Let $\widehat{w}$ be an unbiased estimate of an unknown $w\in R$. Given a function $t(w)$, we show how to choose a function $f_n(w)$ such that for $w^*=\widehat{w} + f_n(w)$, $E\ t\left(w^*\right) =t(w)$. We illustrate this with $t(w)=w^a$ for a given constant $a$. For $a=2$ and $\widehat{w}$ normal, this leads to the convolution equation $c_r=c_r\otimes c_r$

Christopher S. Withers

Saralees Nadarajah

Comptes Rendus Mathématique

Withers, Christopher S.

Nadarajah, Saralees

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Comptes RendusMathématiqueChristopher S. Withers and Saralees NadarajahA note on bias reductionVolume 358, issue 6 (2020), p. 641-644Published online: 2 October 2020https://doi.org/10.5802/crmath.49This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orge-ISSN : 1778-3569Comptes RendusMathématique2020, 358, n 6, p. 641-644https://doi.org/10.5802/crmath.49Statistics / StatistiquesA note on bias reductionChristopher S. Withersa and Saralees Nadarajah∗, ba Industrial Research Limited, Lower Hutt, New Zealandb University of Manchester, Manchester M13 9PL, UKE-mails: kit.withers@gmail.com, mbbsssn2@manchester.ac.ukAbstract. Let ŵ be an unbiased estimate of an unknown w ∈ R. Given a function t (w), we show how to choosea function fn (w) such that for w∗ = ŵ + fn (w), E t(w∗)= t (w). We illustrate this with t (w) = w a for a givenconstant a. For a = 2 and ŵ normal, this leads to the convolution equation cr = cr ⊗ cr .Manuscript received 10th January 2019, accepted 9th April 2020.1. IntroductionLet ŵ be an unbiased estimate of an unknown w ∈ R. Given a function t (w), we show howto choose a function fn(w) such that E t (w∗) = t (w) for w∗ = ŵ + fn(w). We illustrate this inSection 2 with ŵ normal and t (w) = w a for some constant a. For a = 2 this gives the convolutionequation cr = cr ⊗ cr to solve.w∗ is not an estimate since it depends on the unknown w . The method extends to ŵ anystandard estimate of an unknown w ∈ Rp with respect to a given parameter n. That is, E ŵ → was n →∞ and, for r ≥ 1, its r th order cumulants have magnitude n1−r and can be expanded aspower series in n−1:κ(ŵi1 , . . . , ŵir)= ∞∑e=r−1n−e k i1,...,ire (1)for 1 ≤ i1, . . . , ir ≤ p and k i10 = wi1 , where wi is the i th component of w, and the cumulantcoefficients k i1,...,ire are bounded as n →∞, but may depend on w. For p = 1, (1) can be writtenκ (ŵ) =∞∑e=r−1n−e kr,efor r ≥ 1, where k1,0 = w . Cumulant coefficients are the building blocks of analytic methods forstatistical inference. For example, methods for constructing estimates of low bias for any smoothfunction t (w) : Rp → R were given in Mynbaev et al. [3] and Withers and Nadarajah [4–14].∗Corresponding author.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/642 Christopher S. Withers and Saralees NadarajahGiven a sequence a1, a2, . . . , the exponential partial Bell polynomial Bi ,k (a) is defined by( ∞∑j=1a j tj / j !)k/k ! ≡∞∑j=kB j ,k (a)tj / j !for t ∈ R and k = 0,1, . . . . It is tabled on p. 307–308 of Comtet [2] for 1 ≤ r ≤ 12. Given twosequences a1, a2, . . . and b1,b2, . . . , their discrete convolution is defined byar ⊗br =r∑i=1ai br−i .Set δi ,1 = I (i = 1) and (a) j = a!/(a − j )!.2. Adding bias to ŵ to reduce the bias of t (ŵ)Suppose that ŵ ∼N (w, v(w)/n), a normal estimate. Setv = v(w), fn(w) =∞∑i=1ki n−i , (2)where ki = bi /i ! may depend on w . Theorems 1 and 3 show how to choose ki or bi so that for anygiven a,E(w∗)a = w a , w∗ = ŵ + fn(w). (3)Theorem 2.1 considers the case a = 2. Theorem 3 considers the case a = 3. Throughout, we setδ= ŵ −w, ν= v/w2, mn = E w∗ = w + fn(w). (4)Theorem 1. Take w∗ = ŵ + fn(w) withkr =−cr v r /D2r−1,where D = 2w and, for r ≥ 2,c1 = 1, cr = cr ⊗ cr . (5)Then E (w∗)2 = w2.Proof. For w∗ of (3),E(w∗)2 = m2n + v/n = w2 +2w fn(w)+ fn(w)2 + v/n = w2 + ∞∑i=1Ti n−i ,where Ti = Dki + si + vδi ,1, s1 = 0, and for i ≥ 2,si = ki ⊗ki =i−1∑j=1k j ki− j .So, Ti = 0 if we take ki =−(si + vδi ,1)/D for i ≥ 1. This gives the result. Corollary 2 gives an explicit form for cr in (5).Corollary 2. For r ≥ 2,cr =−(1/2r)(−4c1)r /2 = 2r−11 3 · · · (2r −3)/r !.Proof. SetC (t ) =∞∑r=1cr tr ,where c1 is now arbitrary. By (5), C (t ) = c1t +C (t )2. Since C (0) = 0, this givesC (t ) = [1− (1−4c1t )1/2]/2 =− ∞∑r=1(1/2r)(−4c1t )r /2C. R. Mathématique, 2020, 358, n 6, 641-644Christopher S. Withers and Saralees Nadarajah 643which implies the result. Theorem 3. Take w∗ = ŵ + fn(w) withfn(w) = w∞∑j=0A j(c/n3) j/ j !+wu∞∑j=0B j(c/n3) j/ j !−w,where c = 4ν3, a j = (1/2) j /2, u =−νn−1, Bk = ak+1/a1(k +1) andA j =j∑k=0(1/3)k B j ,k (a).Then E (w∗)3 = w3.Proof. For w∗ of (3),E(w∗)3 = m3n +3mn v/n = w3if m3n +3mn v/n −w3 = 0. Set γ= (v/n)3 +w6/4. Since γ> 0, this cubic has one real root given byEquation (3.8.2) of Abramowitz and Stegun [1]:mn = S1/31 +S1/32 ,where S j = w3/2±γ1/2. Suppose that w > 0. (If not, replace w by |w |.) Then for ν of (4),γ1/2 = w3(1+d)1/2/2,where d = cn−3. Furthermore,(1+d)1/2 =∞∑j=0(1/2j)d j , S1 = 1+DforD =∞∑j=1(1/2j)d j /2 =∞∑j=1a j dj / j !.Furthermore,Dk /k ! =∞∑j=kB j ,k (a)dj / j !impliesS1/31 =∞∑k=0(1/3k)Dk =∞∑j=0A j dj / j !.AlsoS2 = 1/2− (1+d)1/2/2 =−∞∑j=1a j dj / j ! =−a1d(1+U )forU =∞∑k=1Bk dk /k !andU j / j ! =∞∑k= jBk, j (B)dk /k !.ThenS1/32 = u(1+U )1/3 = u∞∑j=0(1/3j)U j = u∞∑j=0(1/3) jUj / j ! =∞∑k=0Ck dk /k !,whereCk =k∑j=0(1/3) j Bk, j (B).Hence, for the choice of fn(w), E (w∗)3 = w3. C. R. Mathématique, 2020, 358, n 6, 641-644644 Christopher S. Withers and Saralees NadarajahThe method of Theorems 1 and 3 will not work for t (w) = w5 since there is no explicit solutionto a quintic. However, we now show how to obtain an unbiased or bias-reduced estimate of w afor any a > 0. Set ∆= ŵ −w = w∗−mn . ThenE(w∗)a = E (mn +∆)a = ∞∑j=0(aj)ma− jn µ j (ŵ) =∞∑j=0(a2 j)ma−2 jn µ2 j (ŵ) ,whereµ2 j (ŵ) = N j v j , N0 = 1, N j = 1 3 · · · (2 j −1)for j ≥ 1. By (2),fn(w)k /k ! =∞∑i=kBi ,k (b)n−i /i !.So,(mn/w)a = [1+ fn(w)/w]a = ∞∑k=0(a)k w−k fn(w)k /k ! =∞∑i=0Da,i n−i /i !forDa,i =i∑k=0(a)k w−k Bi ,k (b).This impliesE(w∗)a /w a = ∞∑k=0n−k EkforEk =∑i+ j=k(a2 j)N jνj Da−2 j ,i /i !.So,b1 =−(a −1)w/2, E1 = 0, E(w∗)a = w a +O (n−2) ,b2 = w(a −1)[−(a −1)2 + (a −1)2wν− (a −2)2ν2] , E2 = 0, E (w∗)a = w a +O (n−3) .In this way, we can construct fn(w) so that for any given a > 0 and k ≥ 1, E[ŵ + fn(w)]a =w a +O (n−k).References[1] M. Abramowitz, I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematicaltables, Applied Mathematics Series, vol. 55, U.S. Department of Commerce, 1964.[2] L. Comtet, Advanced combinatorics: the art of finite and infinite expansions, Springer, 1974.[3] K. T. Mynbaev, S. Nadarajah, C. S. Withers, A. S. Aipenova, “Improving bias in kernel density estimation”, Stat. Probab.Lett. 94 (2014), p. 106-112.[4] C. S. Withers, S. Nadarajah, “Analytic bias reduction for k-sample functionals”, Sankhyā, Ser. A 70 (2008), no. 2, p. 186-222.[5] ——— , “The bias and skewness of M-estimators in regression”, Electron. J. Stat. 4 (2010), p. 1-14.[6] ——— , “Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis”, J. Stat. Plann. Inference 141(2011), no. 12, p. 3839-3861.[7] ——— , “Bias reduction for the ratio of means”, J. Stat. Comput. Simulation 81 (2011), no. 12, p. 1799-1816.[8] ——— , “Estimates of low bias for the multivariate normal”, Stat. Probab. Lett. 81 (2011), no. 11, p. 1635-1647.[9] ——— , “Reduction of bias and skewness with applications to second order accuracy”, Stat. Methods Appl. 20 (2011),no. 4, p. 439-450.[10] ——— , “Nonparametric estimates of low bias”, REVSTAT 10 (2012), no. 2, p. 229-283.[11] ——— , “Calibration with low bias”, Stat. Pap. 54 (2013), no. 2, p. 371-379.[12] ——— , “Delta and jackknife estimators with low bias for functions of binomial and multinomial parameters”, J.Multivariate Anal. 118 (2013), p. 138-147.[13] ——— , “Density estimates of low bias”, Metrika 76 (2013), no. 3, p. 357-379.[14] ——— , “Bias reduction when data are rounded”, Stat. Neerl. 69 (2015), no. 3, p. 236-271.C. R. Mathématique, 2020, 358, n 6, 641-644

A note on bias reduction

Numérisation de Documents Anciens Mathématiques

Comptes RendusMathématiqueChristopher S. Withers and Saralees NadarajahA note on bias reductionVolume 358, issue 6 (2020), p. 641-644.<https://doi.org/10.5802/crmath.49>© Académie des sciences, Paris and the authors, 2020.Some rights reserved.This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orgComptes RendusMathématique2020, 358, n 6, p. 641-644https://doi.org/10.5802/crmath.49Statistics / StatistiquesA note on bias reductionChristopher S. Withersa and Saralees Nadarajah∗, ba Industrial Research Limited, Lower Hutt, New Zealandb University of Manchester, Manchester M13 9PL, UKE-mails: kit.withers@gmail.com, mbbsssn2@manchester.ac.ukAbstract. Let ŵ be an unbiased estimate of an unknown w ∈ R. Given a function t (w), we show how to choosea function fn (w) such that for w∗ = ŵ + fn (w), E t(w∗)= t (w). We illustrate this with t (w) = w a for a givenconstant a. For a = 2 and ŵ normal, this leads to the convolution equation cr = cr ⊗ cr .Manuscript received 10th January 2019, accepted 9th April 2020.1. IntroductionLet ŵ be an unbiased estimate of an unknown w ∈ R. Given a function t (w), we show howto choose a function fn(w) such that E t (w∗) = t (w) for w∗ = ŵ + fn(w). We illustrate this inSection 2 with ŵ normal and t (w) = w a for some constant a. For a = 2 this gives the convolutionequation cr = cr ⊗ cr to solve.w∗ is not an estimate since it depends on the unknown w . The method extends to ŵ anystandard estimate of an unknown w ∈ Rp with respect to a given parameter n. That is, E ŵ → was n →∞ and, for r ≥ 1, its r th order cumulants have magnitude n1−r and can be expanded aspower series in n−1:κ(ŵi1 , . . . , ŵir)= ∞∑e=r−1n−e k i1,...,ire (1)for 1 ≤ i1, . . . , ir ≤ p and k i10 = wi1 , where wi is the i th component of w, and the cumulantcoefficients k i1,...,ire are bounded as n →∞, but may depend on w. For p = 1, (1) can be writtenκ (ŵ) =∞∑e=r−1n−e kr,efor r ≥ 1, where k1,0 = w . Cumulant coefficients are the building blocks of analytic methods forstatistical inference. For example, methods for constructing estimates of low bias for any smoothfunction t (w) : Rp → R were given in Mynbaev et al. [3] and Withers and Nadarajah [4–14].∗Corresponding author.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/642 Christopher S. Withers and Saralees NadarajahGiven a sequence a1, a2, . . . , the exponential partial Bell polynomial Bi ,k (a) is defined by( ∞∑j=1a j tj / j !)k/k ! ≡∞∑j=kB j ,k (a)tj / j !for t ∈ R and k = 0,1, . . . . It is tabled on p. 307–308 of Comtet [2] for 1 ≤ r ≤ 12. Given twosequences a1, a2, . . . and b1,b2, . . . , their discrete convolution is defined byar ⊗br =r∑i=1ai br−i .Set δi ,1 = I (i = 1) and (a) j = a!/(a − j )!.2. Adding bias to ŵ to reduce the bias of t (ŵ)Suppose that ŵ ∼N (w, v(w)/n), a normal estimate. Setv = v(w), fn(w) =∞∑i=1ki n−i , (2)where ki = bi /i ! may depend on w . Theorems 1 and 3 show how to choose ki or bi so that for anygiven a,E(w∗)a = w a , w∗ = ŵ + fn(w). (3)Theorem 2.1 considers the case a = 2. Theorem 3 considers the case a = 3. Throughout, we setδ= ŵ −w, ν= v/w2, mn = E w∗ = w + fn(w). (4)Theorem 1. Take w∗ = ŵ + fn(w) withkr =−cr v r /D2r−1,where D = 2w and, for r ≥ 2,c1 = 1, cr = cr ⊗ cr . (5)Then E (w∗)2 = w2.Proof. For w∗ of (3),E(w∗)2 = m2n + v/n = w2 +2w fn(w)+ fn(w)2 + v/n = w2 + ∞∑i=1Ti n−i ,where Ti = Dki + si + vδi ,1, s1 = 0, and for i ≥ 2,si = ki ⊗ki =i−1∑j=1k j ki− j .So, Ti = 0 if we take ki =−(si + vδi ,1)/D for i ≥ 1. This gives the result. Corollary 2 gives an explicit form for cr in (5).Corollary 2. For r ≥ 2,cr =−(1/2r)(−4c1)r /2 = 2r−11 3 · · · (2r −3)/r !.Proof. SetC (t ) =∞∑r=1cr tr ,where c1 is now arbitrary. By (5), C (t ) = c1t +C (t )2. Since C (0) = 0, this givesC (t ) = [1− (1−4c1t )1/2]/2 =− ∞∑r=1(1/2r)(−4c1t )r /2C. R. Mathématique, 2020, 358, n 6, 641-644Christopher S. Withers and Saralees Nadarajah 643which implies the result. Theorem 3. Take w∗ = ŵ + fn(w) withfn(w) = w∞∑j=0A j(c/n3) j/ j !+wu∞∑j=0B j(c/n3) j/ j !−w,where c = 4ν3, a j = (1/2) j /2, u =−νn−1, Bk = ak+1/a1(k +1) andA j =j∑k=0(1/3)k B j ,k (a).Then E (w∗)3 = w3.Proof. For w∗ of (3),E(w∗)3 = m3n +3mn v/n = w3if m3n +3mn v/n −w3 = 0. Set γ= (v/n)3 +w6/4. Since γ> 0, this cubic has one real root given byEquation (3.8.2) of Abramowitz and Stegun [1]:mn = S1/31 +S1/32 ,where S j = w3/2±γ1/2. Suppose that w > 0. (If not, replace w by |w |.) Then for ν of (4),γ1/2 = w3(1+d)1/2/2,where d = cn−3. Furthermore,(1+d)1/2 =∞∑j=0(1/2j)d j , S1 = 1+DforD =∞∑j=1(1/2j)d j /2 =∞∑j=1a j dj / j !.Furthermore,Dk /k ! =∞∑j=kB j ,k (a)dj / j !impliesS1/31 =∞∑k=0(1/3k)Dk =∞∑j=0A j dj / j !.AlsoS2 = 1/2− (1+d)1/2/2 =−∞∑j=1a j dj / j ! =−a1d(1+U )forU =∞∑k=1Bk dk /k !andU j / j ! =∞∑k= jBk, j (B)dk /k !.ThenS1/32 = u(1+U )1/3 = u∞∑j=0(1/3j)U j = u∞∑j=0(1/3) jUj / j ! =∞∑k=0Ck dk /k !,whereCk =k∑j=0(1/3) j Bk, j (B).Hence, for the choice of fn(w), E (w∗)3 = w3. C. R. Mathématique, 2020, 358, n 6, 641-644644 Christopher S. Withers and Saralees NadarajahThe method of Theorems 1 and 3 will not work for t (w) = w5 since there is no explicit solutionto a quintic. However, we now show how to obtain an unbiased or bias-reduced estimate of w afor any a > 0. Set ∆= ŵ −w = w∗−mn . ThenE(w∗)a = E (mn +∆)a = ∞∑j=0(aj)ma− jn µ j (ŵ) =∞∑j=0(a2 j)ma−2 jn µ2 j (ŵ) ,whereµ2 j (ŵ) = N j v j , N0 = 1, N j = 1 3 · · · (2 j −1)for j ≥ 1. By (2),fn(w)k /k ! =∞∑i=kBi ,k (b)n−i /i !.So,(mn/w)a = [1+ fn(w)/w]a = ∞∑k=0(a)k w−k fn(w)k /k ! =∞∑i=0Da,i n−i /i !forDa,i =i∑k=0(a)k w−k Bi ,k (b).This impliesE(w∗)a /w a = ∞∑k=0n−k EkforEk =∑i+ j=k(a2 j)N jνj Da−2 j ,i /i !.So,b1 =−(a −1)w/2, E1 = 0, E(w∗)a = w a +O (n−2) ,b2 = w(a −1)[−(a −1)2 + (a −1)2wν− (a −2)2ν2] , E2 = 0, E (w∗)a = w a +O (n−3) .In this way, we can construct fn(w) so that for any given a > 0 and k ≥ 1, E[ŵ + fn(w)]a =w a +O (n−k).References[1] M. Abramowitz, I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematicaltables, Applied Mathematics Series, vol. 55, U.S. Department of Commerce, 1964.[2] L. Comtet, Advanced combinatorics: the art of finite and infinite expansions, Springer, 1974.[3] K. T. Mynbaev, S. Nadarajah, C. S. Withers, A. S. Aipenova, “Improving bias in kernel density estimation”, Stat. Probab.Lett. 94 (2014), p. 106-112.[4] C. S. Withers, S. Nadarajah, “Analytic bias reduction for k-sample functionals”, Sankhyā, Ser. A 70 (2008), no. 2, p. 186-222.[5] ——— , “The bias and skewness of M-estimators in regression”, Electron. J. Stat. 4 (2010), p. 1-14.[6] ——— , “Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis”, J. Stat. Plann. Inference 141(2011), no. 12, p. 3839-3861.[7] ——— , “Bias reduction for the ratio of means”, J. Stat. Comput. Simulation 81 (2011), no. 12, p. 1799-1816.[8] ——— , “Estimates of low bias for the multivariate normal”, Stat. Probab. Lett. 81 (2011), no. 11, p. 1635-1647.[9] ——— , “Reduction of bias and skewness with applications to second order accuracy”, Stat. Methods Appl. 20 (2011),no. 4, p. 439-450.[10] ——— , “Nonparametric estimates of low bias”, REVSTAT 10 (2012), no. 2, p. 229-283.[11] ——— , “Calibration with low bias”, Stat. Pap. 54 (2013), no. 2, p. 371-379.[12] ——— , “Delta and jackknife estimators with low bias for functions of binomial and multinomial parameters”, J.Multivariate Anal. 118 (2013), p. 138-147.[13] ——— , “Density estimates of low bias”, Metrika 76 (2013), no. 3, p. 357-379.[14] ——— , “Bias reduction when data are rounded”, Stat. Neerl. 69 (2015), no. 3, p. 236-271.C. R. Mathématique, 2020, 358, n 6, 641-644

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A note on bias reduction

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