Métodos de reamostragem - Unicampcnaber/aula_Boot.pdf · 2012. 6. 8. · Title: Métodos de...

Motivacao Exemplos Exemplos

Metodos de reamostragem

Prof. Caio Azevedo

Prof. Caio Azevedo



Notacoes

Seja X = (X1, ...,Xn) uma amostra iid de X ∼ F (, θ), θ ∈ Θ pode

ser vetor.

Seja x = (x1, ..., xn) os valores observados da amostra definida

acima.

Estimador : θ = f (X), estimativa :θ = f (x).

Variancia: Var(θ) e erro-padrao: EP(θ) =

√Var(θ) associados ao

estimador.

Prof. Caio Azevedo



Notacoes cont.

Estimadores para a variancia Var(θ) e o erro-padrao

EP(θ) =

√Var(θ) associados ao estimador.

Estimativas para a variancia Var(θ) e o erro-padrao

EP(θ) =

√Var(θ) do associados ao estimador.

Prof. Caio Azevedo



Exemplo N(µ, σ2)

Seja X1, ...,Xn uma amostra iid de X ∼ N(µ, σ2), (µ, σ2)

desconhecidos e x1, ..., xn valores observados.

Estimador para µ, µ = 1n

∑ni=1 Xi = X . Estimativa

µ = 1n

∑ni=1 xi = x .

Variancia do estimador: Var(µ) = σ2

n . Estimador para a variancia do

estimador Var(θ) = σ2

n .

Estimativa para a variancia do estimador Var(θ) = σ2

n .

Prof. Caio Azevedo



Exemplo N(µ, σ2)

Seja X1, ...,Xn uma amostra iid de X ∼ N(µ, σ2), (µ, σ2)

desconhecidos.

Dispomos de valores observados relativos a uma amostra selecionada

a partir de uma populacao de interesse.

Estimadores otimos (consistentes e nao-viciados),

µ =1

n

n∑i=1

Xi = X , σ2 =1

n − 1

n∑i=1

(Xi − X

)2= S2.

Prof. Caio Azevedo



Cont.

Sabemos que µ ∼ N(µ, σ2/n) e (n−1)σ2

σ2 ∼ χ2(n−1).

Podemos constuir intervalos de confianca (IC) e testar hipoteses de

interesse, com base no resultado acima.

Mesmo que X1, ...,Xn nao tenham distribuicao normal

(aproximadamente) se a sequencia a qual elas pertencem satisfizer o

TCL, os resultados acima sao validos, assintoticamente.

Questao: suponha que queremos fazer inferencia com respeito a

θ = σµ , σ =

√σ2, sem usar resultados assintoticos e sob suposicao de

normalidade.

Prof. Caio Azevedo



Cont.

Estimador natural de θ, θ = σµ , σ =

√σ2.

Qual a distribuicao exata de θ? Qual a expressao para o erro-padrao

associado a este estimador?

Podemos obter uma aproximacao empırica (nao assintotica) de θ

atraves da suposicao de normalidade.

Replicas de Monte Carlo.

Prof. Caio Azevedo



Procedimento

Com base na amostra observada x, estima-se µ e σ2 atraves do

estimadores propostos.

Com as estimativas µ e σ2, gera-se R vezes amostras de tamanho

m, ou seja temos para r = 1, ...,R, m valores simulados de uma

N(µ, σ2), Xr1, ...,Xrm.

Para cada conjunto r de valores simulados, calcula-se

µ∗r = 1m

∑mi=1 xri , (σ∗)2

r = 1m−1

∑mi=1 (xri − µ∗r )2 e θ∗r =

√(σ∗)2

r

µ∗r

.

Assim, os valores θ∗1 , ..., θ∗R serao uma amostra aleatoria de θ.

Prof. Caio Azevedo



Resultados

Uma estimativa (refinada) de θ e θ∗ = 1R

∑Rr=1 θ

∗r (podemos utilizar,

tambem, a estimativa pontual baseada na amostra observada).

Uma estimativa da variancia do estimador proposto e

var(θ) = 1R−1

∑Rr=1

(θ∗ − θ∗r

)2

. Estimativa do vıcio

B(θ) = (θ∗ − θ)

Sejam θ∗(1), θ∗(2), ...., θ

∗(R) as estatısticas de ordem da amostra gerada.

Quantis empıricos podem ser obtidos atraves de um dos algoritmos

propostos por Hyndman and Fan (1996).

Funcao “quantile” no R.

Prof. Caio Azevedo



Amostras Bootstrap 1

n = 10 m = 10 media = −0.09 dp = 183.54

valores

de

nsi

da

de

−10000 −5000 0

0e

+0

04

e−

04

8e

−0

4

n = 30 m = 30 media = 8.37 dp = 164.02

valores

de

nsi

da

de

0 2000 4000 6000 8000 10000

0e

+0

04

e−

04

8e

−0

4

n = 50 m = 50 media = 3.37 dp = 9.24

valores

de

nsi

da

de

−200 −100 0 100 200 300 400

0.0

00

0.0

10

0.0

20

n = 100 m = 100 media = 1.91 dp = 0.44

valores

de

nsi

da

de

2 4 6 8 10

0.0

0.2

0.4

0.6

Prof. Caio Azevedo




n = 200 m = 200 media = 2.62 dp = 0.54

valores

de

nsi

da

de

1 2 3 4 5 6 7 8

0.0

0.2

0.4

0.6

n = 500 m = 500 media = 1.97 dp = 0.19

valores

de

nsi

da

de

1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

n = 800 m = 800 media = 2.09 dp = 0.16

valores

de

nsi

da

de

1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

n = 1000 m = 1000 media = 2.02 dp = 0.14

valores

de

nsi

da

de

1.6 1.8 2.0 2.2 2.4

0.0

1.0

2.0

3.0

Prof. Caio Azevedo



Exemplo beta(a, b)

Suponha uma amostra aleatoria de uma distribuicao beta(a,b).

Com base na amostra observada x, estima-se a e b atraves do

estimadores do metodo dos momentos.

Com as estimativas a e b, gera-se R vezes amostras de tamanho m,

ou seja temos para r = 1, ...,R, m valores simulados de uma

beta(a, b), Xr1, ...,Xrm.

Para cada conjunto r de valores simulados, calcula-se

µ∗r = 1m

∑mi=1 xri , (σ∗)2

r = 1m−1

∑mi=1 (xri − µ∗r )2 e θ∗r =

√(σ∗)2

r

µ∗r

.

Assim, os valores θ∗1 , ..., θ∗R serao uma amostra aleatoria de θ.

Prof. Caio Azevedo




n = 10 m = 10 media = 1.16 dp = 0.29

valores

de

nsi

da

de

0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

n = 30 m = 30 media = 1.44 dp = 0.22

valores

de

nsi

da

de

1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

n = 50 m = 50 media = 0.99 dp = 0.1

valores

de

nsi

da

de

0.8 1.0 1.2 1.4

01

23

4

n = 100 m = 100 media = 1.25 dp = 0.09

valores

de

nsi

da

de

1.0 1.2 1.4 1.6

01

23

4

Prof. Caio Azevedo




n = 200 m = 200 media = 0.97 dp = 0.05

valores

de

nsi

da

de

0.8 0.9 1.0 1.1

02

46

8

n = 500 m = 500 media = 1.18 dp = 0.04

valores

de

nsi

da

de

1.05 1.10 1.15 1.20 1.25 1.30 1.35

02

46

81

0

n = 800 m = 800 media = 1.09 dp = 0.03

valores

de

nsi

da

de

1.00 1.05 1.10 1.15 1.20

02

46

81

01

4

n = 1000 m = 1000 media = 1.09 dp = 0.02

valores

de

nsi

da

de

1.00 1.05 1.10 1.15 1.20

05

10

15

Prof. Caio Azevedo



Bootstrap parametrico

Sob a suposicao de um modelo parametrico F (,θ) e com a amostra

selecionada, estima-se θ atraves de algum metodo apropriado.

Com a estimativa θ, gera-se R vezes uma amostra aleatoria de

tamanho m (m ≤ n) da amostra x = (x1, ..., xn).

Para cada amostra r r = 1, ...,R obtem-se a estimativa desejada θ∗r .

Estimativa Bootstrap θ∗ = 1R

∑Rr=1 θ

∗r

Prof. Caio Azevedo



Caso geral

Suponha agora que nao conhecemos a distribuicao (mesmo uma boa

aproximacao) geradora da amostra

Nao queremos fazer nenhuma suposicao a respeito da distribuicao.

Caso conhecamos ou queiramos fazer alguma suposicao, por algum

motivo, e difıcil de gerar da variavel aleatoria em questao.

Prof. Caio Azevedo



Bootstrap nao-parametrico

Seleciona-se R vezes uma amostra aleatoria (com reposicao) de

tamanho m (m ≤ n) da amostra x = (x1, ..., xn).

Para cada amostra r r = 1, ...,R obtem-se a estimativa desejada θ∗r .

Estimativa Bootstrap θ∗ = 1R

∑Rr=1 θ

∗r

Bootstrap parametrico: conhecemos a distribuicao dos dados F e

simulamos valores a partir dessa distribuicao usando estimativas dos

parametros relacionados a ela.

Bootstrap nao-parametrico: nao conhecemos a distribuicao dos

dados F e simulamos valores a partir da fda empırica.

Prof. Caio Azevedo



Amostras Bootstrap Nao-parametrico Normal (1)

n = 10 m = 10 media = 2.09 dp = 58.83

valores

de

nsi

da

de

−4000 −3000 −2000 −1000 0 1000

0.0

00

00

.00

10

n = 30 m = 30 media = 13.86 dp = 673.85

valores

de

nsi

da

de

−30000 −20000 −10000 0 10000 20000 30000

0.0

00

00

0.0

00

10

n = 50 m = 50 media = 2.29 dp = 1.23

valores

de

nsi

da

de

−40 −30 −20 −10 0 10 20

0.0

00

.05

0.1

00

.15

0.2

0

n = 100 m = 100 media = 2.66 dp = 1.12

valores

de

nsi

da

de

0 10 20 30 40 50

0.0

00

.05

0.1

00

.15

0.2

0

Prof. Caio Azevedo



Amostras Bootstrap Nao-parametrico Normal (2)

n = 200 m = 200 media = 2.76 dp = 0.62

valores

de

nsi

da

de

2 4 6 8

0.0

0.2

0.4

0.6

n = 500 m = 500 media = 2.32 dp = 0.27

valores

de

nsi

da

de

2.0 2.5 3.0 3.5

0.0

0.5

1.0

1.5

n = 800 m = 800 media = 2.13 dp = 0.18

valores

de

nsi

da

de

1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

n = 1000 m = 1000 media = 2.07 dp = 0.15

valores

de

nsi

da

de

1.6 1.8 2.0 2.2 2.4 2.6 2.8

0.0

0.5

1.0

1.5

2.0

2.5

Prof. Caio Azevedo



Amostras Bootstrap Nao-parametrico Beta (1)

n = 10 m = 10 media = 0.54 dp = 0.16

valores

de

nsi

da

de

0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.5

1.0

1.5

2.0

2.5

n = 30 m = 30 media = 0.89 dp = 0.11

valores

de

nsi

da

de

0.6 0.8 1.0 1.2 1.4

01

23

n = 50 m = 50 media = 1.12 dp = 0.12

valores

de

nsi

da

de

0.8 1.0 1.2 1.4 1.6 1.8

0.0

1.0

2.0

3.0

n = 100 m = 100 media = 1.16 dp = 0.08

valores

de

nsi

da

de

0.9 1.0 1.1 1.2 1.3 1.4 1.5

01

23

4

Prof. Caio Azevedo



Amostras Bootstrap Nao-parametrico Beta (2)

n = 200 m = 200 media = 1.03 dp = 0.05

valores

de

nsi

da

de

0.9 1.0 1.1 1.2

01

23

45

67

n = 500 m = 500 media = 1.05 dp = 0.03

valores

de

nsi

da

de

0.95 1.00 1.05 1.10 1.15 1.20

02

46

81

01

2

n = 800 m = 800 media = 1.05 dp = 0.03

valores

de

nsi

da

de

0.95 1.00 1.05 1.10 1.15

05

10

15

n = 1000 m = 1000 media = 1.08 dp = 0.02

valores

de

nsi

da

de

1.00 1.05 1.10 1.15 1.20

05

10

15

Prof. Caio Azevedo



Comparacoes

Bootstrap parametrico: conhecemos a distribuicao dos dados F e

simulamos valores a partir dessa distribuicao usando estimativas dos

parametros relacionados a ela. Dependente: qualidade das

estimativas e do metodo de simulacao escolhido.

Bootstrap nao-parametrico: nao conhecemos a distribuicao dos

dados F e simulamos valores a partir da fda empırica. Com ou sem

reposicao.

Prof. Caio Azevedo



Exemplo: correlacao entre media e variancia amostrais

Sabemos que a corre(X , σ2) = 0, se X ∼ N(µ, σ2).

Contudo, isso nao esperado para outras variaveis aleatorias.

Ademais, o TCL sugere que a distrbuicao conjunta e marginais de X

e σ2, sejam normais.

Prof. Caio Azevedo



Amostras Bootstrap de uma normal, n=10

n = 10 m = 10 media amostral

valores

de

nsi

da

de

−2 −1 0 1

0.0

0.2

0.4

0.6

0.8

n = 10 m = 10 variancia amostral

valores

de

nsi

da

de

0 2 4 6 8

0.0

00

.10

0.2

0

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−2 −1 0 1

02

46

8

n = 10 m = 10 correlacao entre a media e a variancia amostrais

media amostral

vari

an

cia

am

ost

ral

n = 10 m = 10 Densidade bivariada

media amostral

var

am

ost

ral

0.02

0.04

0.06

0.08 0.1

0.12

0.12

0.14

0.16

−2 −1 0 1

02

46

8

Prof. Caio Azevedo





valores

de

nsi

da

de

−0.5 0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8


valores

de

nsi

da

de

2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

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24

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0.5 1.0 1.5

45

67

89


media amostral

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an

cia

am

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media amostral

var

am

ost

ral

0.1

0.2 0.3

0.4 0.5 0.6 0.7

0.5 1.0 1.5

45

67

89

Prof. Caio Azevedo



Amostras Bootstrap de uma gama, n=10


valores

de

nsi

da

de

8 10 12 14 16 18 20 22

0.0

00

.05

0.1

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valores

de

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de

0 20 40 60 80 100

0.0

00

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media amostral

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Prof. Caio Azevedo





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valores

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9 10 11 12 13

20

30

40

50

60


media amostral

vari

an

cia

am

ost

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media amostral

var

am

ost

ral

0.005

0.01

0.015

0.02

0.025

0.03

0.035

9 10 11 12 13

20

30

40

50

60

Prof. Caio Azevedo



Resumo

Queremos estimar θ relativo ou nao a F (, θ). Para isso, dispoe-se de

uma amostra observada de F (; θ).

Define-se um estimador θ, eg, maxima verossimilhanca, momentos,

mınimos quadrados. Sejam Var(θ) e EP(θ) a variancia e o

erro-padrao desse estimador (se conhecidas).

Parametrica (usando-se F (; θ)) ou nao parametricamente (usando a

fda empırica da amostra observada), simula-se R vezes valoes,

xr1, ..., xrm.

Em cada uma das r amostras, calcula-se θ∗1 , ..., θ∗R , estimativas

bootstrap.

Prof. Caio Azevedo



Resumo cont.

Estimativa (refinada) bootstrap θ∗ = 1R

∑Rr=1 θ

∗r

Definimos VarB(θ) = 1R−1

∑Rr=1(θ∗r − θ∗)2 e EPB(θ) =

√VarB(θ),

estimativas bootstrap para variancia e o erro-padra do estimador em

questao.

Prof. Caio Azevedo



Intervalos de confianca

Usar o erro-padrao bootstrap, sob a suposicao de normalidade

(mesmo assintotica) do estimador utilizado.

(θ − z 1+γ2EPB(θ), θ + z 1+γ

2EPB(θ))

P(Z < z 1+γ2

) = 1+γ2 ,Z ∼ N(0, 1), EPB() erro-padrao bootstrap

(estimativa).

Usar quantis empıricos das estimativas bootstrap.

(κ 1−γ2, κ 1+γ

2),

κα quantil de ordem α das estimativas bootstrap.

Prof. Caio Azevedo



Exemplo: Poisson

Dados relativos ao numero de acidentes em um ano em 20

autoestradas estadunidenses.

x = (1, 2, 5, 0, 3, 1, 0, 1, 1, 2, 0, 1, 8, 0, 5, 0, 2, 1, 2, 3).

Objetivos, fazer inferencias sobre µ e θ = σµ .

Considere µ = X , σ =√S2 = S e θ = S

X.

Ha indıcios de superdispersao x = 1, 90 < s2 = 4, 31.

Prof. Caio Azevedo



Frequencias relativas e observadas0

.00

0.0

50

.10

0.1

50

.20

0.2

50

.30

Frequencias relativas esperadas e observadas

numero de acidentes

fre

qu

en

cia

re

lativ

a

0 1 2 3 5 8

●

●

●

●

●

●

●

●

Observada

Esperada

Prof. Caio Azevedo



Distribuicao empırica do estimador

Histogram of v.mu.r

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Prof. Caio Azevedo



Resultados

Estimativas intervalares e pontuais

Metodo Estimativa EP IC(95%)

Assintotico 1,90 0,31 [1,30;2,50]

Assintotico com EP bootstrap 1,90 0,46 [1,01;2,79]

Bootstrap 1,91 0,46 [1,10;2,80]

Prof. Caio Azevedo



Teste de Hipotese

Estabeleca as hipoteses de interesse H0,H1.

Escolha uma estatıstica do teste (digamos T) apropriada.

Observacao: Nao e necessario conhecer a distribuicao da estatistica

sob H0.

Calcule o valor observado da estatıstica tobs .

Gere R amostras da distribuicao sob H0.

Para cada amostras, calcule t∗r , r = 1, ...,R.

Calcule a proporcao de vezes que os valores t∗r foram “mais

extremos” que tobs .

Rejeite H0 de acordo com o nıvel de signficancia α.

Prof. Caio Azevedo



Observacao

“Mais extremo”, depende de H1.

Para H0 : θ = θ0 vs H1 : θ < θ0, mais extremo signfica menor.

Para H0 : θ = θ0 vs H1 : θ > θ0, mais extremo signifca maior.

Para H0 : θ = θ0 vs H1 : θ 6= θ0, mais extremo significa

|bootstrap| > |observado|.

Prof. Caio Azevedo



Cont. Exemplo: Poisson

Suponha que queiramos testar H0 : θ = θ0 vs H1 : θ 6= θ0, θ e a

media da Poisson.

Definse a estatıstica T = X−θ0√X/n

. Valor observado t = −0, 324 (para

θ0 = 2).

Gera-se R vezes amostras (bootstrap nao parametrico) e calcula-se

T para cada amostra t∗1 , ..., t∗R .

Dois p-valores p1 =∑R

r=1 11(|t∗r |>|t|)R e p2 =

∑Rr=1 11(|t∗r |>|t|)+1

R+1 .

p1 = 0, 7642 e p2 = 0, 7642.

Prof. Caio Azevedo



Teste de ajuste a uma distribuicao

Seja X = (X1, ...,Xn) uma amostra iid, supostamente de X ∼ F ().

Seja F1() uma outra FDA.

Para H0 : F = F1 vs H1 : F 6= F1.

Repetir o processo acima usando a estatıstica

D = max(Fn(x)− F (x)), em que Fn() e fda empırica.

Outas estatısticas podem ser usadas.

No exemplo da Poisson, H0 : F ∼ Poisson(1, 9), p1 = 0, 5526 e

p2 = 0, 5527

Prof. Caio Azevedo



Exemplo: Correlacoes entre notas, Matematica e Musica

Dados relativos as notas de n = 12 criancas em testes de

Matematica e musica.

Matematica: x = (60, 58, 73, 51, 54, 75, 48, 72, 75, 83, 62, 52).

Musica: y = (80, 62, 70, 83, 62, 92, 79, 88, 54, 82, 64, 69).

Objetivos, fazer inferencias sobre ρ = correlacao(X ,Y ).

Testes parametricos baseados na correlacao de Pearson amostral e

testes nao-parametricos baseados nas correlacoes de Kendall e

Spearman.

Prof. Caio Azevedo



Histograma das amostras bootstraps

correlacao amostral

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Prof. Caio Azevedo



Resultados

Estimativas intervalares e pontuais: ρ = 0, 169(0, 300).

Metodo IC(95%)

Normal [-0,422; 0,754 ]

Basico [-0,365; 0,808 ]

Percentil [-0,470;0,703 ]

BCa [-0,572;0,652]

Prof. Caio Azevedo



Teste de nulidade H0 : ρ = 0 vs H1 : ρ 6= 0

p-valores: 0,5992(Pearson) ; 0,6793 (Kendall) ; 0,6635 (Spearman).

p-valor bootstrap via estatıstica ρ√n−2√1−ρ

: 0,7250.

Prof. Caio Azevedo



Exemplo: Modelo linear generalizado

Seja Yi ∼ gama(µi , φ), i = 1, ..., n, independentes,

E(Yi ) = µi = exp(X′iβ),V(Yi ) = µi

φ .

Estimar (β, φ).

Hipoteses de interesse H0 : βj = 0 vs H1 : βj 6= 0; H0 : φ = 1 vs

H1 : φ 6= 1.

Conjuntos de dados relativos ao desempenho de 5 tipos de turbina

de aviao. Tempo (em unidade de milhoes de cilcos) ate a perda de

velocidade.

Prof. Caio Azevedo



Boxplot dos dados das turbinas

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Prof. Caio Azevedo



Analise descritiva

Medidas descritivas.

Tipo Media DP CV(%) Min Max

1 10,69 3,27 30,58 3,03 16,84

2 6,05 2,46 40,66 3,19 12,75

3 8,64 2,94 34,03 3,46 13,41

4 9,80 3,13 31,95 5,88 25,46

5 14,71 3,83 26,08 6,43 21,51

Prof. Caio Azevedo



Analise Inferencial

Modelo: Yij ∼ gama(µi , φ), i = 1, 2, 3, 4, 5, j = 1, 2, ..., 10

µi = µ+ αi , α1 = 0

Prof. Caio Azevedo



Analise Inferencial

Estimativas de maxima verossimilhanca.

Parametro Est. EP Wald p-valor (t)

µ 2,37 0,14 16,42 < 0,0001

α2 -0,57 0,20 -2,79 0,0076

α3 -0,22 0,20 -1,05 0,3006

α4 -0,09 0,20 -0,43 0,6704

α5 0,32 0,20 1,56 0,1254

φ = 5, 81(1, 13), [3, 59; 8, 01]

Prof. Caio Azevedo



Analise Inferencial Bootstrap

Estimativas de maxima verossimilhanca (bootstrap).

Parametro Est. EP IC(%95)

µ 2,36 0,15 [2,04;2,62]

α2 -0,58 0,21 [-0,96;-0,16]

α3 -0,22 0,19 [-0,58;0,18]

α4 -0,10 0,23 [-0,51;0,38]

α5 0,32 0,18 [-0,02;0,68]

φ = 6, 85(1, 73), [4, 52; 11, 19]

Prof. Caio Azevedo



Verificacao do ajuste do modelo

Resıduos componente do desvio: sob as validades do modelo devem

apresentar distribuicao aproximadamente normal padrao.

Comparar os quantis dos resıduos, com os resıduos teoricos da

N(0,1).

Visualmente, muitas vezes, e complicado avaliar a proximidade dos

quantis.

Solucao: criar bandas de confianca empıricas.

Prof. Caio Azevedo



Grafico QQplot

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Normal Q−Q Plot

Percentis da N(0,1)

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Prof. Caio Azevedo



Procedimento para a construcao do grafico de envelope

1 Ajustar o modelo de regressao, obtendo as estimativas β, φ e os

resıduos componente do desvio, r1, ..., rn.

2 Com as estimativas originais, obtidas no passo 1, gerar n valores,

Y ∗1 , ...,Y∗n , ajustar novamento o modelo de regressao e calcular os

resıduos componenente do desvio.

3 Repetir o passo 2, m vezes, obtendo-se assim,

r∗ij , i = 1, ..., n; j = 1, ...,m

4 Colocamos cada grupo de n resıduos em ordem crescente,

r∗(i)j , i = 1, ..., n; j = 1, ...,m

Prof. Caio Azevedo



Continuacao

1 Para cada um dos m grupos, obtemos os limites r∗(i)I = minj r∗(i)j e

r∗(i)S = maxj r∗(i)j . Assim, os limites correspondentes ao i-esimo

resıduo serao dados por r∗(i)I e r∗(i)S .

Prof. Caio Azevedo



Grafico QQplot com bandas de confianca

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Percentis da N(0,1)

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Normal Q−Q PlotNormal Q−Q PlotNormal Q−Q Plot

Prof. Caio Azevedo


Métodos de reamostragem - Unicampcnaber/aula_Boot.pdf · 2012. 6. 8. · Title: Métodos de...

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Transcript of Métodos de reamostragem - Unicampcnaber/aula_Boot.pdf · 2012. 6. 8. · Title: Métodos de...