If shrinkage is applied in a clever way, does it always work better for more efficient estimators? Suppose I have two estimators $\widehat{\beta}_1$ and $\widehat{\beta}_2$ that are consistent estimators of the same parameter $\beta_0$ and such that 
$$\sqrt{n}(\widehat{\beta}_1 -\beta_0) \stackrel{d}\rightarrow \mathcal{N}(0, V_1), \quad \sqrt{n}(\widehat{\beta}_2 -\beta_0) \stackrel{d}\rightarrow \mathcal{N}(0, V_2)$$
with $V_1 \leq V_2$ in the p.s.d. sense. Thus, asymptotically $\widehat{\beta}_1$ is more efficient than $\widehat{\beta}_2$. These two estimators are based on different loss functions. 
Now I want to look for some shrinkage techniques to improve finite-sample properties of my estimators. 
Suppose that I found a shrinkage technique that improves the estimator $\widehat{\beta}_2$ in a finite sample and gives me the value of MSE equal to $\widehat{\gamma}_2$. Does this imply that I can find a suitable shrinkage technique to apply to $\widehat{\beta}_1$   that will give me the MSE no greater than $\widehat{\gamma}_2$?  
In other words, if shrinkage is applied cleverly, does it always work better for more efficient estimators? 
 A: Let me suggest an admittedly slightly boring counterexample. Say that $\hat{\beta}_1$ is not just asymptotically more efficient than $\hat{\beta}_2$, but also attains the Cramer Rao Lower Bound. A clever shrinkage technique for $\hat{\beta}_2$ would be:
$$
\hat{\beta}_2^\ast = w \hat{\beta}_2 + (1 - w) \hat{\beta}_1
$$
with $w\in(0,1)$. The asymptotic variance of $\hat{\beta}_2^\ast$ is
$$ 
V^\ast = \mathbb{Avar}(w \hat{\beta}_2 + (1 - w) \hat{\beta}_1) = \mathbb{Avar}(w (\hat{\beta}_2  - \hat{\beta}_1) + \hat{\beta}_1 ) = V_1 + w^2 (V_2 - V_1)
$$
where the last equality uses the Lemma in Hausman's paper. We have 
$$
V_2 - V^\ast = V_2(1-w^2) - V_1(1-w^2) \geq 0
$$
so there is an asymptotic risk improvement (there are no bias terms). So we found a shrinkage technique that gives some asymptotic (and therefore hopefully finite sample) improvements over $\hat{\beta}_2$. Yet, there is no similar shrinkage estimator $\hat{\beta}_1^\ast$ that follows from this procedure.
The point here of course is that the shrinkage is done towards the efficient estimator and is therefore not applicable to the efficient estimator itself. This seems pretty obvious on a high level but I would guess that in a specific example this is not so obvious (MLE and Method of Moments estimator for the uniform distribution may be an example?).
A: This is an interesting question where I want to point out some highlights first.


*

*Two estimators are consistent

*$\hat{\beta}_1$ is more efficient than $\hat\beta_2$ since it
achieves less variation

*Loss functions are not the same

*one shrinkage method is applied to one so that it reduces the variation that by itself ends up a better estimator

*Question: In other words, if shrinkage is applied cleverly, does it
always work better for more efficient estimators?


Fundamentally, it is possible to improve an estimator in a certain framework, such as unbiased class of estimators. However, as pointed out by you, different loss functions makes the situation difficult as one loss function may minimise quadratic loss and the other one minimises the entropy. Moreover, using the word "always" is very tricky since if one estimator is the best one in the class, you cannot claim any better estimator, logically speaking. 
For a simple example (in the same framework), let two estimators, namely a Bridge (penalised regression with $l_p$ norm penalty) and Lasso (first norm penalised likelihood) and a sparse set of parameters namely $\beta$, a linear model $y=x\beta+e$, normality of error term,$e\sim N(0,\sigma^2<\infty)$, known $\sigma$, quadratic loss function (least square errors), and independency of covariates in $x$. Let choose $l_p$ for $p=3$ for the first estimator and $p=2$ for the second estimators. Then you can improve the estimators by choosing $p\rightarrow 1$ that ends up a better estimator with lower variance. Then in this example there is a chance of improving estimator.
So my answer to your question is yes, given you assume the same family of estimators and the same loss function as well as assumptions.
