# Estimating statistical characteristics under interval uncertainty and constraints: Mean, variance, covariance, and correlation

#### Abstract

In many practical situations, we have a sample of objects of a given type. When we measure the values of a certain quantity *x* for these objects, we get a sequence of values *x*_{1}, …, *x _{n}*. When the sample is large enough, then the arithmetic mean

*E*of the values

*x*is a good approximation for the average value of this quantity for all the objects from this class. Other expressions provide a good approximation to statistical characteristics such as variance, covariance, and correlation. ^ The values

_{i }*x*come from measurements, and measurement is never absolutely accurate. Often, the only information that we have about the measurement error is the upper bound Δ

_{i}*on this error. In this case, once we have the measurement result*

_{ i}*x˜*, the condition :

_{i}*x˜*–

_{ i}*x*: ≤ Δ

_{i}*implies that the actual (unknown) value*

_{ i}*x*belongs to the interval

_{i}**x**

*=*

_{i}^{def}[

*x˜*– Δ

_{i}*,*

_{ i}*x˜*+ Δ

_{i}*]. Different values*

_{ i}*x*∈

_{i}**x**

*from the corresponding intervals lead, in general, to different values of sample mean, sample variance, etc. It is therefore desirable to find the range of possible values of these characteristics when*

_{i}*x*∈

_{i}**x**

*. ^ It is known that evaluating such ranges is, in general, NP-hard. The main objective of this thesis is to design feasible (i.e., polynomial-time) algorithms for practically important situations. Several such algorithms are described and proved to be correct.^*

_{i}#### Subject Area

Applied Mathematics|Statistics|Computer Science

#### Recommended Citation

Jalal-Kamali, Ali, "Estimating statistical characteristics under interval uncertainty and constraints: Mean, variance, covariance, and correlation" (2011). *ETD Collection for University of Texas, El Paso*. AAI1513642.

https://digitalcommons.utep.edu/dissertations/AAI1513642