Finding the Minimum Mean Square Estimator for Scalar Parameter w

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SUMMARY

The discussion focuses on finding the Minimum Mean Square Estimator (MMSE) for the scalar parameter \( w \) based on the observation \( z = \ln w + n \). The probability density functions are defined as \( f(w) = 1 \) for \( 0 \leq w \leq 1 \) and \( f(n) = e^{-n} \) for \( n \geq 0 \). The user attempted to derive the estimator using the relationship \( f(z/w) = (f(n)) / g'(n) \) at \( n = z - \ln w \). Clarification is sought regarding the correctness of this approach and the derivation of the MSE.

PREREQUISITES
  • Understanding of Minimum Mean Square Estimation (MMSE)
  • Knowledge of probability density functions (PDFs)
  • Familiarity with scalar random variables and their transformations
  • Basic calculus, particularly differentiation and integration
NEXT STEPS
  • Study the derivation of the MMSE for scalar parameters in statistical estimation
  • Learn about the properties of probability density functions and their applications
  • Explore the concept of I.I.D. (Independent and Identically Distributed) random variables
  • Investigate methods for calculating Mean Square Error (MSE) in estimators
USEFUL FOR

Statisticians, data scientists, and researchers involved in statistical estimation and analysis of random variables will benefit from this discussion.

sant142
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I am not able to understand how to go about this problem:

Find the minimum mean square estimator for the scalar parameter w based
on the scalar observation z = ln w + n where
f(w) =1 if 0<=w<=1;
0 else:

and
f(n) =e^-n if n>= 0;
0 else

I did f(z/w) = (f(n)) /g'(n) at n = z- ln w

Am i wrong?
 
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Hey sant142 and welcome to the forums.

Can you show us what estimator you got (as a function of a random sample I.I.D) and the calculations you obtained for the MSE (and proof that it is MMSE)?
 

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