I have been reading about the James-Stein estimator. It is defined, in this notes, as

$$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$

I have read the proof but I don't understand the following statement>

Geometrically, the James–Stein estimator shrinks each component of $X$ towards the origin...

What does "shrinks each component of $X$ towards the origin" mean exactly? I was thinking of something like $$\|\hat{\theta} - 0\|^2 < \|X - 0\|^2,$$ which is true in this case as long as $(p+2) < \|X\|^2$, since $$\|\hat{\theta}\| = \frac{\|X\|^2 - (p+2)}{\|X\|^2} \|X\|$$.

Is this what people mean when they say "shrink towards zero" because in the $L^2$ norm sense, the JS estimator is closer to zero than $X$?

Update as of 22/09/2017: Today I realized that perhaps I am over-complicating things. It seems like people really mean that once you multiply $X$ by something that is smaller than $1$, namely, the term $\frac{\|X\|^2 - p + 2}{\|X\|^2}$, each component of $X$ will be smaller than it used to be.

I have been reading about the James-Stein estimator. It is defined, in this notes, as

$$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$

I have read the proof but I don't understand the following statement:

Geometrically, the James–Stein estimator shrinks each component of $X$ towards the origin...

What does "shrinks each component of $X$ towards the origin" mean exactly? I was thinking of something like $$\|\hat{\theta} - 0\|^2 < \|X - 0\|^2,$$ which is true in this case as long as $(p+2) < \|X\|^2$, since $$\|\hat{\theta}\| = \frac{\|X\|^2 - (p+2)}{\|X\|^2} \|X\|.$$

Is this what people mean when they say "shrink towards zero" because in the $L^2$ norm sense, the JS estimator is closer to zero than $X$?

Update as of 22/09/2017: Today I realized that perhaps I am over-complicating things. It seems like people really mean that once you multiply $X$ by something that is smaller than $1$, namely, the term $\frac{\|X\|^2 - (p + 2)}{\|X\|^2}$, each component of $X$ will be smaller than it used to be.

I have been reading about the James-Stein estimator. It is defined, in this notes, as

$$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$

I have read the proof but I don't understand the following statement>

Geometrically, the James–Stein estimator shrinks each component of $X$ towards the origin...

What does "shrinks each component of $X$ towards the origin" mean exactly? I was thinking of something like $$\|\hat{\theta} - 0\|^2 < \|X - 0\|^2,$$ which is true in this case as long as $(p+2) < \|X\|^2$, since $$\|\hat{\theta}\| = \frac{\|X\|^2 - (p+2)}{\|X\|^2} \|X\|$$.

Is this what people mean when they say "shrink towards zero" because in the $L^2$ norm sense, the JS estimator is closer to zero than $X$?

I have been reading about the James-Stein estimator. It is defined, in this notes, as

$$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$

I have read the proof but I don't understand the following statement>

Geometrically, the James–Stein estimator shrinks each component of $X$ towards the origin...

What does "shrinks each component of $X$ towards the origin" mean exactly? I was thinking of something like $$\|\hat{\theta} - 0\|^2 < \|X - 0\|^2,$$ which is true in this case as long as $(p+2) < \|X\|^2$, since $$\|\hat{\theta}\| = \frac{\|X\|^2 - (p+2)}{\|X\|^2} \|X\|$$.

Is this what people mean when they say "shrink towards zero" because in the $L^2$ norm sense, the JS estimator is closer to zero than $X$?

Update as of 22/09/2017: Today I realized that perhaps I am over-complicating things. It seems like people really mean that once you multiply $X$ by something that is smaller than $1$, namely, the term $\frac{\|X\|^2 - p + 2}{\|X\|^2}$, each component of $X$ will be smaller than it used to be.

I have been reading about James-Stein estimator. It is defined as, in this notes,

$$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$

I have read the proof but I don't understand the following statement.

Geometrically, the James–Stein estimator shrinks each component of $X$ towards the origin...

What does "shrink each component of $X$ towards the origin" exactly mean? I was thinking of something like $$\|\hat{\theta} - 0\|^2 < \|X - 0\|^2$$ , which is true in this case as long as $(p+2) < \|X\|^2$, since $$\|\hat{\theta}\| = \frac{\|X\|^2 - (p+2)}{\|X\|^2} \|X\|$$.

Is this what people mean when they say "shrink towards zero" because in $L^2$ norm sense, JS estimator is closer to zero than $X$?

I have been reading about the James-Stein estimator. It is defined, in this notes, as

$$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$

I have read the proof but I don't understand the following statement>

Geometrically, the James–Stein estimator shrinks each component of $X$ towards the origin...

What does "shrinks each component of $X$ towards the origin" mean exactly? I was thinking of something like $$\|\hat{\theta} - 0\|^2 < \|X - 0\|^2,$$ which is true in this case as long as $(p+2) < \|X\|^2$, since $$\|\hat{\theta}\| = \frac{\|X\|^2 - (p+2)}{\|X\|^2} \|X\|$$.

Is this what people mean when they say "shrink towards zero" because in the $L^2$ norm sense, the JS estimator is closer to zero than $X$?