# Consistency of an estimator [closed]

I have an estimator for the coefficients of the model $$y=X\beta+\varepsilon$$

with $$y_{n\times1}$$, $$X_{n\times p}$$, $$\beta_{p\times1}$$, $$\varepsilon_{n\times1}$$. The estimator is in the form

$$\hat{\beta}=\left(\frac{1}{n}X^{T}X+I\right)^{-1}\left(\frac{1}{n}X^{T}X+kI\right)\hat{\beta}^{ols}$$ for $$0\le k\le 1$$. It is obvious that $$\hat{\beta}$$ is a consistent estimator if $$k=1$$.

My question?

Is the boundedness of $$k$$ in $$[0,1]$$ an obstacle to define $$k$$ as $$k_{n}$$?

If I can apply $$k\to k_{n}$$, I think I can write if $$k_{n}\to1$$, $$n\to\infty$$ then $$\hat{\beta}$$ is consistent for $$\beta$$. (In this case $$k$$ is not fixed to zero, instead, goes to zero.)

• If, in principle, $k$ may depend on $n$, then boundedness would not be an issue. Take $k_n=\epsilon/n$ for some $0<\epsilon<1$. Dec 4 '20 at 7:46
• Could you explain to us what "define $k$ as $k_n$" means?
– whuber
Dec 7 '20 at 18:02