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Let $X_1\sim N(0,\sigma_1^2), X_2\sim N(0,\sigma_2^2),\dots,X_n\sim N(0,\sigma_n^2)$ where generally $\sigma_1\neq\sigma_2\neq\sigma_3\dots\neq\sigma_n$. What is the distribution of the statistic $$Y = \sqrt{\sum_{k=1}^n X_k^2}?$$

The distribution for $Y^2$ is obtained by a combination of Distribution of sum of squares of normals that have mean zero but not variance one? and General sum of Gamma distributions, but I don't know enough about characteristic functions to see if there's any easy way of incorporating the square root.

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JMJ
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  • The [chi distribution](https://en.m.wikipedia.org/wiki/Chi_distribution) is probably what you want. – Greenparker Apr 02 '16 at 00:29
  • It would be a good idea to clarify whether you want the $X_i$ to be independent: I can't see that in your question at the moment. – Silverfish Apr 02 '16 at 00:30
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    By the way, on this site there's no need to say "thank you" at the end of your post - it might seem rude at first, but it's part of the philosophy of this site ([tour]) to "Ask questions, get answers, no distractions" and it means future readers of your question don't need to read through the pleasantries. – Silverfish Apr 02 '16 at 00:31
  • Yes, I do mean independent. The chi distribution seems good if all of the distributions are standard normal, but does it still work in this more general case? Sorry....I try to be polite in my first posts, before I know a place. – JMJ Apr 02 '16 at 00:34
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    Greenparker was absolutely right for my first typing of the formula, in which I absent mindedly divided each of the $X_i$ by its standard deviation. The currently typed version is the actual statistic I am interested in. – JMJ Apr 02 '16 at 00:39
  • Another way of thinking about this question is, "what is the distribution of the Euclidean norm of a zero centered multivariate normal with diagonal covariance". – Greenparker Apr 02 '16 at 02:04
  • Yes, Indeed this is the application which it has come from. – JMJ Apr 02 '16 at 06:46
  • The distribution of $Y^2$ is found at http://stats.stackexchange.com/questions/72479 (because the $X_i^2$ have Gamma distributions). The form of the answer suggests you won't find a nice closed analytical solution. The saddlepoint approximation suggested in the answer by kjetil b halvorsen might be appealing. – whuber Apr 02 '16 at 21:55
  • Are the variances n=known or unknown? – Michael R. Chernick Sep 30 '18 at 23:30

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