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Results tagged with complete-statistics
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user 276612
A complete statistic T (in some statistical model) is such that for all functions g, if E g(T)=0 for all parameter values, then g is identically zero.
1
vote
Accepted
Gamma distribution: ratio of 2 CSS not containing $\beta$
Let $Y=X/\beta$. Then,
$$f_Y(y)=f_X(y\beta)\beta=\frac{(y\beta)^{\alpha-1}e^{-(\beta y/\beta)}}{\Gamma(\alpha)\beta^\alpha}$$
$$=\frac{y^{\alpha-1}e^{-y}}{\Gamma(\alpha)}$$
Note that the distribution …
2
votes
1
answer
145
views
Gamma distribution: ratio of 2 CSS not containing $\beta$
Let $X_1,...,X_n$ be iid and follow $Gamma(\alpha, \beta)$, where $$f(x,\alpha, \beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$ I already showed that $\overline{X}$ and $X^*=\lef …
0
votes
0
answers
50
views
Conditional distribution of complete sufficient statistics being ancillary of $\alpha$
Regarding the distribution and statistics as described here, I need to show that the conditional distribution of $\overline{X}$ given $X^*=x^*$ does not depend on $\alpha$. I remember my professor us …