![]() A quick caveat: you may have noticed that we could immediately written the second parameter, \(\sigma^2\), in terms of the first and second moments because we know \(Var(X) = E(X^2) - E(X)^2\). Go back and make sure you can follow these steps for what we did here with the Normal. Plug in the sample moments for the moments. Solve for the parameters in terms of the moments. Write the moments of the distribution in terms of the parameters (if you have \(k\) parameters, you will have to write out \(k\) moments). Hopefully you followed what we did here…if not, here’s a checklist that summarizes the process: This is magical! We’ve just written great estimators for our parameters using just data that we’ve sampled. (a) Find the method-of-moments estimator of e (b) Is your estimator in part (a) an unbiased estimator of (c) Given the following 5 observations of X, give a point estimate of : 6.61 7.70 6.98 8.36 7. Looks like we got back to the original parameters. Finding the method of moments estimator example.Thanks for watching //Another method of moments video (finding the MoM estimator based on Kth moment)http. Let X1, X2.Xn be a random sample from a uniform distribution on the interval (-1,0 + 1). Therefore, your MOM estimator is approximately normally distributed with mean equal to 2/3 3/2 theta theta, and variance equal to 4/9 theta2/ (12n) theta2/ (27n). ![]() Where \(\mu^k\) is just our notation for the \(k^) # check if estimates are close: mean(sample_means) mean(sample_var) # 4.939076 # 8.958363 Under the CLT, the sample mean is approximately normal with mean equal to 1.5 theta and variance equal to theta2/ (12n). Recall from probability theory hat the moments of a distribution are given by: This is the first ‘new’ estimator learned in Inference, and, like a lot of the concepts in the book, really relies on a solid understanding of the jargon from the first chapter to nail down. 4.3 Maximum Likelihood Confidence Intervals. ![]() ![]() 2.4.2 Hypothesis Testing for Proportions.2.3.2 Confidence Interval for Proportions.2.3.1 Confidence Interval for the Mean, Unknown Variance. ![]()
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