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Correlation matrix with same pairwise correlation coefficientQuestion

Correlation matrix. Consider 𝑛 random variables with the same pairwise correlation coefficient ${\rho}_{n}$. Find the highest possible value of ${\rho}_{n}$for

a) n=3

b) n=4

c) general, n $\ge $2

HINT: Correlation matrix must be positive semi-definite.

My Workings

This is what I infer from "same pairwise coefficients":

$\left(\begin{array}{ccc}1& {\rho}_{n}& {\rho}_{n}\\ {\rho}_{n}& 1& {\rho}_{n}\\ {\rho}_{n}& {\rho}_{n}& 1\end{array}\right)$

Because a correlation matrix is positive semi-definite, all principal minors have to be positive

For n=3, the principal minors calculations yield

$\mid {H}_{1}\mid $=1

$\mid {H}_{2}\mid $=1-${\rho}_{n}^{2}$ $\ge 0$ $\Rightarrow $${\rho}_{n}$ $\le 1$

$\mid {H}_{3}\mid ={\rho}_{n}^{2}$ - ${\rho}_{n}$$\ge 0$ $\Rightarrow $ ${\rho}_{n}$ $\ge 1$

$\mid {H}_{4}\mid =({1}^{3}$+2${\rho}_{n}^{3}$)-(3${\rho}_{n}^{2}$)$\ge 0$ . I solved for the roots and found 1

Conclusion: For n=3, max ${\rho}_{n}$=1

I did the same method for n=4 and found max ${\rho}_{n}$=1 again

My Problem

Result looks too simple and false. Method is tedious

I have no idea how to do the general case. (by induction ?)

Thank you for your help

INDUCTION ATTEMPT

${H}_{0}$

$\left(\begin{array}{cc}1& {\rho}_{2}\\ {\rho}_{2}& 1\end{array}\right)$

$\Rightarrow -\frac{1}{2-1}\le {\rho}_{2}\le 1$

${H}_{n}$: Pn: Suppose that for a nxn correlation matrix An with same pairwise coefficients ${\rho}_{n}$, $-\frac{1}{n-1}\le {\rho}_{n}\le 1$ holds

${H}_{n+1}$

$-\frac{1}{n-1}\le {\rho}_{n}\le 1$

$\Leftarrow \Rightarrow $$-\frac{1}{(n+1)-1}\le {\rho}_{n+1}\le 1$

$\Leftarrow \Rightarrow $$-\frac{1}{n}\le {\rho}_{n+1}\le 1$

And because for all $-\frac{1}{n}\ge -\frac{1}{n-1}$

$-\frac{1}{n-1}\le {\rho}_{n+1}\le 1$

Correlation matrix. Consider 𝑛 random variables with the same pairwise correlation coefficient ${\rho}_{n}$. Find the highest possible value of ${\rho}_{n}$for

a) n=3

b) n=4

c) general, n $\ge $2

HINT: Correlation matrix must be positive semi-definite.

My Workings

This is what I infer from "same pairwise coefficients":

$\left(\begin{array}{ccc}1& {\rho}_{n}& {\rho}_{n}\\ {\rho}_{n}& 1& {\rho}_{n}\\ {\rho}_{n}& {\rho}_{n}& 1\end{array}\right)$

Because a correlation matrix is positive semi-definite, all principal minors have to be positive

For n=3, the principal minors calculations yield

$\mid {H}_{1}\mid $=1

$\mid {H}_{2}\mid $=1-${\rho}_{n}^{2}$ $\ge 0$ $\Rightarrow $${\rho}_{n}$ $\le 1$

$\mid {H}_{3}\mid ={\rho}_{n}^{2}$ - ${\rho}_{n}$$\ge 0$ $\Rightarrow $ ${\rho}_{n}$ $\ge 1$

$\mid {H}_{4}\mid =({1}^{3}$+2${\rho}_{n}^{3}$)-(3${\rho}_{n}^{2}$)$\ge 0$ . I solved for the roots and found 1

Conclusion: For n=3, max ${\rho}_{n}$=1

I did the same method for n=4 and found max ${\rho}_{n}$=1 again

My Problem

Result looks too simple and false. Method is tedious

I have no idea how to do the general case. (by induction ?)

Thank you for your help

INDUCTION ATTEMPT

${H}_{0}$

$\left(\begin{array}{cc}1& {\rho}_{2}\\ {\rho}_{2}& 1\end{array}\right)$

$\Rightarrow -\frac{1}{2-1}\le {\rho}_{2}\le 1$

${H}_{n}$: Pn: Suppose that for a nxn correlation matrix An with same pairwise coefficients ${\rho}_{n}$, $-\frac{1}{n-1}\le {\rho}_{n}\le 1$ holds

${H}_{n+1}$

$-\frac{1}{n-1}\le {\rho}_{n}\le 1$

$\Leftarrow \Rightarrow $$-\frac{1}{(n+1)-1}\le {\rho}_{n+1}\le 1$

$\Leftarrow \Rightarrow $$-\frac{1}{n}\le {\rho}_{n+1}\le 1$

And because for all $-\frac{1}{n}\ge -\frac{1}{n-1}$

$-\frac{1}{n-1}\le {\rho}_{n+1}\le 1$

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Correlation Function

Let X,Y be random variables. If $\rho (X,Y)=a$ (Correlation), where $a\in (0,1)$, what can be said about the relationship between X and Y? Is it true that Y=bX+c+Z, where Z is a random variable? If it is true then how is $|Z|$ related to the correlation a?

Let X,Y be random variables. If $\rho (X,Y)=a$ (Correlation), where $a\in (0,1)$, what can be said about the relationship between X and Y? Is it true that Y=bX+c+Z, where Z is a random variable? If it is true then how is $|Z|$ related to the correlation a?

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klasyvea 2022-11-06

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Messiah Sutton 2022-11-06

Why is the standardized regression coefficient in a regression model with more than one independent variable not the same as the correlation coefficient between x we interested in and y in a regression model with more than one independent variable?

$$\hat{{\beta}_{i}}=\mathrm{c}\mathrm{o}\mathrm{r}({Y}_{i},{X}_{i})\cdot \frac{\mathrm{S}\mathrm{D}({Y}_{i})}{\mathrm{S}\mathrm{D}({X}_{i})}$$

So

$$\mathrm{c}\mathrm{o}\mathrm{r}({Y}_{i},{X}_{i})=\hat{{\beta}_{i}}\cdot \frac{\mathrm{S}\mathrm{D}({X}_{i})}{\mathrm{S}\mathrm{D}({Y}_{i})}$$

The formula for the standardized regression coefficient is also:

$$standardizedBeta=\hat{{\beta}_{i}}\cdot \frac{\mathrm{S}\mathrm{D}({X}_{i})}{\mathrm{S}\mathrm{D}({Y}_{i})}$$

So shouldn't it be

$$standardizedBeta=\mathrm{c}\mathrm{o}\mathrm{r}({Y}_{i},{X}_{i})$$?

$$\hat{{\beta}_{i}}=\mathrm{c}\mathrm{o}\mathrm{r}({Y}_{i},{X}_{i})\cdot \frac{\mathrm{S}\mathrm{D}({Y}_{i})}{\mathrm{S}\mathrm{D}({X}_{i})}$$

So

$$\mathrm{c}\mathrm{o}\mathrm{r}({Y}_{i},{X}_{i})=\hat{{\beta}_{i}}\cdot \frac{\mathrm{S}\mathrm{D}({X}_{i})}{\mathrm{S}\mathrm{D}({Y}_{i})}$$

The formula for the standardized regression coefficient is also:

$$standardizedBeta=\hat{{\beta}_{i}}\cdot \frac{\mathrm{S}\mathrm{D}({X}_{i})}{\mathrm{S}\mathrm{D}({Y}_{i})}$$

So shouldn't it be

$$standardizedBeta=\mathrm{c}\mathrm{o}\mathrm{r}({Y}_{i},{X}_{i})$$?

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In simple terms, inferential statistics is an approach where you use measurements from the sample of specific subjects as you conduct an experiment. The purpose is to make an outcome based on generalization regarding the greater population of subjects. You may use equations if there are questions that are related to a particular approach. You can get inferential statistics help as we provide a list of answers with good samples to start with. There are related topics like correlation problems that will help you with financial statistics and the coordination of variables