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Additional resources for Basics of Modern Mathematical Statistics: Exercises and Solutions
10) for some functions C. / and B. / on ‚ and a function `. / on R. x/ D log 1= 2 . xŠ/. 13. PÂ ; Â 2 ‚ R/. Let also a function ı W R ‚ ! x Â/ C c; where a; b; c 2 R. 1. x; Â/ is a contrast. 2. Â; 1/. 1. d. 14) Constant c can be chosen arbitrary. 2. 14) in this case yields b D 0. x; Â/ with b D 0 and any constants a and c is a contrast for the Gaussian shift. 14. d. PÂ ; Â 2 ‚ R/. R 1. Xi /: i D1 Show that this estimate can be obtained as the M-estimate for a properly selected function . /. R 2.
0; i2 /. Compute the MLE ÂQ of the parameter Â and prove that this estimate has the following properties: (a) The estimate ÂQ is unbiased: EÂ ÂQ D Â . Â; Â j2 D . Â/ D ; 2 N Â i P 2 i . d. Model 17 Note that random value Xi Â has a normal distribution with zero mean and variance i2 . 6. d. sample with distribution that depends on some parameter Â. Let ÂOn be an estimate of parameter Â. e. ÂOn d Â/ ! Â/2 ; n ! 1: Prove that ÂOn is consistent, P ÂOn ! Â This fact can be briefly formulated as “root-n normality implies consistency”.
In the case of the Poisson distribution, L1 . 1/ Xi Š and L2 . e. j / def Xi D ; j D 1; 2. 17. d. ˛; ˇ/. Find estimators for the multivariate parameter Â using the following approaches: 1. Maximum likelihood approach; 2. Method of moments. 1. n/. n/ . d. n/. ˛; Q ˇ/ Pn ! n/ : 2. d. 18). 18. d. 2 / 1 e jx Â j= : Compute the maximum likelihood estimate for the parameter Â. This model is known as a shift of a Laplace law. n/ and consider two cases. 1. Suppose that n is even. Denote k D n=2 2 N. d.