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
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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
2,5451635
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 |
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2 Answers
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active
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votes
2 Answers
2
active
oldest
votes
active
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active
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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
2,5451635
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 |
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