## #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).

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## #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!!!