Mixture Gaussian distribution quantiles
Let $f_1(x), dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = sum_i w_i f_i(x)$ is also a density function and I call it mixture-Gaussian density.
It is easy to calculate central moments (e.g. mean) of this distribution when we know the central moments of the underlying normal distributions, using linearity of integrals:
$$int x^k g(x) dx = int x^k sum_i w_i f_i(x) dx = sum_i w_i int x^k f_i(x) dx$$
(please correct me if I am wrong).
How can I however calculate the quantiles of the new distribution (e.g. median)? Ideally I would like to get the quantile function, given quantile functions of the underlying normal distributions. Is there closed form solution? If not, what would be an efficient numerical solution?
probability probability-distributions normal-distribution
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
add a comment |
Let $f_1(x), dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = sum_i w_i f_i(x)$ is also a density function and I call it mixture-Gaussian density.
It is easy to calculate central moments (e.g. mean) of this distribution when we know the central moments of the underlying normal distributions, using linearity of integrals:
$$int x^k g(x) dx = int x^k sum_i w_i f_i(x) dx = sum_i w_i int x^k f_i(x) dx$$
(please correct me if I am wrong).
How can I however calculate the quantiles of the new distribution (e.g. median)? Ideally I would like to get the quantile function, given quantile functions of the underlying normal distributions. Is there closed form solution? If not, what would be an efficient numerical solution?
probability probability-distributions normal-distribution
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
add a comment |
Let $f_1(x), dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = sum_i w_i f_i(x)$ is also a density function and I call it mixture-Gaussian density.
It is easy to calculate central moments (e.g. mean) of this distribution when we know the central moments of the underlying normal distributions, using linearity of integrals:
$$int x^k g(x) dx = int x^k sum_i w_i f_i(x) dx = sum_i w_i int x^k f_i(x) dx$$
(please correct me if I am wrong).
How can I however calculate the quantiles of the new distribution (e.g. median)? Ideally I would like to get the quantile function, given quantile functions of the underlying normal distributions. Is there closed form solution? If not, what would be an efficient numerical solution?
probability probability-distributions normal-distribution
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
Let $f_1(x), dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = sum_i w_i f_i(x)$ is also a density function and I call it mixture-Gaussian density.
It is easy to calculate central moments (e.g. mean) of this distribution when we know the central moments of the underlying normal distributions, using linearity of integrals:
$$int x^k g(x) dx = int x^k sum_i w_i f_i(x) dx = sum_i w_i int x^k f_i(x) dx$$
(please correct me if I am wrong).
How can I however calculate the quantiles of the new distribution (e.g. median)? Ideally I would like to get the quantile function, given quantile functions of the underlying normal distributions. Is there closed form solution? If not, what would be an efficient numerical solution?
probability probability-distributions normal-distribution
probability probability-distributions normal-distribution
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265111
265111
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
Since $F(x)$ for normal distribution is strictly increasing, you can map from quantile function to the density function to obtain $mu_i$ and $sigma^2_i forall i$. You obtain the entire density function of $g(x)$ (which depends on only two moments) using $w_i$, $mu_i$ and $sigma^2_i forall i$. You could then map density function to the quantile function.
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
add a comment |
As far as I know there is no closed form solution. Bisection is a conservative way of solving it numerically, and coding up Newton's method should be no trouble, as all the constituent parts have explicit formulas.
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f325428%2fmixture-gaussian-distribution-quantiles%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Since $F(x)$ for normal distribution is strictly increasing, you can map from quantile function to the density function to obtain $mu_i$ and $sigma^2_i forall i$. You obtain the entire density function of $g(x)$ (which depends on only two moments) using $w_i$, $mu_i$ and $sigma^2_i forall i$. You could then map density function to the quantile function.
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
add a comment |
Since $F(x)$ for normal distribution is strictly increasing, you can map from quantile function to the density function to obtain $mu_i$ and $sigma^2_i forall i$. You obtain the entire density function of $g(x)$ (which depends on only two moments) using $w_i$, $mu_i$ and $sigma^2_i forall i$. You could then map density function to the quantile function.
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
add a comment |
Since $F(x)$ for normal distribution is strictly increasing, you can map from quantile function to the density function to obtain $mu_i$ and $sigma^2_i forall i$. You obtain the entire density function of $g(x)$ (which depends on only two moments) using $w_i$, $mu_i$ and $sigma^2_i forall i$. You could then map density function to the quantile function.
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
Since $F(x)$ for normal distribution is strictly increasing, you can map from quantile function to the density function to obtain $mu_i$ and $sigma^2_i forall i$. You obtain the entire density function of $g(x)$ (which depends on only two moments) using $w_i$, $mu_i$ and $sigma^2_i forall i$. You could then map density function to the quantile function.
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
answered Mar 9 '13 at 12:40
BravoBravo
2,5451635
2,5451635
add a comment |
add a comment |
As far as I know there is no closed form solution. Bisection is a conservative way of solving it numerically, and coding up Newton's method should be no trouble, as all the constituent parts have explicit formulas.
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
add a comment |
As far as I know there is no closed form solution. Bisection is a conservative way of solving it numerically, and coding up Newton's method should be no trouble, as all the constituent parts have explicit formulas.
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
add a comment |
As far as I know there is no closed form solution. Bisection is a conservative way of solving it numerically, and coding up Newton's method should be no trouble, as all the constituent parts have explicit formulas.
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
As far as I know there is no closed form solution. Bisection is a conservative way of solving it numerically, and coding up Newton's method should be no trouble, as all the constituent parts have explicit formulas.
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,69631128
9,69631128
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f325428%2fmixture-gaussian-distribution-quantiles%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
