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 1 2 , ,..., n y y y where ( 1) i P Y ! " " , ( 0) 1 i P Y ! " " # a) Show that the probability function for i Y can be written 1 ( ) (1 ) , 0,1 y y p y y ! ! # " # " We have 0 1 1 0 (0) (1 ) 1 , (0) (1 ) p p ! ! ! ! ! ! " # " # " # " 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 ! . 1 1 1 1 2 2 1 1 ( ) ( ) ( ) ... ( ) (using independence) (1 ) ... (1 ) (1 ) n n i i n n y y y y y n y L P Y y P Y y P Y y ! ! ! ! ! ! ! # # # " " $ " $ $ " " # $ $ # % % " # and the log-likelihood function is ( ) ln ( ) ln( ) ( )ln(1 ) i i 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 (1 ) i i i i y n y y n y dl d l d d ! ! ! ! ! ! ! ! # # " # " # # # # % % % % so to find the MLE, set ( ) 0 dl d ! ! " and solve. We have 0 ˆ ˆ 1 i i 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 1 2 ˆ ˆ ( , ,..., ) n y y y ! ! " is a sufficient statistic. We can express ( ) l ! as ˆ ˆ ( ) [ ln( ) (1 )ln(1 )] l n ! ! ! ! ! " & # # so that ( ) l ! depends on 1 2 , ,..., n y y y only through ˆ ! . Hence 1 2 ˆ ˆ ( , ,..., ) n y y y ! ! " is a sufficient statistic. e) Find [ ] E ! ! and [ ] Var ! ! . We have 1 n i i Y Y n n ! " " " % ! where ~ ( , ) Y binom n ! . Hence [ ] [ ] (1 ) [ ] , [ ] E Y Var Y E Var n n n ! ! ! ! ! # " " " " ! ! . f) Suppose 20 1 20, 12 i i n y " " " % . What is the relative likelihood function ( ) R ! ?
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2 ( ) 1 ( ) ( ) ( ) ˆ ˆ ˆ ( ) 1 i i 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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