# Point Estimation Parameters

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Published on November 3, 2008

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7-1 Introduction : 7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These methods utilize the information contained in a sample from the population in drawing conclusions. Statistical inference may be divided into two major areas: Parameter estimation Hypothesis testing 7-1 Introduction :  Definition 7-1 Introduction 7-1 Introduction :  7-1 Introduction 7-1 Introduction :  7-1 Introduction 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.1 Unbiased Estimators Definition 7-2 General Concepts of Point Estimation :  Example 7-1 7-2 General Concepts of Point Estimation 7-2 General Concepts of Point Estimation :  Example 7-1 (continued) 7-2 General Concepts of Point Estimation 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.3 Variance of a Point Estimator Definition Figure 7-1 The sampling distributions of two unbiased estimators 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.3 Variance of a Point Estimator Theorem 7-1 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.4 Standard Error: Reporting a Point Estimate Definition 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.4 Standard Error: Reporting a Point Estimate 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation Example 7-2 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation Example 7-2 (continued) 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator Definition 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator 7-2 General Concepts of Point Estimation : 7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator Figure 7-2 A biased estimator that has smaller variance than the unbiased estimator 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Definition Definition 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-4 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation 7-3.2 Method of Maximum Likelihood Definition 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-6 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-6 (continued) 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Figure 7-3 Log likelihood for the exponential distribution, using the failure time data. (a) Log likelihood with n = 8 (original data). (b) Log likelihood if n = 8, 20, and 40. 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-9 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-9 (continued) 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Properties of the Maximum Likelihood Estimator 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation The Invariance Property 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-10 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Complications in Using Maximum Likelihood Estimation It is not always easy to maximize the likelihood function because the equation(s) obtained from dL()/d = 0 may be difficult to solve. It may not always be possible to use calculus methods directly to determine the maximum of L(). 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Example 7-11 7-3 Methods of Point Estimation : 7-3 Methods of Point Estimation Figure 7-4 The likelihood function for the uniform distribution in Example 7-11. 7-4 Sampling Distributions : 7-4 Sampling Distributions Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population. Definition 7-5 Sampling Distributions of Means : 7-5 Sampling Distributions of Means Theorem 7-2: The Central Limit Theorem 7-5 Sampling Distributions of Means : 7-5 Sampling Distributions of Means Figure 7-6 Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter (1978).] Slide 36:  Example 7-13 7-5 Sampling Distributions of Means : 7-5 Sampling Distributions of Means Figure 7-7 Probability for Example 7-13. 7-5 Sampling Distributions of Means : 7-5 Sampling Distributions of Means Definition

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