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Unformatted text preview: ion • • • Suppose we are given the data X1, X2, ...., Xn and we have already hypothesized the family of distribution from which these observations might come from • • • Possibly from histograms or bar plots Now, we need to estimate the parameters of the hypothesize distribution Two main methods: Method of Moment (MoM): Choose the parameters to match the moments Maximum Likelihood Estimation (MLE): Choose the parameters that maximize the likelihood function 17 Method of Moments (MoM) • • Very general and can be applied in many cases where other estimation methods fail. Example: Suppose we have 5 observations of interarrival times: 3, 1, 4, 3, and 8. Assume that we have hypothesized that these data come from an exponential distribution with some parameter λ • • • Question: How do we determine λ? Note: An exponential distribution with parameter λ has a mean 1/ λ Key Idea: Under MoM, we choose the parameter λ so that the mean (aka ﬁrst moment) matches with the observed sample mean! • Observed sample mean = (3+1+4+3+8)/5 = 19/5
1 ˆ λ MoM Estimator: = 19 5 ⇔ ˆ λ= 5 19 This method works for any distribution having a single parameter. Can do we extend this to distributions with multiple parameters?
18 General MoM • For k = 1, 2, 3....., the k moment of • Given observations X , ...., X , the the random variable X is deﬁned by: estimated sample k moment is n k 1 k mk = (Xi ) µk = E X n i=1 • Suppose the observations X , ...., X are hypothesized to be i.i.d. samples
th 1 n th 1 n from a distribution whose p.d.f. f(⋅) has p unknown parameters. • • Step 1: Estimate the ﬁrst p moments using the data. Let m1, m2, ..., mp denote these estimated sample moments. • • • Note: The estimated moments m1, m2, ..., mp are p numbers Step 2: Analytically compute the ﬁrst p moments of the hypothesized p.d.f. Let μ1, μ2, ..., μp denote these exact moments. Each μk will be an expression involving the p unknown parameters For many common distributions, the expression for the moments are well-documents. So, you can just look them up!
19 • Step 3: Solve the following system of p equations for the p unknowns parameters: µk = mk k = 1,...
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This note was uploaded on 10/26/2010 for the course OR&IE 5580 at Cornell University (Engineering School).