 0 Recall Maximum likelihood estimator of is Method of moments estimator of is Def 1 n ln X i n i 1 1 X 1 1 X X E X 1 1 An estimator is said to be unbiased for if E for all unbiased biased unbiased large variance small variance small variance d Is unbiased for That is does E equal 1 1 E ln X 1 ln x f X x dx ln x x 0 1 dx Integration by parts Choice of u b L A T E b b u dv u v a v du a a 1 u ln x dv 1 1 du 1 x v x dx 1 1 E ln X 1 ln x x 0 1 1 x 0 x 1 1 x 1 1 dx x 1 1 1 1 x dx 0 0 x 1 1 1 1 1 1 dx x dx x 1 0 1 0 Therefore E n n E ln X i n i 1 n i 1 1 that is is an unbiased estimator for dx 1 dx ln x x ogarithmic lgebraic rigonometric xponential 1 Jensen s Inequality If g is convex on an open interval I and X is a random variable whose support is contained in I and has finite expectation then E g X g E X If g is strictly convex then the inequality is strict unless X is a constant random variable E X 2 E X 2 Var X 0 E etX MX t 1 1 E E X X for a positive random variable X E X 3 E X 3 for a non negative random variable X E ln X ln E X for a positive random variable X E X e t E X E X et for a non negative random variable X e Is unbiased for That is does E equal Since g x 1 x x 1 x 1 0 x 1 is strictly convex and X is not a constant random variable by Jensen s Inequality E E g X g E X is NOT an unbiased estimator for 6 Let X 1 X 2 X n be a random sample of size n from a population with mean and variance 2 Show that the sample mean X and the sample variance S 2 are unbiased for and 2 respectively X X 1 X 2 X n n E X1 X2 Xn n E X Var X E X 2 Var X E X 2 2 2 Var X 1 X 2 X n n 2 2 n 2 2 E X Var X E X 2 2 n S2 1 1 2 2 Xi X X i2 n X n 1 n 1 E S2 1 2 E X i2 n E X n 1 2 2 1 2 2 2 n n n 1 n For an estimator of define the Mean Squared Error of by MSE E 2 E 2 E 2 Var bias 2 Var 7 Let X 1 X 2 X n be a random sample of size probability density function f X x f X x a 1 1 1 x 1 1 1 X x2 x3 1 x dx 4 6 2 3 1 1 3 3 X Is an unbiased estimator for Justify your answer E E 3 X 3 E X 3 3 3 an unbiased estimator for c from a distribution with Obtain the method of moments estimator of E X x b 1 x 2 n Find Var x3 x4 1 x dx E X x 8 2 6 1 2 1 2 2 Var X 1 3 3 2 Var 9 Var X 9 1 1 1 3 3 2 9 2 3 2 n n MSE 3 2 n 8 Let X 1 X 2 be a random sample of size density function f X x f X x 1 x 2 n 2 from a distribution with probability 1 x 1 1 1 Find the maximum likelihood estimator of L 1 x1 x 2 2 x1 x 2 1 x1 1 x 2 4 2 2 L is a parabola with vertex at Case 1 a x 1 x 2 0 x1 x 2 b 2 x1 x 2 2a Parabola has its antlers up Subcase 1 x 1 0 x 2 0 The vertex is the minimum x x2 Vertex 1 0 2 x1 x 2 Maximum of L on 1 1 is at 1 Subcase 2 x 1 0 x 2 0 x x2 Vertex 1 0 2 x1 x 2 Maximum of L on 1 1 is at 1 Case 2 a x 1 x 2 0 Parabola has its antlers down x x2 Vertex is at 1 2 x1 x 2 The vertex is the maximum Subcase 1 x x2 x1 1 That is x 2 1 2 x1 x 2 2 x1 1 Maximum of L on 1 1 is at 1 Subcase 2 x x2 x1 1 That is x 2 1 2 x1 x 2 2 x1 1 Maximum of L on 1 1 is at 1 Subcase 3 x x2 1 1 1 2 x1 x 2 Maximum of L on 1 1 is at Pink 1 Purple 1 Green X1 X 2 2X1 X 2 X1 X 2 2X1 X 2 9 Let X 1 X 2 X n be a random sample from the distribution with probability density function f x 4 x 3 e x a 4 Obtain the maximum likelihood estimator of n x i4 3 L 4 xi e i 1 n 0 x 0 n ln L n ln ln 4 x i3 x i4 i 1 i 1 n n 4 x 0 i 1 i ln L n n X i4 i 1 b Find E X k k 4 E Xk x k 4 x 3 e x dx 4 u x4 du 4 x 3 dx 0 u 0 c k 4 e u du 1 k 1 k 4 4 Find the method of moments estimator of E X E X1 X 1 25 1 4 1 0 9064 1 1 1 25 1 4 4 1 4 1 4 1 1 25 X 4 0 675 X 4
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