Chapter 8. Introduction to Statistical Inference

Transcription

1 Chapter 8. Introduction to Statistical Inference Point Estimation Statistical inference is to draw some type of conclusion about one or more parameters(population characteristics). Now you know that a sample from the distribution of population is useful in making inferences about the population. Two important problems in statistical inference are estimation and tests of hypothesis. There are two types of statistical estimation methods. One is Point Estimation and the other is Confidence interval Estimation. For notational convenience, we use the Greek letter θ for the parameter of interest. For example, θ might denote population mean µ or population standard deviation σ. Point estimation Recall that a parameter is a numerical characteristic of the population and a statistic is a numerical characteristic of random sample from the population of interest. A statistic is any function of random variables. Estimator: Any statistic(function of random variables) whose values are used to estimate θ Estimate: A value of estimator calculated from particular sample Example: Sample mean X n n X i, where X, X 2,..., X n are random variables is an estimator of a population mean µ and a x n n x i calculated from particular sample x, x 2,..., x n is an estimate of the population mean µ.

2 Definition: A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. Point estimate results in a number that we consider as a plausible value of θ. The symbol ˆθ ( theta hat ) is used to denote both the estimator of θ and the point estimate resulting from a given sample. Example : Consider the following 20 observations on dielectric breakdown voltage for pieces of epoxy resin Suppose that the distribution of dielectric breakdown voltage for pieces of epoxy resin follows normal distribution with unknown mean µ and unknown variance σ 2. A point estimator of the unknown mean µ is ˆµ X n n X i and the point estimate calculated from the above observations using this point estimator is ˆµ x 20 n x i ( )/ (x i x) 2 20 {(24.46 A point estimator of the unknown variance σ 2 is ˆσ 2 S 2 n n (X i X) 2 and the point estimate calculated from the above observations using this point estimator is ˆσ 2 n 27.93) 2 + ( ) 2 + ( ) ( ) 2 + ( ) 2 }

3 2 Properties of Point Estimator There might be several criteria for finding a best estimator among many possible estimators. Unbiased and minimum variance properties are the two main criteria for a best estimator. Unbiased estimator A point estimator ˆθ is said to be an unbiased estimator of θ if E(ˆθ) θ for every possible value of θ. If ˆθ is not unbiased estimator, we call it biased estimator. That is, the difference E(ˆθ) θ is called the bias of ˆθ. (Example 2) Let X, X 2,..., X n be a random sample from a population with mean µ and variance σ 2. Then, the sample mean X n n X i is an unbiased estimator of the population mean µ and the sample variance ˆσ 2 S 2 n n (X i X) 2 is an unbiased estimator of the population variance σ 2. (sol) First we show that E( X) µ. E( X) E( n X i ) n E( X i ) n n E(X i ) µ, since E(X i ) µ n nµ µ 3

4 For the unbiasedness of S 2, E(S 2 ) E( n (X i X) 2 ) n E( Xi 2 n X 2 ) n [E( Xi 2 ) ne( X 2 )] n [ E(Xi 2 ) ne( X 2 )] n [ (σ 2 + µ 2 ) n(σ 2 /n + µ 2 )] n [nσ2 + nµ 2 σ 2 nµ 2 ] n [nσ2 σ 2 ] n [(n ) σ2 ] σ 2 Therefore, X and S 2 are unbiased estimators of µ and σ 2, respectively. Minimum Variance Unbiased Estimation Among all estimators of θ that are unbiased, the one ˆθ that has minimum variance is called the minimum variance unbiased estimator(mvue) of θ. (Example 3) Let X, X 2,..., X n be a random sample from a normal distribution with parameters µ and σ. Then the estimator ˆµ X is the MVUE for µ. Standard error is the standard deviation of a statistic. When we report the value of a point estimate from sample, some indication of its precision should be given. The usual measure of precision is the standard error of the estimator used. 4

5 (Example 4) In example 3, the standard error of X is V ( X) σ2 /n σ/ n. 3 Methods of Point Estimation Method of Moments Maximum Likelihood Estimation 5