Sufficient Statistic for Shape Parameter in Gamma Distribution | UPSC ISS 2024 Paper-2 | Problem-23
Автор: RitwikMath
Загружено: 2025-10-22
Просмотров: 58
Описание:
This video explains how to find a sufficient statistic for the shape parameter \(\theta\) of a Gamma distribution with PDF:
\[
f(x; \theta) = \frac{1}{\Gamma(\theta) \mu^\theta} x^{\theta - 1} e^{-x/\mu}, \quad x 0,
\]
where \(\mu\) is known. By analyzing the likelihood function of a sample \(X_1, X_2, \ldots, X_n\), it shows that the likelihood depends on the data only through the product
\[
T = \prod_{i=1}^n X_i.
\]
Thus, by the factorization theorem, \(T\) is a sufficient statistic for \(\theta\). This illustrates sufficient statistics identification in exponential family distributions, fundamental in statistical inference and relevant for UPSC preparations.
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