Point Estimation by MLE Lesson 5

Transcription

1 Poit Estimatio b MLE Lesso 5

2 Review Defied Likelihood Maximum Likelihood Estimatio Step : Costruct Likelihood Step : Maximize fuctio Take Log of likelihood fuctio Take derivative of fuctio Set derivative 0 Solve for parameter

3 Poit estimatio Goal: 'fit' a model to data Sigle best value of a parameter that fids most support i the data set Cotrasts with iterval estimatio (e.g. 95% cofidece iterval) Examples: N SS x x i N N SS xi x N

4 Maximum Likelihood Step : Write a likelihood fuctio describig the likelihood of the observatio Step : Fid the value of the model parameter that maximized the likelihood dl 0 d

5 a L l L l a a0 0. da a l L ML e

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7 A secod data poit Suppose a secod plat dies at da 4 Step : Defie the likelihood L Pr (a, a ρ) Pr (a a, ρ) Pr (a ρ) Pr (a ρ) Pr (a ρ) Exp(a ρ) Exp(a ρ) Assume measuremets are idepedet

8 Step : Fid the maximum a L l L l a a a a da a a l L ML e e a

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11 A whole data set Step : Defie Likelihood L Pr a, a,, a Pr ai Exp ai Assume measuremets are idepedet

12 Step : Fid the maximum L e ai l L l ai l ai l L ML ai 0 ai / a

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14 Mortalit Alterate data First example based o kowig times idividuals died What if istead we ol kew the fial outcome N umber of idividual plats umber that survived Wat to fid the MLE for q probabilit of survivig to the ed of the expm't

15 Step : Write the likelihood L Biom N, N

16 Step : Write the likelihood L Biom N, Step : Fid the maximum ll l N l N

17 Step : Write the likelihood L Biom N, Step : Fid the maximum ll l N l ll N N

18 Step : Write the likelihood L Biom N, N Step : Fid the maximum ll l N l ll N N

19 Step : Write the likelihood L Biom N, N Step : Fid the maximum ll l N l ll N N N ML

20 Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N

21 Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N P survive to T P dead b T

22 Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N P survive to T P dead b T T e e T Expoetial CDF

23 Mortalit combiig age of death ad survival till ed Step Write the likelihood L Biom N, N P survive to T P dead b T T e e T L Biom N, e T T N e

24 Step : Fid the maximum Le T e T N

25 Step : Fid the maximum Le ll T e T N T N l e T

26 Step : Fid the maximum Le ll ll T e T N T N l e T T Te T N T e

27 Step : Fid the maximum Le ll ll ML T e T N T N l e T T Te T N T e l / N T

28 Survival Aalsis L Biom N, Died N Survived

29 Survival Aalsis L Biom N, L i i N

30 Survival Aalsis L Biom N, L i i Mortalit at time a L e ai N

31 Survival Aalsis L Biom N, L i i Mortalit at time a L e Cesored at time c ai k e ci N

32 Survival Aalsis L Biom N, N L i i Mortalit at time a L e Survived to time T Cesored at time c a i k e c i e T k

33 L e a i k e c i e T k

34 L e a i k e c i e T k k ll l ai c i T k

35 L e a i k e c i e T k k ll l ai c i T k k ll ai c i T k 0

36 L e a i k e c i e T k k ll l ai c i T k k ll ai c i T k 0 ML k ai c i T k

37 MLE for the Normal L N i, [ i exp ]

38 L ll [ i exp ] l l i

39 L ll l L exp [ i ] l l i i 0

40 L ll l L exp [ i ] l l i i i 0

41 L exp [ i ] l l i ll l L i ML i 0

42 Normal Variace ll l l i

43 Normal Variace ll l L l l i 3 i 0

44 Normal Variace ll l L l l i 3 i 0 3 i

45 Normal Variace ll l L l l i 3 i 0 3 i i

46 Populatio Growth Rate t t t e t t e r r t l t l t r t x t x t r t t ~ N 0, x t ~ N x t r,

47 Step : Write Likelihood T L N x t x t r, t [ T T exp x t x t r t ]

48 Step : Write Likelihood T L N x t x t r, t [ T T exp x t x t r t Step : Maximize Likelihood Take logs T l L T l x t x t r t ]

49 Solve for mea T l L T l x t x t r t T l L x t x t r 0 r t

50 Solve for mea T l L T l x t x t r t T l L x t x t r 0 r t T T r x t x t t

51 Solve for mea T l L T l x t x t r t T l L x t x t r 0 r t T T r x t x t t x T x 0 r ML T

52 Solve for variace T l L T l x t x t r t

53 Solve for variace T l L T l x t x t r t T l L T 3 x t x t r 0 t

54 Solve for variace T l L T l x t x t r t T l L T 3 x t x t r 0 t ML T x t x t r T t

55 Homework You wat to kow the desit of fish i a set of experimetal pods You observe the followig couts i te pods: 5,6,7,3,6,5,8,4,4,3 What is our process model? What is our data model? Solve for the aaltical MLE What is the estimate for this populatio?