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I seek a generalization of the idea and answer found in Stochastic order of generalized chi-square distributions. Given that $χ^2_a=∑^n_{i=1}a_iZ^2_i$ where $Z_i∼N(0,1)$ are i.i.d. $\forall i$. The answerer found through a majorization argument and Schur-concavity of the variable's CDF that

$$1/a≺1/b⟹P(χ_a≤t)≥P(χ_b≤t) \quad ∀t≥0.$$

So the notation is clear, I start with the general definition of the generalized chi square variable from https://en.wikipedia.org/wiki/Generalized_chi-squared_distribution. We first have

\begin{align*} \tilde\chi^2(\tilde w,\tilde k,\tilde\lambda,s,m)=\sum_{i=1}^n w_i\chi^{2}(k_i,\lambda_i)+sZ+m, \end{align*} where $Z$ is standard normal and independent from $\chi^{2}(k_i,\lambda_i)$, which is non-central chi-square with $k_i$ degrees of freedom and non-centrality parameter $\lambda_i$.

Now instead of summing $n$ variables like in the most general case, I reduce this to just one variable with these parameters, but not involving $Z$ (i.e. $s=0$) and setting $k=1$. Now for two specifications of this variable to be stochastically ordered, this gives \begin{align*} \chi^2_1\left(w_1,k_1:=1,\lambda_1,s_1:=0,m_1\right) \\ \chi^2_2\left(w_2,k_2:=1,\lambda_2,s_2:=0,m_2\right) \end{align*}

So now instead of only using scale parameters $w_1$ and $w_2$ to show $\chi^2_1$ dominates $\chi^2_2$ like in the original problem, I ask:

Does there exist an inequality-based relation between parameter sets $\left(w_1,\lambda_1,m_1\right)$ and $\left(w_2,\lambda_2,m_2\right)$ such that

$$P(χ_1≤t)≥P(χ_2≤t) \quad ∀t≥0.$$

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  • $\begingroup$ Thanks for your interest. You will find I edited the post so everything should be more clear. I apologize if the setup is a bit wordy but I'm not an expert in the related notation. In the old post $t$ was time and $f,g,h$ were specific to a paper I'm writing for a stochastic optimization problem for prediction with linear models. These functions are interesting, but I don't believe they are central to the stochastic dominance question. Does this help? $\endgroup$ Commented yesterday

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$\newcommand\la\lambda$The random variables (r.v.'s) in question are of the form $X_{w,m,\la}:=wX_\la+m$, where $X_\la\sim\chi_1^2(\la)$.

First, trivial observations. Obviously, $X_{w,m,\la}$ is increasing in $m$ and hence stochastically increasing in $m$. Also, because $X_\la\ge0$, $X_{w,m,\la}$ is increasing in $w$ and hence stochastically increasing in $w$.

It is a bit less trivial that $X_\la$ is stochastically increasing in $\la$, and hence $X_{w,m,\la}$ is stochastically increasing in $\la$ for each pair $(w,m)$ such that $w\ge0$. One way to show this is to recall that the noncentral chi-squared distribution $\chi_1^2(\la)$ is the distribution of the r.v. $Y_{1+N_{\la/2}}$, where $Y_n$ has the central chi-squared with $n$ degrees of freedom (d.f.'s) and $N_{\la/2}$ is a r.v. with the Poisson distribution with mean $\la/2$. The precise meaning of the latter informal statement is that, for each nonnegative integer $k$, the conditional distribution of the r.v. $Y_{1+N_{\la/2}}$ given that $N_{\la/2}=k$ is the distribution of $Y_{1+k}$ -- that is, the central chi-squared distribution with $1+k$ d.f.'s. It remains to note the following:

  • $Y_n$ is stochastically increasing in $n$, because $Y_n$ equals $Z_1^2+\cdots+Z_n^2$ in distribution, where $Z,Z_1,Z_2,\ldots$ are independent standard normal r.v.'s.
  • A Poisson r.v. $N_{\mu}$ with mean $\mu$ is stochastically increasing in $\mu$, say because for any real $\mu$ and $\nu$ such that $0<\mu<\nu$, the r.v. $N_{\nu}$ equals $N'_{\mu}+N''_{\nu-\mu}$ in distribution (and $N''_{\nu-\mu}\ge0$), where $N'_{\mu}$ and $N''_{\nu-\mu}$ are independent Poisson r.v.'s with respective means $\mu$ and $\nu-\mu$.

A more direct (but more calculational) way to prove that that $X_\la$ is stochastically increasing in $\la$ is as follows. Without loss of generality (wlog), $$X_\la=(Z+\mu)^2,$$ where $\mu:=\sqrt\la$. It remains to note that, for each real $t\ge0$, $$p(\mu):=P(X_\la\le t^2)=P(\mu-t\le Z\le\mu+t)$$ and $p'(\mu)=g(\mu+t)-g(\mu-t)\le0$, where $g$ is the standard normal p.d.f.

It follows that, more generally, $X_{d,\la}$ is stochastically increasing in $\la$ for each natural $d$, where $X_{d,\la}$ is a r.v. with the noncentral chi-squared distribution $\chi_d^2(\la)$ with $d$ d.f.'s. Indeed, if $d\ge2$, then wlog $$X_{d,\la}=(Z_1+\mu)^2+Z_2^2+\cdots+Z_d^2,$$ again with $\mu:=\sqrt\la$.

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    $\begingroup$ This makes a lot of sense now. Thanks for your time. $\endgroup$ Commented 17 hours ago

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