TABLE OF CONTENTS The study or investigation of random phenomena usually requires the knowledge of its statistical properties, that is, its probabilistic description. Most often this description is not available and it must be discovered from experimental measurements. The measurements produce a sample of N data values { x 1, x 2 , , x N}. Mathematical operations are performed on the sample in order to determine the statistical properties. These operations are called estimators and this entire process is called estimation. An important aspect of estimation is its accuracy, which can be quite difficult to determine. This section will introduce the notion of sample moments and some general characteristics of estimation. All of this will be exemplified by focusing on estimating several useful moments of a data sample.
The procedure for estimating many properties is often obtained by directly translating the mathematical definition of its theoretical counterpart. Consider approximating the general expectation operator of equation 4.8 with an infinite sum. Divide the real line into intervals of length ? x with boundary points d i, then
Notice that a very important assumption is made. All of the sample values of x have the same pdf that is, they are identically distributed. Since data points may equal boundary values, the intervals are half-open and
Using the interpretation of relative frequency, then
where N i; is the number of measurements in the interval...
