I want to estimate a parameter $k$, from my data $(y_1,y_2, ....,y_n)$ which are independent and identically distributed.
$k=a$, $\;$$\;$$\;$$\;$$\;$$\;$ if $y_i$ is uniformly distributed on $[a,b]$ or $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$ $\;$$\;$ $\;$
$k=a+1$, $\;$ if $y_i$ is uniformly distributed on $[a+1, b+1]$.
My question is this: What are the problems of estimation and inference in my model? The discussions I have seen so far do not answer my question
Note: I must consider at least 2 different consistent estimators of $k$
Thanks.