Lecture Note Statistical Inference

Course Title: Statistical Inference Course code: Stat 3052 Credit: 5 EtCTS Credit hours: 3 (3Lecture hrs+2 hrs tutorial) Instructor's Name: Kenenisa T. (MSc.) Email: kenenisatadesse@

April, 2020 Jimma, Ethiopia

Statistical Inference(Stat-3052)

Outline

1 Chapter 0: Preliminaries Definitions of Some Basic Terms Sampling Distribution What is Statistical Inference? What is Statistical Inference? What is Statistical Inference? What is Statistical Inference?

2 Chapter 1: Parametric Point Estimation Methods of Finding Parametric Point Estimators Methods of Finding Parametric Point Estimators Methods of Finding Parametric Point Estimators Methods of Finding Parametric Point Estimators Maximum Likelihood (ML) Method Properties of MLE Properties of MLE Properties of MLE Properties of MLE Method of Moments Properties of Point Estimators Properties of Point Estimators Properties of Point Estimators Unbiased Estimators Unbiased Estimators Mean Square Error (MSE) of an Estimator Efficiency of an Estimator

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Chapter 0: Preliminaries

The aim of statistical inference is to make certain determinations with regard to the unknown constants known as parameter(s) in the underlying distribution. With the intention of emphasizing the importance of the basic concepts, we begin with a review of the definitions of terms related to random sampling distribution of some estimators in the preliminary chapter. The first step in statistical inference is Point Estimation, in which we compute a single value (statistic) from the sample data to estimate a population parameter. General concept of point estimators, different methods of finding estimators and clarification of their properties are discussed in Chapter 1. Then proceed to Interval Estimation, a method of obtaining, at a given level of confidence (or probability), two statistics which include within their range an unknown but fixed parameter are discussed in Chapter 2. In Chapter 3 we discuss a 2nd major area of statistical inference is Testing of Hypotheses. The significance of the differences between estimated parameters from two or more samples are also included in this chapter; such as the significance the difference of two population means. Nonparametric methods that does not based on sampling distributions are discussed in Chapter 4 (Group Work to be presented by Students).

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Definitions of Some Basic Terms

Population refers to all elements of interest characterized by a distribution F with some parameter, say (where is the set of its possible values called the parameter space). Sample is the set of data X1, . . . , Xn, selected subset of the population, n is sample size. Remember, to use sample data for inference, needs to be representative of population for the question(s) of interest or our study. For X1, . . . , Xn, a random sample (independent and identically distributed, iid) from a distribution with cumulative distribution function (cdf ) F (x; ). The cdf admits a probability mass function (pmf) in the discrete case and a probability density function (pdf) in the continuous case, in either case, write this function as f (x; ). A parameter is a number associated with a population characteristic.? value unknown. It is usually assumed to be fixed but unknown. Thus, we estimate the parameter using sample information. Examples of population parameters: sample mean (?) and population variance (2). A statistic or estimate is a number computed from a sample. A statistic estimates a parameter and it changes with each new sample. A statistic is any function of the observations in a random sample, (no parameter in the function).

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