  In many signal processing applications of Kalman filter (KF) and its variants
and extensions, accurate estimation of extreme states is often of great
importance. When the observations used are uncertain, however, KF suffers from
conditional bias (CB) which results in consistent under- and overestimation of
extremes in the right and left tails, respectively. Recently, CB-penalized KF,
or CBPKF, has been developed to address CB. In this paper, we present an
alternative formulation based on variance-inflated KF to reduce computation and
algorithmic complexity, and describe adaptive implementation to improve
unconditional performance. For theoretical basis and context, we also provide a
complete self-contained description of CB-penalized Fisher-like estimation and
CBPKF. The results from 1-dimensional synthetic experiments for a linear system
with varying degrees of nonstationarity show that adaptive CBPKF reduces root
mean square error at the extreme tail ends by 20 to 30% over KF while
performing comparably to KF in the unconditional sense. The alternative
formulation is found to approximate the original formulation very closely while
reducing computing time to 1.5 to 3.5 times of that for KF depending on the
dimensionality of the problem. Adaptive CBPKF hence offers a significant
addition to the dynamic filtering methods for general application in signal
processing when accurate estimation of extremes is of importance.
