Cramer row inequality

The Cramer-Rao Inequality provides us with a lower bound on the variance of an unbiased estimator for a parameter.
The Cramer-Rao Inequality Let X = (X1,X2,. . ., Xn) be a random sample from a distribution with d.f. f(x|θ), where θ is a scalar parameter. Under certain regularity conditions on f(x|θ), for any unbiased estimator φˆ (X) of φ (θ)

1. Cramér - Rao Inequality   Subject:- PBD-2802 ADVANCE STATISTICAL METHODS Presented by : Vashu Gupta 2020-PBD-012

2. Contents :-

3. Introduction:- The Cramer-Rao Inequality provides us with a lower bound on the variance of an unbiased estimator for a parameter. The Cramer-Rao Inequality Let X = (X1,X2,. . ., Xn) be a random sample from a distribution with d.f. f(x| θ), where θ is a scalar parameter. Under certain regularity conditions on f(x| θ), for any unbiased estimator φˆ ( X) of φ (θ) Where

4. Continue… Most Efficient estimator :- Note that 0 ≤ eff(φˆ) ≤ 1 for all θ ∈ Θ

5. Continue… The quantity I(θ) = E(S 2 (X)) is called the Fisher information on the parameter θ contained in the sample X. The Fisher information is in general, a function of the value of θ. The higher the Fisher information i.e. the more information there is in the sample, the smaller the CRLB and consequently the smaller the variance of the most efficient estimator.

6. Basic Conditions :- We further make the following assumptions, which are known as the Regularity conditions for Cramer-Roo Inequality. The parameter space e is a non-degenerate open interval on the real line For almost all , and for all exists, the exceptional set, if any, is independent of

7. Continue… The range of Integration is independent of the parameter , so that is differentiable under integral sign. The conditions of uniform convergence of integrals are satisfied so that differentiation under the integral sign is valid. exists and is positive for all .

8. Conditions for the Equality sign in Cramer-Rao Inequality The sign of equality will hold in C.R. Inequality if and only if the sign of equality holds in The sign o f equality will hold in by Cauchy Schwartz inequality, if and only if the variables and are proportional to each other

9. Example of Cramer-Rao inequality:- If we have in Poisson distribution so find the minimum variance unbiased estimator. Solution:- firstly we have pdf of Poisson distribution is And find likelihood function:

10. Continue… Let log in both side, Differentiating in both side- =

11. Continue… Now we can say that, S is a MUVE of . ={CRLB} Cramer Rao lower bond

12. References:- https://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/162/Handouts/CramerRaoHandout08.pdfhttps://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/162/Handouts/CramerRaoHandout08.pdf https://www.dcpehvpm.org/E-Content/Stat/FUNDAMENTAL%20OF%20MATHEMATICAL%20STATISTICS-S%20C%20GUPTA%20&%20V%20K%20KAPOOR.pdf

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