1 Maximum likelihood review questions Set 5 revised November 25 2010 Happy Thanksgiving See also the student questions and answers that I distributed fall 2007 1 Specify a statistical or econometric model with at least one parameter a model of some real world process Be as speci c as possible Assume a random sample from the population implied by your model Describe in words and with functional notation how you would derive the maximum likelihood estimate of the parameter s Then derive the ml parameter estimate s Do not choose the CLR model as your example Brie y describe what in general is good and bad about maximum likelihood estimation Then brie y describe the realisim of your model 2 Consider a zoo that has only chimps and elephants where poopj is the weight in pounds of poop animal j produces on a given day Sj 1 if the animal is a elephant and zero otherwise The amount of poop animal j produces on a given day is a draw from a Poisson distribution some animals on some days are constipated with parameter j where j Sj The director of the zoo likes only chimps but knows that chimps like to ride on the backs of elephants so has 90 chimps and 10 elephants lots of chimps can ride one elephant Write down the density function for poop for this population Explain why this is the density function Derive the expected amount of poop produced per day Comment brie y on the likelihood of the poop being Poisson distributed rather than normally distributed answer to the rst part The distribution function f poop is what is called a mixture distribution or a contagious distribution In this case a mixture of two densities one for chimps and the one for elephants poop for For elephants the distribution function is fe poop e poop poop x 0 1 2 and 0 otherwise For chimps it is fc poop e poop At this zoo chimps appear with probability pc 9 and elephants appear poop poop with probability pe 1 So f poop 1 e 9 e p p for x 0 1 2 and 0 otherwise This is an application of the following theorem If one has a sequence fo x fX 1 x of density functions X and a sequence po p1 such that pi 0 and pi 1 then f x pi fi x The expected amount of poop is 9 1 1 0 0 1 the simple weighted average On a given day an animal can t poop a negative amount The Poisson is consistent with this restriction the normal is inconsistent with this restriction the normal assocates positive probability with negative amounts of poop The Poisson assumes poop is discretely distributed animals poop in units of 0 1 2 but not 1 34 units The normal allows all animals to 1 poop in other than integer amounts The latter is more likely On the other hand the Poisson allows for the possibility that on a give day the animal produces no poop the normal does not Second part of question Assume on a randomly chosen day Jane the zookeeper independently collects poop from two animals and the poops weighs 3 pounds and 1 pound Write down the likelihood function for this sample and explain in words why you have written down the correct function answer to the second part We want to nd the values of and that maximixe the probability of a sample of 3 and 1 We know p p 9 e p So Pr 3 that Pr poop f poop 1 e p 3 3 1 e 9 e 3 And Pr 1 1 e 9 e 3 So the probability of observing this sample of two independent observa 3 3 tions is Pr 3 Pr 1 1 e 9 e 3 1 e 3 9 e L j3 1 The maximum likelihood estimates are those values of a and aml and ml that maximizes this function L j3 1 Third part of question Now assume that God informs you that the maximimum likelihood estimates of are either 1 1 or 1 4 What are the maximum likelihood estimates answer to part 3 L 1 1 j3 1 1 e 1 1 1 3 1 1 3 9 e 3 1 1 e 1 1 1 3 1 4 1 4 3 L 1 4 j3 1 1 e 3 2 1 9 e 1 1 2 622 7 10 2 and 1 3 9 e 3 1 1 e 1 4 1 4 9 e 1 1 2 315 1 10 The second set generate a larger probability likelihood of observing the sample so the ml estimates are 1 and 4 Note the maximum likelihood estimate of the expected amount of amount of poop per animal is 1 0 0 1 1 0 1 4 1 4 3 Assume that the rv X has a uniform distribution 1 if a x b b a fX x a b 0 if otherwise where b a Assume a random sample of three observations 9 4 and 12 Derive the maximum likelihood estimates of the parameters a and b Show all of your work and explain all of your step The parameter b 12 otherwise it would be impossible to observe a 12 The parameter a 9 otherwise it would be impossible to observe a 9 the observations must be in the range where there is positive density Can we be more speci c about the maximum likelihood estimates of a and b Yes The liklihood function is L 9 4 12 a b 1 b a 2 1 b a 3 b 1 b a a 3 1 b a And ln L 3 ln b a This is maximized by minimizing b a subject to the constraints that a 9 and b 12 So aml 9 and bml 12 4 Consider the density function 2 a2 x fX x a 0 if 0 x a otherwise Find the maximum likelihood estimators of the mean and variance 2X of this function assuming some random sample X1 X2 Xn where n is the sample size How do you know you have the ml estimators of and 2 X Za 2 Note that a2 x 08x 0 and a22 x dx 1 so this is a density function 0 Graphically if a 3 f x 0 625 0 5 0 375 0 25 0 125 0 0 0 5 1 1 5 2 2 5 3 x And if a 10 3 f x 0 2 0 15 0 1 0 05 0 0 2 5 5 7 5 10 x f x 0 2 0 15 0 1 0 05 0 0 2 5 5 7 5 10 x One proceeds by rst nding the ml estimator of the parameter a assuming some random sample X1 X2 Xn Then one plugs aml into the formula for the mean and the formula for the variance of this density function The rst thing to note is that fX x a only has positive density in the range 0 x a …