Confidence Interval For The Weighted Sum Of Two Binomial Proportions - - PowerPoint PPT Presentation

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Confidence Interval For The Weighted Sum Of Two Binomial Proportions - - PowerPoint PPT Presentation

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Confidence Interval For The Weighted Sum Of Two Binomial Proportions Wojciech Zieli nski Department of Econometrics and Statistics SGGW XLII Wi slana Konferencja Statystyka Matematyczna B edlewo, 2 grudnia 2016 Wojciech Zieli

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Confidence Interval For The Weighted Sum Of Two Binomial Proportions

Wojciech Zieli´ nski

Department of Econometrics and Statistics SGGW XLII Wi´ slana Konferencja „Statystyka Matematyczna” B˛ edlewo, 2 grudnia 2016

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Probability θ of success is estimated.

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Probability θ of success is estimated. A sample of size n is drawn and successes are counted.

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Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Let ξ be the number of successes.

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Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Let ξ be the number of successes. ξ is a random variable binomially distributed.

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Probability θ of success is estimated. A sample of size n is drawn and successes are counted. Let ξ be the number of successes. ξ is a random variable binomially distributed. The statistical model ({0, 1, . . . , n}, {Bin(n, θ), θ ∈ (0, 1)})

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UMV (θ) is ˆ θc = ξ

n

Variance of that estimator equals D2

θ ˆ

θc = θ(1 − θ) n for all θ.

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The Clopper-Pearson symmetric confidence interval at confidence level γ for θ has the form (θc

L(u), θc U(u)), where

θc

L(u) =

  • for u = 0,

B−1 n(1 − u) + 1, nu; 1+γ

2

  • for u > 0,

θc

U(u) =

  • 1

for u = 1, B−1 n(1 − u), nu + 1; 1−γ

2

  • for u < 0.

Here B−1(·, ·; ·) is the quantile of the beta distribution.

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Suppose θ = w1θ1 + w2θ2 w1, w2 ∈ (0, 1) w1 + w2 = 1

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Motivation

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Motivation Suppliers

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Motivation Suppliers DECROUEZ, G. & ROBINSON, A. P . (2012) Aust. N. Z. J. Stat.

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Motivation Suppliers DECROUEZ, G. & ROBINSON, A. P . (2012) Aust. N. Z. J. Stat. Stratification

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Je˙ zeli ζn jest zmienn ˛ a losow ˛ a o rozkładzie Bin(n, π), to (∀π) (∀ε > 0) (∃N(π)) (∀n > N(π)) sup

x∈R

  • P
  • ζn − nπ
  • nπ(1 − π)

≤ x

  • − Φ(x)
  • < ε

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Je˙ zeli ζn jest zmienn ˛ a losow ˛ a o rozkładzie Bin(n, π), to (∀π) (∀ε > 0) (∃N(π)) (∀n > N(π)) sup

x∈R

  • P
  • ζn − nπ
  • nπ(1 − π)

≤ x

  • − Φ(x)
  • < ε

ale nie zachodzi (∀ε > 0) (∃N) (∀n > N) (∀π) sup

x∈R

  • P
  • ζn − nπ
  • nπ(1 − π)

≤ x

  • − Φ(x)
  • < ε

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Je˙ zeli ζn jest zmienn ˛ a losow ˛ a o rozkładzie Bin(n, π), to (∀π) (∀ε > 0) (∃N(π)) (∀n > N(π)) sup

x∈R

  • P
  • ζn − nπ
  • nπ(1 − π)

≤ x

  • − Φ(x)
  • < ε

ale nie zachodzi (∀ε > 0) (∃N) (∀n > N) (∀π) sup

x∈R

  • P
  • ζn − nπ
  • nπ(1 − π)

≤ x

  • − Φ(x)
  • < ε
  • R. Zieli´

nski, Applicationes Mathematicae (2004)

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θ = w1θ1 + w2θ2, w1, w2 ∈ (0, 1), w1 + w2 = 1

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θ = w1θ1 + w2θ2, w1, w2 ∈ (0, 1), w1 + w2 = 1 Let n = n1 + n2.

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θ = w1θ1 + w2θ2, w1, w2 ∈ (0, 1), w1 + w2 = 1 Let n = n1 + n2. There are two random variables ξ1 ∼ Bin(n1, θ1), ξ2 ∼ Bin(n2, θ2),

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θ = w1θ1 + w2θ2, w1, w2 ∈ (0, 1), w1 + w2 = 1 Let n = n1 + n2. There are two random variables ξ1 ∼ Bin(n1, θ1), ξ2 ∼ Bin(n2, θ2), Let ˆ θw = w1 ξ1 n1 + w2 ξ2 n2

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The estimator ˆ θw is an unbiased estimator of θ.

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The estimator ˆ θw is an unbiased estimator of θ. For a given θ there are infinitely many θ1 and θ2 giving θ. Hence we average with respect to θ1 assuming the uniform distribution

  • f θ1 on the interval (aθ, bθ), where

aθ = max

  • 0, θ − w2

w1

  • ; bθ = min
  • 1, θ

w1

  • .

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Eθˆ θw = Eθ

  • w1

ξ1 n1 + w2 ξ2 n2

  • =

1 bθ − aθ bθ

aθ

w1 n1 Eθ1ξ1 + w2 n2 E θ−w1θ1

w2

ξ2

  • dθ1

= 1 bθ − aθ bθ

aθ

w1 n1 n1θ1 + w2 n2 n2 θ − w1θ1 w2

  • dθ1

= θ.

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Let ˆ θw = u be observed. The (symmetric) confidence interval for θ at confidence level γ is (θw

L(u), θw U(u)), where

θw

L(u) =

  • for u = 0,

max{θ : Pθ{ˆ θw < u}} = 1+γ

2

for u > 0, θw

U(u) =

  • 1

for u = 1, min{θ : Pθ{ˆ θw ≤ u}} = 1−γ

2

for u < 1. .

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Pθ{ˆ θw ≤ u} = Pθ

  • w1

ξ1 n1 + w2 ξ2 n2 ≤ u

  • =

1 L(θ) b(θ)

a(θ) n2

  • i2=0

Pθ1

  • ξ1 ≤ n1

w1

  • u − w2

i2 n2

  • Pθ2 {ξ2 = i2} dθ1

θ2 = (θ − w1θ1)/w2

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Pθ{ˆ θw < u} = Pθ

  • w1

ξ1 n1 + w2 ξ2 n2 < u

  • =

1 L(θ) b(θ)

a(θ) n2

  • i2=0

Pθ1

  • ξ1 ≤ n1

w1

  • u − w2

i2 n2

  • − 1
  • Pθ2 {ξ2 = i2} dθ1

θ2 = (θ − w1θ1)/w2

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For given θ ∈ (0, 1), the expected length of the confidence interval equals lw(θ) =

  • u∈U

(θw

U(u) − θw L(u)) Pθ{ˆ

θw = u}1(θw

L(u),θw U (u))(θ),

where U =

  • u = w1

k1 n1 + w2 k2 n2 : k1 = 0, 1, . . . , n1, k2 = 0, 1, . . . , n2

  • and

Pθ{ˆ θw = u} = Pθ

  • w1

ξ1 n1 + w2 ξ2 n2 = u

  • =

1 L(θ) b(θ)

a(θ) n2

  • i2=0

Pθ1

  • ξ1 = n1

w1

  • u − w2

i2 n2

  • P θ−w1θ1

w2

{ξ2 = i2} dθ1.

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For given θ ∈ (0, 1), the expected length of the Clopper-Pearson confidence interval equals lc(θ) =

n

  • u=1

(θc

U(u) − θc L(u))

n u

  • θu(1 − θ)n−u1(θc

L(u),θc U(u))(θ), Wojciech Zieli´ nski Confidence Interval For...

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For comparison of lc(θ) and lw(θ) it is enough to compare F −1

c

(γ) − F −1

c

(1 − γ) and F −1

w (γ) − F −1 w (1 − γ), where Fc and

Fw are the cdf’s of ˆ θc and ˆ θw, respectively.

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For comparison of lc(θ) and lw(θ) it is enough to compare F −1

c

(γ) − F −1

c

(1 − γ) and F −1

w (γ) − F −1 w (1 − γ), where Fc and

Fw are the cdf’s of ˆ θc and ˆ θw, respectively. Since distributions of ˆ θc and ˆ θw have the same support (0, 1) are unimodal have the same expected value hence, for the length comparison it is sufficient to compare the variances of ˆ θc and ˆ θw.

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D2

θ ˆ

θc = θ(1 − θ) n D2

θ ˆ

θw =             

θ(3n1+3nw1−6n1w1−2nθ) 6n1(n−n1)

, 0 < θ < w1,

nw2

1−3n1w1+6n1θ(1−θ)

6n1(n−n1)

, w1 ≤ θ ≤ 1 − w1,

(1−θ)(3n1+3nw1−6n1w1−2n(1−θ)) 6n1(n−n1)

, 1 − w1 < θ < 1. (for w1 ≤ 0.5)

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D2

0.5ˆ

θc = 1 4n and D2

0.5ˆ

θw = 1 4n f(1 − 2w1) + 2w2

1/3

f(1 − f) (where f = n1/n)

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D2

0.5ˆ

θc = 1 4n and D2

0.5ˆ

θw = 1 4n f(1 − 2w1) + 2w2

1/3

f(1 − f) (where f = n1/n) f∗ =

  • 1 +
  • 1.5 − 3w1 + w2

1

w1 −1

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D2

0.5ˆ

θc = 1 4n and D2

0.5ˆ

θw = 1 4n f(1 − 2w1) + 2w2

1/3

f(1 − f) (where f = n1/n) f∗ =

  • 1 +
  • 1.5 − 3w1 + w2

1

w1 −1 w1 0.10 0.20 0.30 0.40 0.50 f∗ 0.0833 0.1710 0.2653 0.3710 0.5000

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Concluding remarks.

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Concluding remarks. The confidence interval for the probability of success using information on the non homogeneity of the sample is better, i.e. shorter, than the standard Clopper-Pearson confidence interval.

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Concluding remarks. The confidence interval for the probability of success using information on the non homogeneity of the sample is better, i.e. shorter, than the standard Clopper-Pearson confidence interval. Closed formulae for such confidence intervals are not

  • available. Nevertheless, for given w1, n, n1 and observed

ξ1 and ξ2 the confidence interval may be easily obtained with standard mathematical software (for example Mathematica, MathLab etc.).

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Concluding remarks. The confidence interval for the probability of success using information on the non homogeneity of the sample is better, i.e. shorter, than the standard Clopper-Pearson confidence interval. Closed formulae for such confidence intervals are not

  • available. Nevertheless, for given w1, n, n1 and observed

ξ1 and ξ2 the confidence interval may be easily obtained with standard mathematical software (for example Mathematica, MathLab etc.). The results may be generalized to the case of arbitrary w1 and w2, for example w1 = 1 = −w2, i.e. the difference of the probabilities of success.

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