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* (i) State and prove the factorisation criterion for sufficient statistics, in the Case or discrete random variables. (ii) A linear function y = Ax + B with unknown coefficients A and B is repeatedly measured at distinct points x, .... x: first n, times at x, then n, times at x, and so on; and finally n times at x. The result of the ith measurement series is a sample Ya. ...., Yin,, i = 1, ..., k. The errors of all measurements are independent normal variables, with mean zero and variance 1. You are asked to estimate A and B from the whole sample yj. 1sjsn, lsisk. Prove that the maximum likelihood and the least squares estimators of (A, B) coincide and find these. Denote by A the maximum likelihood estimator of A and by B the maximum likelihood estimator of B. Find the distribution of (A, B).

* (i) State and prove the factorisation criterion for sufficient statistics, in the Case or discrete random variables. (ii) A linear function y = Ax + B with unknown coefficients A and B is repeatedly measured at distinct points x, .... x: first n, times at x, then n, times at x, and so on; and finally n times at x. The result of the ith measurement series is a sample Ya. ...., Yin,, i = 1, ..., k. The errors of all measurements are independent normal variables, with mean zero and variance 1. You are asked to estimate A and B from the whole sample yj. 1sjsn, lsisk. Prove that the maximum likelihood and the least squares estimators of (A, B) coincide and find these. Denote by A the maximum likelihood estimator of A and by B the maximum likelihood estimator of B. Find the distribution of (A, B).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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* (i) State and prove the factorisation criterion for sufficient statistics, in the case or discrete random variables. (ii) A linear function y= Ax + B with unknown coefficients A and B is repeatedly measured at distinct points x,, ..., xg: first n, times at x, then n, times at x, and so on; and finally n times at x. The result of the ith measurement series is a sample Ya..... Yin, » i= 1, ..., k. The errors of all measurements are independent normal variables, with mean zero and variance 1. You are asked to estimate A and B from the whole sample yij, 1sisn,, lsisk. Prove that the maximum likelihood and the least squares estimators of (A, B) coincide and find these. Denote by A the maximum likelihood estimator of A and by B the maximum likelihood estimator of B. Find the distribution of (Â, B).
Transcribed Image Text:* (i) State and prove the factorisation criterion for sufficient statistics, in the case or discrete random variables. (ii) A linear function y= Ax + B with unknown coefficients A and B is repeatedly measured at distinct points x,, ..., xg: first n, times at x, then n, times at x, and so on; and finally n times at x. The result of the ith measurement series is a sample Ya..... Yin, » i= 1, ..., k. The errors of all measurements are independent normal variables, with mean zero and variance 1. You are asked to estimate A and B from the whole sample yij, 1sisn,, lsisk. Prove that the maximum likelihood and the least squares estimators of (A, B) coincide and find these. Denote by A the maximum likelihood estimator of A and by B the maximum likelihood estimator of B. Find the distribution of (Â, B).
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