Is it possible to prove that a $Beta(\alpha,\beta)$ distributed random variable is always stochastically dominated by a $Beta(\alpha',\beta')$ if $\alpha'>\alpha$ and $\beta'<\beta$? I've read it in an article but I couldn't find any bibliography about. The proof is straightforward whenever $\alpha'+\beta'=\alpha+\beta$ using the Beta-Binomial identity and exploiting the fact that for $n ≥ m, q ≥ p ⇒ Bin(n, q) \succeq Bin(m, p)$. But I couldn't prove this fact whenever such a bound isn't enforced.
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$\newcommand\al\alpha\newcommand\be\beta$Let $f_{\al,\be}$ be the pdf of the beta distribution with parameters $\al,\be$. Here we have a monotone likelihood ratio, in the sense that $$\frac{f_{\al',\be'}(x)}{f_{\al,\be}(x)} =c_{\al,\be,\al',\be'}\, x^{\al'-\al}(1-x)^{\be'-\be}$$ is increasing in $x\in(0,1)$ if $\al'>\al$ and $\be'<\be$, where $c_{\al,\be,\al',\be'}$ is a positive real number not depending on $x$. So, the desired stochastic domination follows.
