  Instead of minimizing the sum of all $n$ squared residuals as the classical
least squares (LS) does, Rousseeuw (1984) proposed to minimize the sum of $h$
($n/2 \leq h < n$) smallest squared residuals, the resulting estimator is
called least trimmed squares (LTS). The idea of the LTS is simple but its
computation is challenging since no LS-type analytical computation formula
exists anymore. Attempts had been made since its presence, the feasible
solution algorithm (Hawkins (1994)), fastlts.f (Rousseeuw and Van Driessen
(1999)), and FAST-LTS (Rousseeuw and Van Driessen (2006)), among others, are
promising approximate algorithms. The latter two have been incorporated into R
function ltsReg by Valentin Todorov. These algorithms utilize combinatorial- or
subsampling- approaches. With the great software accessibility and fast speed,
the LTS, enjoying many desired properties, has become one of the most popular
robust regression estimators across multiple disciplines. This article proposes
analytic approaches -- employing first-order derivative (gradient) and
second-order derivative (Hessian matrix) of the objective function. Our
approximate algorithms for the LTS are vetted in synthetic and real data
examples. Compared with ltsReg -- the benchmark in robust regression and
well-known for its speed, our algorithms are comparable (and sometimes even
favorable) with respect to both speed and accuracy criteria. Other major
contributions include (i) originating the uniqueness and the strong and Fisher
consistency at empirical and population settings respectively; (ii) deriving
the influence function in a general setting; (iii) re-establishing the
asymptotic normality (consequently root-n consistency) of the estimator with a
neat and general approach.
