Lecture19_Solutions

Lecture19_Solutions - Lecture 17 Exercises 1. Suppose in a...

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1 Lecture 17 Exercises 1. Suppose in a sequence of Bernouilli trials, we observe 12 , ,..., n y yy where ( 1) i PY ! "" , ( 0) 1 i " " # a) Show that the probability function for i Y can be written 1 ( ) (1 ) , 0,1 p y y !! # " # " We have 0 1 1 0 (0) ) 1 , (0) ) pp " # " # " # " as required. b) Construct the likelihood function () L and the log-likelihood function l The likelihood function is the probability of the observed data, as a function of . 11 1 1 2 2 1 1 ( ) ( ) ( ) ... ( ) (using independence) ) ... ) ) nn ii y n y L P Y y P Y y P Y y # # # " " $ " $ $ " " # $ $ # %% "# and the log-likelihood function is ( ) ln ( ) ln( ) ( )ln(1 ) l L y n y " " & # # % % c) Find the MLE ˆ . Be sure to show that it is a maximum. We have 2 2 2 2 ( ) ( ) , 1 ) i i i i y n y y n y dl d l dd ## " # " # # % % % % so to find the MLE, set 0 dl d " and solve. We have 0 ˆˆ 1 y n y # #" # or ˆ i y n " % . Since the second derivative at ˆ is negative, we know that we have found a maximum of the log likelihood. d) Show that ( , ,..., ) n y y y " is a sufficient statistic. We can express l as ( ) [ ln( ) (1 )ln(1 )] ln " & # # so that l depends on , ,..., n y y y only through ˆ . Hence ( , ,..., ) n y y y " is a sufficient statistic. e) Find [ ] E ! and [ ] Var ! . We have 1 n i i Y Y " % ! where ~ ( , ) Y binom n . Hence [ ] [ ] ) [ ] , [ ] E Y Var Y E Var n n n # " " " " . f) Suppose 20 1 20, 12 i i ny " % . What is the relative likelihood function ( ) R ?
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2 ( ) 1 ( ) ( ) ( ) ˆ ˆ ˆ ( ) 1 ii y n y L R L ! !! # # %% "" # g) A plot of ( ) R is shown below. What does the plot tell us about an hypothesized value 0.4 " ? From the plot, we see that (0.4) 0.2 R and so the observed data is about 5 times more probable if .6 " than if .4 " so that .4 " is somewhat implausible, given the observed data.
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This note was uploaded on 07/13/2010 for the course STAT 330 taught by Professor Paulasmith during the Spring '08 term at Waterloo.

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Lecture19_Solutions - Lecture 17 Exercises 1. Suppose in a...

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