Maximum Likelihood Estimator for a function

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SUMMARY

The discussion centers on deriving the Maximum Likelihood Estimator (MLE) for the given density function f(x) = ABB/xB+1, where A > 0 and B > 0. The attempt at a solution reveals that after differentiating the log-likelihood function ln f(x1,...,xn) with respect to A, the equation Bn/A = 0 indicates that there is no valid MLE for A. Participants question the presence of multiple variables "x" in the density function and seek clarification on the problem statement regarding the estimation of A.

PREREQUISITES
  • Understanding of Maximum Likelihood Estimation (MLE)
  • Familiarity with probability density functions
  • Knowledge of logarithmic differentiation
  • Basic statistics concepts, particularly related to estimators
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  • Learn about the implications of non-existence of MLEs in statistical models
  • Explore the derivation of log-likelihood functions for various distributions
  • Investigate the role of parameters in probability density functions
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Statisticians, data scientists, and students studying statistical estimation methods, particularly those focusing on Maximum Likelihood Estimation and its applications in probability theory.

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Homework Statement



Consider the following density function:
f(x) = ABB/xB+1; A<= x, zero elsewhere, where A > 0 and B> 0

Homework Equations




The Attempt at a Solution



f(x1,...,xn)= ABnBn(x1...xn)B+1

ln f(x1,...,xn)= Bn ln A + n ln B + (B+1)ln((x1...xn)

After differentiating with respect to A and setting to 0, Bn/A= 0. Therefore there is no maximum likelihood estimator for A?
 
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Where did all of the different "x"s come from? You seem to have your original density function to a power. Why?

Also please state the entire problem. Are you asking for the "maximum likelihood estimator of A?
 

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