CHAPTER 8

Statistical Errors in Basic Estimates

As noted in Chapter 4, the descriptive properties of a random variable cannot be precisely determined from sample data. Only estimates of the parameters of interest can be obtained from a finite sample of observations. The accuracy of certain basic parameter estimates is discussed in Chapter 4 for the case of data in the form of discrete independent observations of sample size *N*. In this chapter, the accuracy of parameter estimates is developed for data in the form of continuous time history records of record length *T*. It is assumed that the data are single sample records from continuous stationary (ergodic) random processes with arbitrary mean values. Statistical error formulas are developed for

Mean value estimates

Mean square value estimates

Probability density function estimates

Correlation function estimates

Autospectral density function estimates

Attention in this chapter and the next chapter is restricted to those errors that are due solely to statistical considerations. Other errors associated with data acquisition and processing are covered in Chapter 10.

**8.1 DEFINITION OF ERRORS**

Referring to Section 4.1, the accuracy of parameter estimates based on sample values can be described by a mean square error defined as

where is an estimator for *ϕ*. Expanding Equation (8.1) yields

Note that the middle term in the above expression ...