Mixing
Mixture Gaussian distribution quantiles
$begingroup$
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
265112
$endgroup$
add a comment |
$begingroup$
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
265112
$endgroup$
add a comment |
$begingroup$
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
265112
$endgroup$
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
265112
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265112
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265112
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265112
asked Mar 9 '13 at 11:00
GrzenioGrzenio
265112
265112
add a comment |
add a comment |
2 Answers
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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
2,5551635
$endgroup$
add a comment |
$begingroup$
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
9,72631128
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add a comment |
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2 Answers
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2 Answers
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active
oldest
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$begingroup$
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
2,5551635
$endgroup$
add a comment |
$begingroup$
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
2,5551635
$endgroup$
add a comment |
$begingroup$
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
2,5551635
$endgroup$
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
2,5551635
answered Mar 9 '13 at 12:40
BravoBravo
2,5551635
answered Mar 9 '13 at 12:40
BravoBravo
2,5551635
answered Mar 9 '13 at 12:40
BravoBravo
2,5551635
2,5551635
add a comment |
add a comment |
$begingroup$
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
9,72631128
$endgroup$
add a comment |
$begingroup$
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
9,72631128
$endgroup$
add a comment |
$begingroup$
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
9,72631128
$endgroup$
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
9,72631128
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,72631128
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,72631128
answered Aug 10 '17 at 11:36
kimchi loverkimchi lover
9,72631128
9,72631128
add a comment |
add a comment |
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