Consistent estimator for parameter from Rayleigh distribution.

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SUMMARY

The discussion centers on the maximum likelihood estimator (MLE) for the parameter α of the Rayleigh distribution, defined by the formula \(\hat{\alpha}=\sqrt{\frac{\sum_{i=1}^{n}x_i^2}{2n}}\). The user seeks assistance in proving that this estimator is consistent. A key point raised is the necessity to demonstrate that \(E[X^2] = 2\alpha^2\) to establish the consistency of the estimator.

PREREQUISITES
  • Understanding of maximum likelihood estimation (MLE)
  • Familiarity with the Rayleigh distribution
  • Knowledge of statistical expectations and properties
  • Basic proficiency in mathematical proofs
NEXT STEPS
  • Study the properties of the Rayleigh distribution, focusing on its moments
  • Learn about the consistency of estimators in statistical theory
  • Review the derivation of expected values for continuous random variables
  • Explore examples of maximum likelihood estimation in statistical software
USEFUL FOR

Statisticians, data scientists, and researchers working with statistical estimators, particularly those focusing on the Rayleigh distribution and its applications in reliability engineering and signal processing.

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Welcome

Using MLE I found that estimator [tex]\alpha[/tex] parametr from Rayleigh distribution is described by formula
[tex]\hat{\alpha}=\sqrt{\frac{\sum_{n}^{i=1}x_i^2}{2n}}[/tex]
but I can't proof that this estimator is consistent estimator.

Would you be mind and help me with my problem.
 
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Wouldn't you just need to show that E[X^2] = 2a^2 ?
 

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