#StackBounty: #statistical-significance #joint-distribution Significant Test for Joint Probability Distributions

Bounty: 50

I am familiar with significant tests and distance measure between standard probability distributions. However, I am looking for a significance test between joint probability distributions (JPD) for a set of ecological data. (The data is discrete and categorical).

As an example, we have the following two joint probability distributions, ${P_1}$ and ${P_2}$. (Categories in rows; attributes in columns).

$${P_1}=
begin{pmatrix}
0.203 & 0.203 & 0.020 \
0.033 & 0.229 & 0.033 \
0.059 & 0.072 & 0.150 \
end{pmatrix}
$$

$${P_2}=
begin{pmatrix}
0.159 & 0.051 & 0.025 \
0.080 & 0.239 & 0.051 \
0.040 & 0.188 & 0.167 \
end{pmatrix}
$$

with the respective sample sizes being ${N_{P_1}=153}$; ${N_{P_2}=276}$

The respective, if not obvious, hypotheses are: ${{H_0}: {P_1}={P_2}}$ and ${{H_A}: {P_1}neq{P_2}}$.

Is there a significance test for this situation? My assumption there is such a test (after all, there seems to be a test for everything) and that I have simply not come across yet. Any pointers would be most welcome.

(Note: I have already done a range of other tests (e.g. Chi-squared test, entropy) for each JPD. I have also calculated the Hellinger distance between each JPD. This question relates explicitly to a significance test between two JPD).


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!

#StackBounty: #probability #distributions #joint-distribution #symmetry #exchangeability When $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1…

Bounty: 100

Consider a bivariate distribution function $P: mathbb{R}^2rightarrow [0,1]$. I have the following question:

Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that
$$
exists text{ a random vector $(X_0,X_1,X_2)$ such that }
$$

$$
(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)sim P
$$


Remarks:

(I) $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ does not imply that some of the random variables among $X_1, X_2, X_0$ are degenerate.

For example, $(X_1-X_0, X_1-X_2)sim (X_2-X_0, X_2-X_1)sim (X_0-X_1, X_0-X_2)$ is implied by $(X_0, X_1, X_2)$ exchangeable.

(II) The symbol "$sim$" denotes "DISTRIBUTED AS"


My thoughts: among the necessary conditions, I would list the following: let $P_1,P_2$ be the two marginals of $P$. Then it should be that
$$
begin{cases}
P_1 text{ is symmetric around zero, i.e., $P_1(a)=1-P_1(-a)$ $forall a in mathbb{R}$}\
P_2 text{ is symmetric around zero, i.e., $P_2(a)=1-P_2(-a)$ $forall a in mathbb{R}$}\
end{cases}
$$

Should $P$ be as well symmetric at zero?

Are these conditions also sufficient? If not, what else should be added to get an exhaustive set of sufficient and necessary conditions?


Get this bounty!!!