Slide 1Example: Point EstimationProcedure: Point EstimationDefinition: Point EstimationExample 6.2: Point EstimationUnbiased EstimatorsUnbiased estimatorExamples: Point EstimationEstimators with Minimum VariancePrincipal of Minimum Variance Unbiased EstimationEstimators with Minimum VarianceBest Estimators for μExample 6.9( 6.2): Estimate of errorCopyright © Cengage Learning. All rights reserved. 6Point EstimationExample: Point EstimationSuppose that we want to find the proportion, p, of bolts that are substandard in a large manufacturing plant. To test the bolt, you destroy the bolt so you do not want to check all of the bolts to see if they fail.What is a good point estimator of p, ?ppProcedure: Point Estimation1. Define the r.v. and determine its distribution (random sample).2. For the parameter of interest, determine the appropriate statistic and its formula (estimator),3. Calculate the statistic from the data (estimate).Suppose that bolt numbers 5, 13, 24 are substandard out of 25 bolts, what is the value of ?ppDefinition: Point EstimationA 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. The selected statistic, is called the point estimator.•Example 6.2: Point EstimationAssume the dielectric breakdown voltage for pieces of epoxy resin is normally distributed. We want to estimate the mean μ of the breakdown voltage. We randomly check 20 breakdown voltages (below). Which point estimators could be used to estimate μ?24.46 25.61 26.25 26.42 26.6627.15 27.31 27.54 27.74 27.9427.98 28.04 28.28 28.49 28.5028.87 29.11 29.13 29.50 30.88Unbiased Estimatorshttp://www.weibull.com/DOEWeb/unbiased_and_biased_estimators.htmUnbiased estimatorFigure 6.1The pdf’s of a biased estimator and an unbiased estimator for a parameter Examples: Point EstimationFor a binomial distribution with parameters n and p with p unknown, Is the estimator of the sample proportion , an unbiased estimator of p?For normal distribution with mean  and variance 2, given a random sample of size n, X1, …., Xn.Is the sample mean , an unbiased estimator of ?Xˆpn=1 nX XXn+ +=LEstimators with Minimum VarianceFigure 6.3Graphs of the pdf’s of two different unbiased estimatorsPrincipal of Minimum Variance Unbiased EstimationAmong all estimators of  that are unbiased, choose the one that has minimum variance. The resulting is called the minimum variance unbiased estimator (MVUE) of .•Estimators with Minimum VarianceIs a biased estimator always the best estimator?Best Estimators for μDistr cdfBest EstimatorNormal- < x < Cauchy- < x < Uniform-c  x – μ  c0elseX%XeX2 2(x ) /(2 )1f(x) e2m ss p-=21f(x)[1 (x ) ]p m=+ -12cExample 6.9( 6.2): Estimate of errorAssume the dielectric breakdown voltage for pieces of epoxy resin is normally distributed. Here s = 1.462, n = 20. What is the standard error of the best estimator of