Estimating sample mean under interval uncertainty and constraint on sample variance

• Published in: International journal of approximate reasoning Vol. 52; no. 8; pp. 1136 - 1146
• Main Authors: Koshelev, Misha, Jalal-Kamali, Ali, Longpré, Luc
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01-11-2011; Elsevier
• Subjects: Applied sciences; Approximation; Artificial intelligence; Computer science; control theory; systems; Exact sciences and technology; Imprecise probabilities; Intervals; Learning and adaptive systems; Reasoning; Robust statistics; Samples; Statistical analysis; Statistical methods; Statistics under interval uncertainty; Uncertainty; Variance; Robust statistics; Statistics under interval uncertainty; Imprecise probabilities; Mean; Statistical analysis; Probabilistic approach; Constraint; Statistics under constraints; Artificial intelligence; Interval arithmetic; Variance; Confidence interval
• ISSN: 0888-613X; 1873-4731
• DOI: 10.1016/j.ijar.2011.06.002

► Described situations when we need to estimate mean under bounds on variance. ► Found a formula for the mean under interval uncertainty and bound on variance ► Designed an algorithm for the mean under interval uncertainty and bound on variance
Traditionally, practitioners start a statistical analysis of a given sample
x
1,
…
,
x
n
by computing the sample mean
E and the sample variance
V. The sample values
x
i
usually come from measurements. Measurements are never absolutely accurate and often, the only information that we have about the corresponding measurement errors are the upper bounds Δ
i
on these errors. In such situations, after obtaining the measurement result
x
˜
i
, the only information that we have about the actual (unknown) value
x
i
of the
ith quantity is that
x
i
belongs to the interval
x
i
=
[
x
˜
i
-
Δ
i
,
x
˜
i
+
Δ
i
]
. Different values
x
i
from the corresponding intervals lead, in general, to different values of the sample mean and sample variance. It is therefore desirable to find the range of possible values of these characteristics when
x
i
∈
x
i
.
Often, we know that the values
x
i
cannot differ too much from each other, i.e., we know the upper bound
V
0 on the sample variance
V
:
V
⩽
V
0. It is therefore desirable to find the range of
E under this constraint. This is the main problem that we solve in this paper.