Bounding the Cokurtosis
$begingroup$
Summary
I have four random variables $A_1$, $A_2$, $B_1$, $B_2$ and want to bound their Cokurtosis
$K(A_1, A_2, B_1, B_2) := Eleft((A_1-E(A_1))(A_2-E(A_2))(B_1-E(B_1))(B_2-E(B_2)) right)$.
I am interested in a bound for $K$ deptendent on the expected values, variances, and correlation coefficients of the random variables or the kurtoses of the individual random variables - somewhat like here for the covariance. Though a general bound would suffice, tighter bounds exploiting the details below would be preferred.
Details
$A_1$ and $A_2$ are i.i.d. normally distributed with mean $mu_A$ and
variance $sigma^2_A$.
$B_1$ and $B_2$ are identically distributed with mean $mu_B$ and variance $sigma^2_B$.- $0leq B_1, B_2 leq 1$
- The covariance between one of the $A$ variables and one of the $B$ variables is $C_{AB}$.
- The covariance between the $B$ variables is $C_{BB}$.
My ultimate goal is to approximate the covariance $Cov(A_1 B_1, A_2 B_2)$, but I found I would the cokurtosis fo this purpose.
My thoughts so far
I am trying to consider a "worst case", in which the expected value is maximized. Since $0leq B_1, B_2 leq 1$, it is $|B_1-E(B_1)|leq 1$.
Then by Jensen's inequality,
$K(A_1, A_2, B_1, B_2) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| cdot |B_1-E(B_1)| cdot |B_2-E(B_2)| right) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| right) = frac{2 sigma^2_a}{pi}.$
For the last equality I used that $A_1$ and $A_2$ are independent and that the mean of the half-normal distribution is known.
My Questions
- Do you see any issues with my derivation?
- Do you have an idea of how to obtain a possibly sharper bound? In Bohrnstedt & Goldberger (1969) I have read that the third moment vanishes for multivariate normal distributions. Can something similar be exploited here?
Bohrnstedt & Goldberger cite
Anderson, T. W. An Introduction to Multivariate Statistical Analysis.
New York: John Wiley and Sons, 1958.
on page 39, but I do not have access to that book.
statistics upper-lower-bounds expected-value
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
$endgroup$
add a comment |
$begingroup$
Summary
I have four random variables $A_1$, $A_2$, $B_1$, $B_2$ and want to bound their Cokurtosis
$K(A_1, A_2, B_1, B_2) := Eleft((A_1-E(A_1))(A_2-E(A_2))(B_1-E(B_1))(B_2-E(B_2)) right)$.
I am interested in a bound for $K$ deptendent on the expected values, variances, and correlation coefficients of the random variables or the kurtoses of the individual random variables - somewhat like here for the covariance. Though a general bound would suffice, tighter bounds exploiting the details below would be preferred.
Details
$A_1$ and $A_2$ are i.i.d. normally distributed with mean $mu_A$ and
variance $sigma^2_A$.
$B_1$ and $B_2$ are identically distributed with mean $mu_B$ and variance $sigma^2_B$.- $0leq B_1, B_2 leq 1$
- The covariance between one of the $A$ variables and one of the $B$ variables is $C_{AB}$.
- The covariance between the $B$ variables is $C_{BB}$.
My ultimate goal is to approximate the covariance $Cov(A_1 B_1, A_2 B_2)$, but I found I would the cokurtosis fo this purpose.
My thoughts so far
I am trying to consider a "worst case", in which the expected value is maximized. Since $0leq B_1, B_2 leq 1$, it is $|B_1-E(B_1)|leq 1$.
Then by Jensen's inequality,
$K(A_1, A_2, B_1, B_2) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| cdot |B_1-E(B_1)| cdot |B_2-E(B_2)| right) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| right) = frac{2 sigma^2_a}{pi}.$
For the last equality I used that $A_1$ and $A_2$ are independent and that the mean of the half-normal distribution is known.
My Questions
- Do you see any issues with my derivation?
- Do you have an idea of how to obtain a possibly sharper bound? In Bohrnstedt & Goldberger (1969) I have read that the third moment vanishes for multivariate normal distributions. Can something similar be exploited here?
Bohrnstedt & Goldberger cite
Anderson, T. W. An Introduction to Multivariate Statistical Analysis.
New York: John Wiley and Sons, 1958.
on page 39, but I do not have access to that book.
statistics upper-lower-bounds expected-value
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
$endgroup$
add a comment |
$begingroup$
Summary
I have four random variables $A_1$, $A_2$, $B_1$, $B_2$ and want to bound their Cokurtosis
$K(A_1, A_2, B_1, B_2) := Eleft((A_1-E(A_1))(A_2-E(A_2))(B_1-E(B_1))(B_2-E(B_2)) right)$.
I am interested in a bound for $K$ deptendent on the expected values, variances, and correlation coefficients of the random variables or the kurtoses of the individual random variables - somewhat like here for the covariance. Though a general bound would suffice, tighter bounds exploiting the details below would be preferred.
Details
$A_1$ and $A_2$ are i.i.d. normally distributed with mean $mu_A$ and
variance $sigma^2_A$.
$B_1$ and $B_2$ are identically distributed with mean $mu_B$ and variance $sigma^2_B$.- $0leq B_1, B_2 leq 1$
- The covariance between one of the $A$ variables and one of the $B$ variables is $C_{AB}$.
- The covariance between the $B$ variables is $C_{BB}$.
My ultimate goal is to approximate the covariance $Cov(A_1 B_1, A_2 B_2)$, but I found I would the cokurtosis fo this purpose.
My thoughts so far
I am trying to consider a "worst case", in which the expected value is maximized. Since $0leq B_1, B_2 leq 1$, it is $|B_1-E(B_1)|leq 1$.
Then by Jensen's inequality,
$K(A_1, A_2, B_1, B_2) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| cdot |B_1-E(B_1)| cdot |B_2-E(B_2)| right) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| right) = frac{2 sigma^2_a}{pi}.$
For the last equality I used that $A_1$ and $A_2$ are independent and that the mean of the half-normal distribution is known.
My Questions
- Do you see any issues with my derivation?
- Do you have an idea of how to obtain a possibly sharper bound? In Bohrnstedt & Goldberger (1969) I have read that the third moment vanishes for multivariate normal distributions. Can something similar be exploited here?
Bohrnstedt & Goldberger cite
Anderson, T. W. An Introduction to Multivariate Statistical Analysis.
New York: John Wiley and Sons, 1958.
on page 39, but I do not have access to that book.
statistics upper-lower-bounds expected-value
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
$endgroup$
Summary
I have four random variables $A_1$, $A_2$, $B_1$, $B_2$ and want to bound their Cokurtosis
$K(A_1, A_2, B_1, B_2) := Eleft((A_1-E(A_1))(A_2-E(A_2))(B_1-E(B_1))(B_2-E(B_2)) right)$.
I am interested in a bound for $K$ deptendent on the expected values, variances, and correlation coefficients of the random variables or the kurtoses of the individual random variables - somewhat like here for the covariance. Though a general bound would suffice, tighter bounds exploiting the details below would be preferred.
Details
$A_1$ and $A_2$ are i.i.d. normally distributed with mean $mu_A$ and
variance $sigma^2_A$.
$B_1$ and $B_2$ are identically distributed with mean $mu_B$ and variance $sigma^2_B$.- $0leq B_1, B_2 leq 1$
- The covariance between one of the $A$ variables and one of the $B$ variables is $C_{AB}$.
- The covariance between the $B$ variables is $C_{BB}$.
My ultimate goal is to approximate the covariance $Cov(A_1 B_1, A_2 B_2)$, but I found I would the cokurtosis fo this purpose.
My thoughts so far
I am trying to consider a "worst case", in which the expected value is maximized. Since $0leq B_1, B_2 leq 1$, it is $|B_1-E(B_1)|leq 1$.
Then by Jensen's inequality,
$K(A_1, A_2, B_1, B_2) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| cdot |B_1-E(B_1)| cdot |B_2-E(B_2)| right) leq Eleft(|A_1-mu_A| cdot |A_2-mu_A| right) = frac{2 sigma^2_a}{pi}.$
For the last equality I used that $A_1$ and $A_2$ are independent and that the mean of the half-normal distribution is known.
My Questions
- Do you see any issues with my derivation?
- Do you have an idea of how to obtain a possibly sharper bound? In Bohrnstedt & Goldberger (1969) I have read that the third moment vanishes for multivariate normal distributions. Can something similar be exploited here?
Bohrnstedt & Goldberger cite
Anderson, T. W. An Introduction to Multivariate Statistical Analysis.
New York: John Wiley and Sons, 1958.
on page 39, but I do not have access to that book.
statistics upper-lower-bounds expected-value
statistics upper-lower-bounds expected-value
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
asked Dec 13 '18 at 13:56
SamufiSamufi
1154
1154
add a comment |
add a comment |
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