Ch7 Estimation

Ch7 Estimation - Stat1801 Probability and Statistics Foundations of Actuarial Science Chapter VII 7.1 Fall 2010-2011 Estimation Point Estimation

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Chapter VII Estimation § 7.1 Point Estimation θ : unknown parameter of the population, or the corresponding pdf () x f ) : a point estimator of . It is a statistic (function of the items of a random sample) and is a random variable. An observed value of ) is a point estimate of . Example 7.1 Suppose we are interested in a normal population with unknown mean μ and variance 2 σ , i.e. we only know (or assume) that the population distribution resembles the normal bell-shaped curve, but have no idea in the location and spread of the distribution. Then and 2 are the unknown parameters. We may draw a random sample from this population and calculate the statistics X and 2 S , and then use their values to estimate and 2 . Hence we may denote X = ) , 2 2 S = ) as the point estimators of and 2 . § 7.2 Desirable Properties of Estimator The notations ˆ and refer to different quantities, with ˆ as the sample statistic that can be observed from the sample while is the population parameter that is unobservable. Usually for any parameter, there exist many different methods to do the estimation. For example, based on a random sample from , 0 U , a reasonable estimator of is the maximum of the sample, i.e. ( ) n X X X ,..., , max ˆ 2 1 = . However, since the population mean is 2 = , it is also reasonable to estimate by using the sample mean, i.e. X 2 ˆ = . P.140
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 In order to compare the performance of different estimators, some desirable properties of estimation are discussed below. § 7.2.1 Bias of Estimator Definition The bias of an estimator θ ) of the parameter is defined as ( ) ( ) θθ = ) ) E bias Note that the expectation is taken with respect to the sampling distribution of ) . We say that ) overestimates if ( ) 0 > ) bias ; underestimates if ( ) 0 < ) bias . Unbiased Estimator The estimator ) is called an unbiased estimator of the parameter if ( ) = ) E for all the values of in the parameter space, i.e. the estimator is unbiased if ( ) 0 ˆ = bias for any . Example 7.2 Suppose we have a random sample { } n X X X ,..., , 2 1 from a distribution with mean μ and variance 2 σ . 1. Sample mean is an unbiased estimator of the population mean as ( ) = X E for all . 2. Sample variance () X S 2 is an unbiased estimator of the population variance 2 = = n i i X n 1 2 1 1 as ( ) 2 2 = S E for all 2 . P.141
Background image of page 2
Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Example 7.3 Suppose . Consider the two estimators ( θ , 0 ~ ,..., , 2 1 U X X X iid n ) X 2 ˆ 1 = and ( ) n X X X ,..., , max ˆ 2 1 2 = . Obviously is unbiased as 1 ˆ ( ) () μ = × = = = 2 2 2 2 ˆ 1 X E E . For , consider the cdf of . For 2 ˆ 2 ˆ < < y 0, w e h a v e () ( ) n n n n y y X P y X P y X P y X y X y X P y X X X P y P y F = = = = = L K 2 1 2 1 2 1 2 ˆ , , , ,..., , max ˆ 2 The pdf of is therefore given by 2 ˆ () () < < = = otherwise 0 0 for 1 ' ˆ ˆ 2 2 y ny y F y f n n and hence 1 1 ˆ 0 1 0 2 + = + = = + n n n ny dy ny E n n n n .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

Page1 / 25

Ch7 Estimation - Stat1801 Probability and Statistics Foundations of Actuarial Science Chapter VII 7.1 Fall 2010-2011 Estimation Point Estimation

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online