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Statistics

Statistics. Statistics: 1. Model 2. Estimation 3. Hypothesis test.            . Estimation Estimat. Definition:

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Statistics

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  1. Statistics Statistics: 1. Model 2. Estimation 3. Hypothesis test lecture 7

  2.             . . . . . . EstimationEstimat Definition: A (point) estimate of a parameter, , in the model is a “guess”at what can be (based on the sample). The corresponding random variable Q is called an estimator. ^ ^ parameter estimat estimator Sample Population ^   lecture 7

  3. EstimationUnbiased estimate Definition: An estimator Q is said to be unbiased if E(Q) = q ^ ^ lecture 7

  4. Confidence interval for the mean • Example: • In a sample of 20 chocolate bars the amount of calories has been measured: • the sample mean is 224 calories • How certain are we that the population mean  is close to 224? • The confidence interval (CI) for m helps us here! lecture 7

  5. Confidence interval for Known variance Let x be the average of a sample consisting of n observations from a population with mean  and variance 2 (known). A (1-)100% confidence interval for  is given by From the standard normal distribution N(0,1) • We are (1-)100% confidentthat the unknown parameter  lies in the CI. • We are (1-)100% confidentthat the error we make by using x as an estimate of  does not exceed (from which we can find n for a given error tolerance). lecture 7

  6. Confidence interval for Known variance • Confidence interval for known variance is a results of the • Central Limits Theorem.The underlying assumptions: • sample size n > 30, or • the corresponding random var. is (approx.) normally distributed • (1 - )100% confidence interval : • typical values of :  =10% ,  = 5% ,  = 1% • what happens to the length of the CI when n increases? lecture 7

  7. Confidence interval for Known variance • Problem: • In a sample of 20 chocolate bars the amount of calories has been measured: • the sample mean is 224 calories • In addition we assume: • the corresponding random variable is approx. normally distributed • the population standard deviation  = 10 • Calculate 90% and 95% confidence interval for  • Which confidence interval is the longest? lecture 7

  8. Confidence interval for Unknown variance Let x be the mean and s the sample standard deviation of a sample of n observations from a normal distributed population with mean  and unknown variance. A (1-)100 % confidence interval for  is given by From the t distribution with n-1 degrees of freedom t(n -1) • Not necessarily normally distributed, just approx. normal distributed. • For n > 30 the standard normal distribution can be used instead of the t distribution. • We are (1-)100% confidentthat the unknown  lies in the CI. lecture 7

  9. Confidence interval for Unknown variance • Problem: • In a sample of 20 chocolate bars the amount of calories has been measured: • the sample mean is 224 calories • the sample standard deviation is 10 • Calculate 90% and 95% confidence intervals for  • How does the lengths of these confidence intervals compare to those we obtained when the variance was known? lecture 7

  10. Confidence interval for Using the computer • MATLAB: If x contains the data we can obtain a (1-alpha)100%confidence interval as follow: • mean(x) + [-1 1] * tinv(1-alpha/2,size(x,1)-1) * std(x)/sqrt(size(x,1)) • where • size(x,1)is the size n of the sample • tinv(1-alpha/2,size(x,1)-1) = ta/2(n-1) • std(x) = s (sample standard deviation) • R: • mean(x) + c(-1,1) * qt(1-alpha/2,length(x)-1) * sd(x)/sqrt(length(x)) lecture 7

  11. Confidence interval for s2 • Example: • In a sample of 20 chocolate bars the amount of calories has been measured: • sample standard deviation is10 • How certain are we that the population variance2 is close to 102? The confidence interval for 2 helps us answer this! lecture 7

  12. Confidence interval for s2 Let s be the standard deviation of a sample consisting of n observations from a normal distributed population with variance 2. A (1-) 100% confidence interval for 2 is given by From 2 distribution with n-1 degrees of freedom We are (1-)100%confident that the unknown parameter 2 lies in the CI. lecture 7

  13. Confidence interval for s2 • Problem: • In a sample of 20 chocolate bars the amount of calories has been measured: • sample standard deviation is10 • Find the 90% and 95% confidence intervals for s2 lecture 7

  14. Confidence interval for s2Using the computer MATLAB: If x contains the data we can obtain a (1-alpha)100%confidence interval for s2 as follow: (size(x,1)-1)*std(x)./ chi2inv([1-alpha/2 alpha/2],size(x,1)-1) R: (length(x)-1)*sd(x))/ qchisq(c(1-alpha/2,alpha/2), length(x)-1) lecture 7

  15. Maximum Likelihood EstimationThe likelihood function Assume that X1,...,Xnare random variables with joint density/probability function where q is the parameter (vector) of the distribution. Considering the above function as a function of q given the data x1,...,xnwe obtain the likelihood function lecture 7

  16. Maximum Likelihood EstimationThe likelihood function Reminder: If X1,...,Xnare independent random variables with identical marginal probability/ density function f(x;q), then the joint probability / density function is Definition: Given independent observations x1,...,xnfrom the probability / density function f(x;q) the maximum likelihood estimate (MLE) q is the value of q which maximizes the likelihood function lecture 7

  17. Maximum Likelihood EstimationExample Assume that X1,...,Xnare a sample from a normal population with mean m and variance s2. Then the marginal density for each random variable is Accordingly the joint density is The logarithm of the likelihood function is lecture 7

  18. Maximum Likelihood EstimationExample We find the maximum likelihood estimates by maximizing the log-likelihood: which implies . For s2 we have which implies Notice , i.e. the MLE is biased! lecture 7

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