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Let's say I have a statistic X which is distributed as a mixture of two chi-square distributions; $$X \sim 0.5\chi^2_1 + 0.5\chi^2_2.$$ I'm wondering how I can calculate the critical value(and p-value) for this statistic $X$ at a given significance level $\alpha$?

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    $\begingroup$ If the dfs of your two $\chi^2$ differ by 1, you could use emdbook::dchibarsq(). Related, but no answer: Mixture of two chi-squared distributions $\endgroup$
    – Stephan Kolassa
    Commented Aug 2, 2022 at 20:20
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    $\begingroup$ The p-value is easy, but the critical value has to be computed using numerical methods. See stats.stackexchange.com/a/411671/919 or stats.stackexchange.com/a/496147/919 for the details and working general-purpose code. $\endgroup$
    – whuber ♦
    Commented Aug 2, 2022 at 21:14
  • $\begingroup$ @whuber Which method would you recommend to use to find the p-value? $\endgroup$
    – Pame
    Commented Aug 5, 2022 at 10:16
  • $\begingroup$ The methods advocated in my posts on the subject: namely, use a well-crafted univariate root finder. $\endgroup$
    – whuber ♦
    Commented Aug 5, 2022 at 11:51

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