Are there superefficient statistics that shrink toward the true parameter value, in probability?

Stein (1964) defined a superefficient estimator for the univariate normal variance with population mean unknown. (It's unrelated to James-Stein; see here for a summary.) He gave an indicator variable of when, in probability, the sample variance is an overestimate. When it is, one shrinks the estimate toward 0, by dividing by n + 1. The procedure shrinks the estimate toward its parameter value, in probability, but only incidentally: it's really shrinking toward 0, and the parameter happens to lie between the two points. But suppose the indicator variable implied whether the estimate is more probably an underestimate or an overestimate, with the procedure applying a correction in the indicated direction. This version would shrink the estimate toward the latent parameter's value, without specifying a manifest point. For all superefficient statistics of which I'm aware, the shrinkage point is fixed, either by the researcher (e.g., James-Stein) or by design (often at 0, as with the LASSO ). Are there examples that shrink toward a non-fixed point? Say, via empirical Bayes? The closest I've found is a case where the endpoints of the CI for the t -statistic are shifted toward "the experimenter's best prior estimate" (p. 1512).