Examples Method Theory Simulations Conclusion A general procedure to combine estimators Fr´ ed´ eric Lavancier Laboratoire de Math´ ematiques Jean Leray University of Nantes Joint work with Paul Rochet (University of Nantes)
Examples Method Theory Simulations Conclusion Introduction Let θ be an unknown quantity in a statistical model. Consider a collection of k estimators T 1 , ..., T k of θ . Aim: combining these estimators to obtain a better estimate.
Examples Method Theory Simulations Conclusion Some examples 1 The method 2 Theoretical results 3 Simulations : back to the examples 4 Conclusion 5
Examples Method Theory Simulations Conclusion Some examples 1 The method 2 Theoretical results 3 Simulations : back to the examples 4 Conclusion 5
Examples Method Theory Simulations Conclusion Example 1 : mean and median Let x 1 , . . . , x n be n i.i.d realisations of an unknown distribution on the real line. Assume this distribution is symmetric around some parameter θ ( θ ∈ R ). Two natural choices to estimate θ : the mean T 1 = ¯ x n the median T 2 = x ( n / 2) The idea to combine these two estimators comes from Pierre Simon de Laplace. In the Second Supplement of the Th´ eorie Analytique des Probabilit´ es (1812), he wrote : ” En combinant les r´ esultats de ces deux m´ ethodes, on peut obtenir un r´ esultat dont la loi de probabilit´ e des erreurs soit plus rapidement d´ ecroissante.” [ In combining the results of these two methods, one can obtain a result whose probability law of error will be more rapidly decreasing. ]
Examples Method Theory Simulations Conclusion Example 1 : mean and median Let x 1 , . . . , x n be n i.i.d realisations of an unknown distribution on the real line. Assume this distribution is symmetric around some parameter θ ( θ ∈ R ). Two natural choices to estimate θ : the mean T 1 = ¯ x n the median T 2 = x ( n / 2) The idea to combine these two estimators comes from Pierre Simon de Laplace. In the Second Supplement of the Th´ eorie Analytique des Probabilit´ es (1812), he wrote : ” En combinant les r´ esultats de ces deux m´ ethodes, on peut obtenir un r´ esultat dont la loi de probabilit´ e des erreurs soit plus rapidement d´ ecroissante.” [ In combining the results of these two methods, one can obtain a result whose probability law of error will be more rapidly decreasing. ]
Examples Method Theory Simulations Conclusion Example 1 : mean and median Laplace considered the combination λ 1 ¯ x n + λ 2 x ( n / 2) with λ 1 + λ 2 = 1. 1. He proved that the asymptotic law of this combination is Gaussian (in 1812)! 2. Minimizing the asymptotic variance in λ 1 , λ 2 , he concluded that if the underlying distribution is Gaussian, then the best combination is to take λ 1 = 1 and λ 2 = 0. for other distributions, the best combination depends on the distribution: ” L’ignorance o` u l’on est de la loi de probabilit´ e des erreurs des observations rend cette correction impraticable” [ When one does not know the distribution of the errors of observation this correction is not feasible. ]
Examples Method Theory Simulations Conclusion Example 1 : mean and median Laplace considered the combination λ 1 ¯ x n + λ 2 x ( n / 2) with λ 1 + λ 2 = 1. 1. He proved that the asymptotic law of this combination is Gaussian (in 1812)! 2. Minimizing the asymptotic variance in λ 1 , λ 2 , he concluded that if the underlying distribution is Gaussian, then the best combination is to take λ 1 = 1 and λ 2 = 0. for other distributions, the best combination depends on the distribution: ” L’ignorance o` u l’on est de la loi de probabilit´ e des erreurs des observations rend cette correction impraticable” [ When one does not know the distribution of the errors of observation this correction is not feasible. ]
Examples Method Theory Simulations Conclusion Example 1 : mean and median Laplace considered the combination λ 1 ¯ x n + λ 2 x ( n / 2) with λ 1 + λ 2 = 1. 1. He proved that the asymptotic law of this combination is Gaussian (in 1812)! 2. Minimizing the asymptotic variance in λ 1 , λ 2 , he concluded that if the underlying distribution is Gaussian, then the best combination is to take λ 1 = 1 and λ 2 = 0. for other distributions, the best combination depends on the distribution: ” L’ignorance o` u l’on est de la loi de probabilit´ e des erreurs des observations rend cette correction impraticable” [ When one does not know the distribution of the errors of observation this correction is not feasible. ]
Examples Method Theory Simulations Conclusion Example 1 : mean and median Laplace considered the combination λ 1 ¯ x n + λ 2 x ( n / 2) with λ 1 + λ 2 = 1. 1. He proved that the asymptotic law of this combination is Gaussian (in 1812)! 2. Minimizing the asymptotic variance in λ 1 , λ 2 , he concluded that if the underlying distribution is Gaussian, then the best combination is to take λ 1 = 1 and λ 2 = 0. for other distributions, the best combination depends on the distribution: ” L’ignorance o` u l’on est de la loi de probabilit´ e des erreurs des observations rend cette correction impraticable” [ When one does not know the distribution of the errors of observation this correction is not feasible. ] Is it possible to estimate λ 1 and λ 2 ?
Examples Method Theory Simulations Conclusion Example 2 : Boolean model The standard Boolean model Γ is the union of random discs where - the centres come from a homogeneous Poisson point process with intensity ρ - the radii are independently distributed according to some probability law µ Examples on [0 , 1] 2 when µ = U ([0 . 01 , 0 . 1]) and ρ = 50 , 100 , 200
Examples Method Theory Simulations Conclusion Example 2 : Boolean model Assume the law of the radii µ is known and we want to estimate ρ . Denote ˆ p = | Γ ∩ W | / | W | the volume fraction on the observation window W . First estimator : denoting R ∼ µ p ) / ( π E ( R 2 )) T 1 = − log(1 − ˆ Second estimator : Let n + ν be the number of lower tangent points of Γ in direction ν ( ν ∈ [0 , 2 π ]). ρ ν = n + ν / | W | ˆ 1 − ˆ p is a consistent estimator of ρ . Sample K independent directions ν i uniformly on [0 , 2 π ], then K T 2 = 1 � ˆ ρ ν i K i =1 ρ ν , ∀ ν (Molchanov(1995)) T 2 has a smaller asymptotic variance than ˆ
Examples Method Theory Simulations Conclusion Example 2 : Boolean model Assume the law of the radii µ is known and we want to estimate ρ . Denote ˆ p = | Γ ∩ W | / | W | the volume fraction on the observation window W . First estimator : denoting R ∼ µ p ) / ( π E ( R 2 )) T 1 = − log(1 − ˆ Second estimator : Let n + ν be the number of lower tangent points of Γ in direction ν ( ν ∈ [0 , 2 π ]). ρ ν = n + ν / | W | ˆ 1 − ˆ p is a consistent estimator of ρ . Sample K independent directions ν i uniformly on [0 , 2 π ], then K T 2 = 1 � ˆ ρ ν i K i =1 ρ ν , ∀ ν (Molchanov(1995)) T 2 has a smaller asymptotic variance than ˆ
Examples Method Theory Simulations Conclusion Example 2 : Boolean model Assume the law of the radii µ is known and we want to estimate ρ . Denote ˆ p = | Γ ∩ W | / | W | the volume fraction on the observation window W . First estimator : denoting R ∼ µ p ) / ( π E ( R 2 )) T 1 = − log(1 − ˆ Second estimator : Let n + ν be the number of lower tangent points of Γ in direction ν ( ν ∈ [0 , 2 π ]). ρ ν = n + ν / | W | ˆ 1 − ˆ p is a consistent estimator of ρ . Sample K independent directions ν i uniformly on [0 , 2 π ], then K T 2 = 1 � ˆ ρ ν i K i =1 ρ ν , ∀ ν (Molchanov(1995)) T 2 has a smaller asymptotic variance than ˆ
Examples Method Theory Simulations Conclusion Example 2 : Boolean model n + ν : number of lower tangent points in direction ν . Example : for ν = π/ 4, we obtain n + ν = 6
Examples Method Theory Simulations Conclusion Example 2 : Boolean model Simulation study on 10 3 replications on [0 , 1] 2 when µ = U ([0 . 01 , 0 . 1]) and ρ = 50 , 100 , 200 Boxplots of T 1 and T 2 160 90 350 80 140 300 70 120 250 60 100 50 200 40 150 80 30 100 60 T 1 T 2 T 1 T 2 T 1 T 2 ρ = 50 ρ = 100 ρ = 200 Which one to choose? Can we combine them to get a better estimate?
Examples Method Theory Simulations Conclusion Example 2 : Boolean model Simulation study on 10 3 replications on [0 , 1] 2 when µ = U ([0 . 01 , 0 . 1]) and ρ = 50 , 100 , 200 Boxplots of T 1 and T 2 160 90 350 80 140 300 70 120 250 60 100 50 200 40 150 80 30 100 60 T 1 T 2 T 1 T 2 T 1 T 2 ρ = 50 ρ = 100 ρ = 200 Which one to choose? Can we combine them to get a better estimate?
Examples Method Theory Simulations Conclusion Example 3 : Thomas cluster process A Thomas cluster process is a Poisson cluster process with 3 parameters κ : intensity of the Poisson process of cluster centres (i.e. the parents) µ : expected number of points per cluster (i.e. the offsprings) σ : given the parents, each offspring is distributed according to a Gaussian law centred at his parent and with standard deviation σ . Examples on [0 , L ] 2 with L = 1 , 2 , 3 ( κ = 10, µ = 10, σ = 0 . 05)
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