Slide 1overviewParameter estimationParameter estimationMethod of momentsMethod of momentsMethod of maximum likelihoodPROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapter 9: Statistical Inference1OVERVIEWWe saw in Chapter 8 how to use statistics to estimate some distribution properties like mean, median, quartiles, and measures of variability (variance, standard deviation)We will now look at more general methods that will allow us to estimate more parametersWe will also look at a method to measure the confidence in our estimateWe’ll do this by actually producing an interval for the parameterWe’ll see how to use these methods to test hypotheses2PARAMETER ESTIMATIONWe will now generalize our methods for estimating parameters of populations by analyzing a sample from that populationWhich is the best way to do this?Example: The number of defects on a computer chip is a rare event, and can be modeled by a Poisson distribution. What is a good choice for the frequency parameter, namely ?We could take a sample of chips, count the number of defects, and estimate the mean with Recall that for Poisson, the frequency is the mean, i.e. But it is also true that the variance is the frequency for this distribution ()We can estimate the variance … but which will give the best approximation for ?•<3PARAMETER ESTIMATIONAs another example, suppose we think the underlying population distribution is a Gamma distributionThe parameters for the distribution, namely and , are not equal to any of the parameters we estimated in Chapter 8How do we find these?There are multiple methods to solve these problemsThis is known as statistical inference – inferring information about the underlying population distribution by using information from a sample of that populationWe will consider two popular methods that can be used to estimate parameters: Method of Moments and Method of Maximum Likelihood•<4METHOD OF MOMENTSThe k-th population moment is defined as . This is a measure of the actual population’s distributionThe k-th sample moment is given by the equation . This is an estimate of the population moment that is drawn from a sample When we have the usual mean and sample mean we saw in Chapter 8The second central moment is the variance•<5METHOD OF MOMENTSHow can sample moments be used?The method of moments uses sample moments to estimate parameters of the population distributionThe sample mean was just one exampleAnd occasionally the moments give us important information about the underlying distribution6METHOD OF MAXIMUM LIKELIHOODBasic idea: Given that we have observed a sample from a population, we find the parameters for the presumed underlying distribution that maximize the probability (likelihood) for this to happenIn other words, what parameters will make it most likely that the distribution will produce the sample we observed?The method is slightly different for discrete and continuous
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