Distribution of Maximun Likelihood Estimator

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SUMMARY

The discussion focuses on determining the maximum likelihood estimator (MLE) for the parameter \( b \) in a Pareto distribution defined by the probability density function \( f(x|b) = \frac{a b^a}{x^{a+1}} \) for \( x \geq b \). The MLE is established as \( b' = \min(X_i) \) for \( i = 1, 2, \ldots, n \), leveraging the monotonicity of the pdf with respect to \( b \). The participants explore how to demonstrate that \( b' \) follows a Pareto distribution by considering the cumulative distribution function (CDF) and the independence of the random variables involved.

PREREQUISITES
  • Understanding of Pareto distribution and its properties
  • Familiarity with maximum likelihood estimation (MLE) techniques
  • Knowledge of cumulative distribution functions (CDF) and probability concepts
  • Basic statistical inference and asymptotic distribution theory
NEXT STEPS
  • Study the derivation of the CDF for the Pareto distribution
  • Learn about the properties of minimum order statistics in random samples
  • Explore the concept of asymptotic distributions in MLE
  • Investigate the implications of independence in probability distributions
USEFUL FOR

Statisticians, data scientists, and researchers involved in statistical modeling, particularly those working with Pareto distributions and maximum likelihood estimation techniques.

Tranquillity
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Hey guys how are you? I have the following question:

Let X1,X2,...,Xn be a random sample from a Pareto distribution having pdf
f(x|b)= (a*b^a)/x^(a+1) where x>=b (1)

Determine the maximum likelihood estimator for b, say b' on (0,infinity) and by considering P(b'>x) or otherwise show that b' has the Pareto distribution with pdf given by (1) but with a replaced by an.


My attempt: I found the MLE as b'=min Xi where 1<=i<=n, since our pdf is monotonically increasing w.r.t b.

After that I know how to find the asymptotic distribution of the MLE using the formula including the expected information but then we say that MLE follows a normal distribution for large n.

How do I show that the MLE follows a Pareto distribution in this case? I am so struggled, any help would be much appreciated!

P.S The hint tells us to consider P(b'>x) but how can I find P(min Xi >x) and why should it help me?
 
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Hint: min Xi > x if and only if X1>x and X2>x etc.

If you write the distribution's cdf instead of pdf it'll be much easier.
 
Let say P(b'>x) = P(Xmin>x)= P(X1>x, X2>x,...,Xn>x)= P(X1>x) * P(X2>x) * ...*P(Xn>x)= [P(X1>x)]^n (since they are independent) But then what is P(X1>x)? Have I used correctly the independence?
 

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