On the inadmissibility of Watterson s estimate
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1 On the inadmissibility of Watterson s estimate A. Futschik F. Gach Institute of Statistics and Decision Support Systems University of Vienna ISDS-Kolloquium, 2007
2 Outline 1 Motivation 2 Inadmissibility of Watterson s estimator 3 Inadmissibility of unbiased estimators 4 The genetic context
3 Outline 1 Motivation 2 Inadmissibility of Watterson s estimator 3 Inadmissibility of unbiased estimators 4 The genetic context
4 Outline 1 Motivation 2 Inadmissibility of Watterson s estimator 3 Inadmissibility of unbiased estimators 4 The genetic context
5 Outline 1 Motivation 2 Inadmissibility of Watterson s estimator 3 Inadmissibility of unbiased estimators 4 The genetic context
6 A Poisson process Let S be the number of cars arriving during a fixed known time interval [0, l] Then S is Poisson distributed with parameter λl, so that E λ (S) = λl. Therefore, a natural estimator of λ is ˆλ = S l
7 A Poisson process on a random interval Let [0, L] be the random time interval while the train crossing is closed. S counts the number of cars that arrive during [0, L]. S L = l is Poisson distributed with parameter λl, so that E λ (S) = E(E λ (S L)) = E(λL) = λ E(L) On supposing we know E(L), a natural unbiased estimator of λ is ˆλ W = S E(L), the so-called Watterson estimator.
8 The main idea Watterson s estimator ˆλ W is of the form cs, with c = 1 E(L). Consider estimators ˆλ c of the form Take the mean squared error ˆλ c = cs, c 0. MSE λ (ˆλ c ) = E λ ((ˆλ c λ) 2 ), λ (0, ) as performance measure. Find a constant c such that the corresponding estimator ˆλ c has minimal mean squared error.
9 Finding the best linear estimator 1 The number of events S: E λ (S) = µ 1 λ Var λ (S) = µ 1 λ + (µ 2 µ 2 1 )λ2 2 The linear estimators ˆλ c : ( ) MSE λ (ˆλ c ) = µ 2 λ 2 + µ 1 λ c 2 2µ 1 λ 2 c + λ 2 3 The best linear estimator λ BLE : λ BLE = 1 µ 2 µ λ S
10 Finding the best linear estimator 1 The number of events S: E λ (S) = µ 1 λ Var λ (S) = µ 1 λ + (µ 2 µ 2 1 )λ2 2 The linear estimators ˆλ c : ( ) MSE λ (ˆλ c ) = µ 2 λ 2 + µ 1 λ c 2 2µ 1 λ 2 c + λ 2 3 The best linear estimator λ BLE : λ BLE = 1 µ 2 µ λ S
11 Finding the best linear estimator 1 The number of events S: E λ (S) = µ 1 λ Var λ (S) = µ 1 λ + (µ 2 µ 2 1 )λ2 2 The linear estimators ˆλ c : ( ) MSE λ (ˆλ c ) = µ 2 λ 2 + µ 1 λ c 2 2µ 1 λ 2 c + λ 2 3 The best linear estimator λ BLE : λ BLE = 1 µ 2 µ λ S
12 Finding the best linear estimator 1 The number of events S: E λ (S) = µ 1 λ Var λ (S) = µ 1 λ + (µ 2 µ 2 1 )λ2 2 The linear estimators ˆλ c : ( ) MSE λ (ˆλ c ) = µ 2 λ 2 + µ 1 λ c 2 2µ 1 λ 2 c + λ 2 3 The best linear estimator λ BLE : λ BLE = 1 µ 2 µ λ S
13 An admissible linear estimator 4 In order to take charge of the infeasibility of λ BLE, set c s = µ 1 µ 2 and call ˆλ s = c s S = S E(L) + Var(L) E(L) the shrinked Watterson estimator. 5 The admissibility of ˆλ s : MSE λ (ˆλ c ) MSE λ (ˆλ s ) = ( ) = µ 2 c 2 2µ 1 c + µ2 1 µ 2 = p(c)λ 2 + q(c)λ λ 2 + ( µ 1 c 2 µ3 1 µ 2 2 ) λ
14 An admissible linear estimator 4 In order to take charge of the infeasibility of λ BLE, set c s = µ 1 µ 2 and call ˆλ s = c s S = S E(L) + Var(L) E(L) the shrinked Watterson estimator. 5 The admissibility of ˆλ s : MSE λ (ˆλ c ) MSE λ (ˆλ s ) = ( ) = µ 2 c 2 2µ 1 c + µ2 1 µ 2 = p(c)λ 2 + q(c)λ λ 2 + ( µ 1 c 2 µ3 1 µ 2 2 ) λ
15 Main result Let (X l ) l L be a Poisson process with rate λ > 0 on an interval [0, L] of random length L. Let S count the events. Suppose that E(L) and Var(L) are both finite. Then 1 The Watterson estimator is inadmissible. 2 The shrinked estimator ˆλ W = S E(L) ˆλ s = S E(L) + Var(L) E(L) is uniformly better than every estimator cs with c > c s ; in particular, ˆλ s is uniformly better than ˆλ W.
16 Main result Let (X l ) l L be a Poisson process with rate λ > 0 on an interval [0, L] of random length L. Let S count the events. Suppose that E(L) and Var(L) are both finite. Then 1 The Watterson estimator is inadmissible. 2 The shrinked estimator ˆλ W = S E(L) ˆλ s = S E(L) + Var(L) E(L) is uniformly better than every estimator cs with c > c s ; in particular, ˆλ s is uniformly better than ˆλ W.
17 Main result Let (X l ) l L be a Poisson process with rate λ > 0 on an interval [0, L] of random length L. Let S count the events. Suppose that E(L) and Var(L) are both finite. Then 1 The Watterson estimator is inadmissible. 2 The shrinked estimator ˆλ W = S E(L) ˆλ s = S E(L) + Var(L) E(L) is uniformly better than every estimator cs with c > c s ; in particular, ˆλ s is uniformly better than ˆλ W.
18 Main result 3 Furthermore, ˆλ s is admissible in the class of all estimators of the form cs, c 0, where c does not depend on the parameter λ.
19 A more general setting Let ˆλ be an unbiased estimator for the unknown parameter λ Λ R. Take the mean squared error as performance measure. Is it possible to improve ˆλ uniformly over Λ by just multiplying it by a constant c R?
20 Finding the best linear estimator 1 The linear estimators ˆλ c = cˆλ: MSE λ (ˆλ c ) = (λ 2 + Var λ (ˆλ))c 2 2λ 2 c + λ 2 2 The best linear estimator λ BLE : λ BLE = λ 2 ˆλ λ 2 + Var λ (ˆλ) = c(λ) ˆλ 0 c(λ) < 1
21 Finding the best linear estimator 1 The linear estimators ˆλ c = cˆλ: MSE λ (ˆλ c ) = (λ 2 + Var λ (ˆλ))c 2 2λ 2 c + λ 2 2 The best linear estimator λ BLE : λ BLE = λ 2 ˆλ λ 2 + Var λ (ˆλ) = c(λ) ˆλ 0 c(λ) < 1
22 Finding the best linear estimator 1 The linear estimators ˆλ c = cˆλ: MSE λ (ˆλ c ) = (λ 2 + Var λ (ˆλ))c 2 2λ 2 c + λ 2 2 The best linear estimator λ BLE : λ BLE = λ 2 ˆλ λ 2 + Var λ (ˆλ) = c(λ) ˆλ 0 c(λ) < 1
23 Existence of admissible linear estimators We may improve ˆλ if and only if 1 c(λ) is bounded away from 0. This is the case iff ( ) λ Λ : Var λ (ˆλ) > K λ 2 for some K > 0 A factor leading to an admissible estimator in the class {cˆλ : c R} of linear estimators is c s = sup λ Λ c(λ).
24 Existence of admissible linear estimators We may improve ˆλ if and only if 1 c(λ) is bounded away from 0. This is the case iff ( ) λ Λ : Var λ (ˆλ) > K λ 2 for some K > 0 A factor leading to an admissible estimator in the class {cˆλ : c R} of linear estimators is c s = sup λ Λ c(λ).
25 Inadmissibility via uniform shrinkage Let ˆλ be an unbiased estimator of the unknown parameter λ Λ R such that λ Λ : Var λ (ˆλ) exists and Var λ (ˆλ) > 0. 1 If condition ( ) holds, then there is an estimator of the form cˆλ, c R, which is uniformly better than ˆλ with respect to the mean squared error.
26 Inadmissibility via uniform shrinkage Let ˆλ be an unbiased estimator of the unknown parameter λ Λ R such that λ Λ : Var λ (ˆλ) exists and Var λ (ˆλ) > 0. 1 If condition ( ) holds, then there is an estimator of the form cˆλ, c R, which is uniformly better than ˆλ with respect to the mean squared error.
27 Inadmissibility via uniform shrinkage 2 An optimal uniform shrinkage factor is given by c s = sup λ Λ c(λ), with c(λ) denoting the pointwise optimal shrinkage constant, which is obtained by minimizing the function in c. c MSE λ (cˆλ) 3 If ( ) is violated, then we may not improve ˆλ by uniform shrinkage.
28 Inadmissibility via uniform shrinkage 2 An optimal uniform shrinkage factor is given by c s = sup λ Λ c(λ), with c(λ) denoting the pointwise optimal shrinkage constant, which is obtained by minimizing the function in c. c MSE λ (cˆλ) 3 If ( ) is violated, then we may not improve ˆλ by uniform shrinkage.
29 Back to Watterson 1 For the Watterson estimator the pointwise optimal shrinkage constant is given by c(λ) = 1 µ µ 2 µ 1 1 λ, λ (0, ). 2 c s = sup λ (0, ) c(λ) = lim λ c(λ) = µ2 1 µ 2 3 ˆλ s = c sˆλ W = µ 1 S µ 2 S = E(L)+ Var(L) E(L)
30 Back to Watterson 1 For the Watterson estimator the pointwise optimal shrinkage constant is given by c(λ) = 1 µ µ 2 µ 1 1 λ, λ (0, ). 2 c s = sup λ (0, ) c(λ) = lim λ c(λ) = µ2 1 µ 2 3 ˆλ s = c sˆλ W = µ 1 S µ 2 S = E(L)+ Var(L) E(L)
31 Genes code for hereditary characteristics The total hereditary information of any organism is carried by RNA or DNA molecules. It is encoded by the four bases A, G; C, U/T. This totality of hereditary information is called genome. Genes are regions in the DNA that control a discrete hereditary characteristic, e.g. our blood group. Example (Genetic code) A-T-G-G-T-G-C-A-C-C-T-G-A-C-T-C-C-T-G-A-G-G-A-G-A-G-G
32 Genes code for hereditary characteristics The total hereditary information of any organism is carried by RNA or DNA molecules. It is encoded by the four bases A, G; C, U/T. This totality of hereditary information is called genome. Genes are regions in the DNA that control a discrete hereditary characteristic, e.g. our blood group. Example (Genetic code) A-T-G-G-T-G-C-A-C-C-T-G-A-C-T-C-C-T-G-A-G-G-A-G-A-G-G
33 Inheritance of genetic information The genetic information is passed on from the parental to the filial generation. This involves the process of DNA replication, which leads to predominantly identical copies of the genome.
34 The coalescent tree MRCA generations
35 Mutations on the coalescent A mutation on a gene is a change in the sequence of its bases. Example (Mutation) A-T-A-T-C-C-T-T A-T-G-T-C-C-T-T Let u be the probability that a fixed DNA sequence is altered by a mutation within one generation. With N denoting the total population size, we call θ = 2Nu the scaled mutation parameter. We provide any given coalescent tree with a Poisson process with parameter θ/2.
36 The parameter θ regulates the mutations. Observations θ regulates the diversity of the sample and is the parameter of interest. The mutational process on the coalescent can abstractly be viewed as a Poisson process on a random interval. Suppose each mutation affects a new position in the DNA sequences. Then the number of segregating sites S corresponds to the number of mutations on the coalescent. Example A-T-A-T-C-C-T-T A-T-G-T-C-C-T-T T-T-G-T-C-T-T-T
37 Watterson versus shrinked Watterson
38 Appendix Related work Futschik, A., and Gach, F. (2007) On the inadmissibility of Watterson s estimate. Submitted article. Gach, F. (2007) The coalescent: A stochastic approach to evolution. Magisterarbeit, Wien. Tavaré, S. (2004) Ancestral inference in population genetics. In: Lecture Notes in Mathematics 1837, Springer, Berlin, 2004.
39 Appendix Related work Futschik, A., and Gach, F. (2007) On the inadmissibility of Watterson s estimate. Submitted article. Gach, F. (2007) The coalescent: A stochastic approach to evolution. Magisterarbeit, Wien. Tavaré, S. (2004) Ancestral inference in population genetics. In: Lecture Notes in Mathematics 1837, Springer, Berlin, 2004.
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