Some General Concepts of Point Estimation

6.1 Some General Concepts of Point Estimation

  Statistical inference is almost always directed toward drawing some type of conclu- sion about one or more parameters (population characteristics). To do so requires that an investigator obtain sample data from each of the populations under study. Conclusions can then be based on the computed values of various sample quantities. For example, let m (a parameter) denote the true average breaking strength of wire connections used in bonding semiconductor wafers. A random sample of n ⫽ 10 connections might be made, and the breaking strength of each one determined,

  resulting in observed strengths x 1 ,x 2 ,...,x 10 . The sample mean breaking strength x could then be used to draw a conclusion about the value of . Similarly, if s m 2 is the

  variance of the breaking strength distribution (population variance, another parame-

  ter), the value of the sample variance s 2 can be used to infer something about s 2 .

  When discussing general concepts and methods of inference, it is convenient to have a generic symbol for the parameter of interest. We will use the Greek letter for this purpose. The objective of point estimation is to select a single number, based on sample data, that represents a sensible value for . Suppose, for example, that the u parameter of interest is m, the true average lifetime of batteries of a certain type. A

  random sample of n ⫽ 3 batteries might yield observed lifetimes (hours) x 1 ⫽ 5.0,

  x 2 ⫽ 6.4, x 3 ⫽ 5.9. The computed value of the sample mean lifetime is ⫽ 5.77, and x it is reasonable to regard 5.77 as a very plausible value of —our “best guess” for the value of based on the available sample information.

  Suppose we want to estimate a parameter of a single population (e.g., m or s) based on a random sample of size n. Recall from the previous chapter that before data

  is available, the sample observations must be considered random variables (rv’s) X 1 ,

  X 2 ,...,X n . It follows that any function of the X i ’s—that is, any statistic—such as the sample mean or sample standard deviation S is also a random variable. The same is true if available data consists of more than one sample. For example, we can represent

  tensile strengths of m type 1 specimens and n type 2 specimens by X 1 ,...,X m and

  Y 1 ,...,Y n , respectively. The difference between the two sample mean strengths is

  X ⫺ , the natural statistic for making inferences about Y 1 ⫺ m 2 , the difference

  between the population mean strengths.

  DEFINITION

  A point estimate of a parameter is a single number that can be regarded as

  a sensible value for . A point estimate is obtained by selecting a suitable sta- tistic and computing its value from the given sample data. The selected statis- tic is called the point estimator of . u

  In the battery example just given, the estimator used to obtain the point estimate of m was , and the point estimate of m was 5.77. If the three observed lifetimes had

  instead been x 1 ⫽ 5.6, x 2 ⫽ 4.5, and x 3 ⫽ 6.1, use of the estimator X would have

  resulted in the estimate x ⫽ (5.6 ⫹ 4.5 ⫹ 6.1)3 ⫽ 5.40. The symbol (“theta hat”) uˆ is customarily used to denote both the estimator of and the point estimate resulting u from a given sample. Thus ˆ5X is read as “the point estimator of m is the sample

  Following earlier notation, we could use ⌰ˆ (an uppercase theta) for the estimator, but this is cumber-

  some to write.

  6.1 Some General Concepts of Point Estimation

  mean .” The statement “the point estimate of m is 5.77” can be written concisely as X m ˆ 5 5.77 . Notice that in writing uˆ 5 72.5 , there is no indication of how this point estimate was obtained (what statistic was used). It is recommended that both the esti- mator and the resulting estimate be reported.