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# Note for MATH 552 at UA-Math Stats WApplictn II

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Alabama - Tuscaloosa taught by a professor in Fall. Since its upload, it has received 21 views.

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Date Created: 02/06/15
Point Estimation August 19 2011 1 De nition A point estimate of a parameter 9unknown is a single number computed from the sample data that can be regarded as an educated guess for the value of the parameter 9 Example a Suppose we want to estimate the proportion p of all crashes that results in no damages in a population of automobiles from a certain manufacturer In a sample of 25 cars 15 cars result in such crashes what is the estimate of the proportion p 9 b 20 observations on the dielectric breakdown voltage for pieces of epoxy resin yield the following statistcis sample mean X 27793 and the median M 27960 What is the estimate of the mean u of the population of interest Remark A point estimate is obtained by selecting a suitable statistic and computing its value from the sample data The selected statistic is called the point estimator De nition MSE accuracy and precision of the estimator The mean square error of an estimator 9 of 9 is the quantity A A MSE9 E9 7 92l It measures the accuracy and the precision of the estimator De nition A point estimator 9 is said to be unbiased estimator of9 if 9 for every possible value of 9 otherwise 9 is said to be biased and the di erence E9 7 9 is called the bias of 9 Example Suppose that X the reaction time to a certain stimulus has a uniform distribution on the interval 09 where 9 is unknown What is the estimator of9 9 Is it unbiased 9 Can you nd an unbiased estimator based on the rst one 9 Given a random sample of n 5 reaction times 11 42 12 17 13 24 14 3 9 15 13 Find the estimate of 9 Proposition 1 When X is a binomial random variable with parameters n and p the sample proportion p Xn is an unbiased estimator of p Proposition 2 If X1X2 r Xn is a random sample from a distribution with mean u and variance 02 then X is an unbiased estimator of u and S2 is an unbiased estimator of 02 If in addition the distribution is continuous and symmetric then the median M and any trimmed mean are also unbiased estimators of u Principle of Minimum variance Unbiased Estimation A Among all estimators of 9 that are unbiased choose the one that has minimum variance The resulting 9 is called the the minimum variance unbiased estimator MVUE of 9 Theorem Let XlLX2 Xn be a random sample from the normal distribution with parameters a and 0 Then the estimator i X is the MVUE for u So in example lb X should be used to estimate a because the underlining population has a normal distribution Remark In general the best estimator for it depends crucially on which distribution is being sampled De nition The standard error of an estimator9 is its standard deviation Var9 If the standard error itself involves unknown parameters whose values can be estimated substitution of these estimate into Var9 yields the estimated standard error of the estimatorand is denoted by Var9 Example Assuming that theibreakdown voltage is normally distributed fi X is the best unbiased estimator of u Find the standard error ofX in the following 2 cases a the value ofa is known to be 15 b the value ofa is unknown and the sample standard deviation is s L462 Homework 61pages 240243 5 4 15

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