The constant b is needed to make the estimator
consistent for the parameter of interest. For example if the
observations are randomly sampled from a normal
distribution, by including b = 1:4826, the MADn will
estimate s , the standard deviation. With constant b = 1,
MADn will estimate 0:75 , and this is known as MAD.
2.2.2 Tn
Suitable for asymmetric distribution, [14] proposed Tn, a
scale known for its highest breakdown point like MADn.
However, this estimator has more plus points compared
to MADn. It has 52% efficiency, making it more efficient
thanMADn. It also has a continuous and bounded influence
function. Furthermore, the calculation of Tn is much easier
than the other scale estimators.
Given as
Tn = 1:3800
1
h
hå
k=1
{med
i̸=j
|xi−x j|}
(k)
where h = [n2
+1]
Tn has a simple and explicit formula that guarantees
uniqueness. This estimator also has 50% breakdown point.
2.2.3 LMSn
LMSn is also a scale estimator with 50% breakdown point
which is based on the length of the shortest half sample as
shown below:
LMSn = c′{min
i
|x(i+h−1)
−x(i)
|}
given x(1)
≤ x(2)
≤ : : : ≤ x(n) are the ordered data and
h = [n2
+ 1] . The default value of c′ is 0.7413 which
achieves consistency at Gaussian distributions. LMSn has
influence function which is similar to MAD [13] and its
efficiency equals to that of the MAD as well [2].
3 Empirical Investigations
Since this paper deals with robust method where
sensitivity to small changes is the main concern,
manipulating variables could help in identifying the
robustness of each method. Four variables (listed below)
were manipulated to create conditions which are known
to highlight the strengths and weaknesses of the
procedure.
(1)Number of Groups: Investigations were done on four
unbalanced completely randomized groups design
since previous researches have looked into these
designs ([9]; [12]; [19]).