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Volumn 17, Issue 2, 2011, Pages 507-529

An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

Author keywords

Completeness; Differences and derivatives of higher order; Fourier coefficients; Orthogonal polynomials; Parseval identity; Rodrigues inversion formula; Rodrigues type formula; Stein type identity; Variance bounds

Indexed keywords


EID: 79953877267     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/10-BEJ282     Document Type: Article
Times cited : (26)

References (34)
  • 1
    • 0012416859 scopus 로고
    • The construction of uniformly minimum variance unbiased estimators for exponential distributions
    • MR0285064
    • Abbey, J.L. and David, H.T. (1970). The construction of uniformly minimum variance unbiased estimators for exponential distributions. Ann. Math. Statist. 41 1217-1222. MR0285064
    • (1970) Ann. Math. Statist. , vol.41 , pp. 1217-1222
    • Abbey, J.L.1    David, H.T.2
  • 2
    • 42249111232 scopus 로고    scopus 로고
    • The discrete Mohr and Noll inequality with applications to variance bounds
    • MR2428867
    • Afendras, G., Papadatos, N. and Papathanasiou, V. (2007). The discrete Mohr and Noll inequality with applications to variance bounds. Sankhyā 69 162-189. MR2428867
    • (2007) Sankhyā , vol.69 , pp. 162-189
    • Afendras, G.1    Papadatos, N.2    Papathanasiou, V.3
  • 4
    • 0030607383 scopus 로고    scopus 로고
    • General characterization theorems based on versions of the Chernoff inequality and the Cox representation
    • DOI 10.1016/0378-3758(95)00191-3
    • Alharbi, A.A. and Shanbhag, D.N. (1996). General characterization theorems based on versions of the Chernoff inequality and the Cox representation. J. Statist. Plann. Inference 55 139-150. MR1423963 (Pubitemid 126167208)
    • (1996) Journal of Statistical Planning and Inference , vol.55 , Issue.2 , pp. 139-150
    • Alharbi, A.A.1    Shanbhag, D.N.2
  • 5
    • 16244404689 scopus 로고    scopus 로고
    • Multiplicative renormalization and generating functions I
    • Asai, N., Kubo, I. and Kuo, H. (2003). Multiplicative renormalization and generating functions I. Taiwanese J. Math. 7 89-101. MR1961041 (Pubitemid 41393471)
    • (2003) Taiwanese Journal of Mathematics , vol.7 , Issue.1 , pp. 89-101
    • Asai, N.1    Kubo, I.2    Kuo, H.-H.3
  • 6
    • 7444251626 scopus 로고    scopus 로고
    • Multiplicative renormalization and generating functions II
    • Asai, N., Kubo, I. and Kuo, H. (2004). Multiplicative renormalization and generating functions II. Taiwanese J. Math. 8 593-628. MR2105554 (Pubitemid 41393385)
    • (2004) Taiwanese Journal of Mathematics , vol.8 , Issue.4 , pp. 593-628
    • Asai, N.1    Kubo, I.2    Kuo, H.-H.3
  • 7
    • 0001828623 scopus 로고
    • On a certain class of orthogonal polynomials
    • MR0003852
    • Beale, F.S. (1941). On a certain class of orthogonal polynomials. Ann. Math. Statist. 12 97-103. MR0003852
    • (1941) Ann. Math. Statist. , vol.12 , pp. 97-103
    • Beale, F.S.1
  • 8
    • 0002914602 scopus 로고
    • Density questions in the classical theory of moments
    • VI, MR0638619
    • Berg, C. and Christensen, J.P.R. (1981). Density questions in the classical theory of moments. Ann. Inst. Fourier (Grenoble) 31 VI 99-114. MR0638619
    • (1981) Ann. Inst. Fourier (Grenoble) , vol.31 , pp. 99-114
    • Berg, C.1    Christensen, J.P.R.2
  • 9
    • 0015954998 scopus 로고
    • The convergence of Bhattacharyya bounds
    • MR0381097
    • Blight, B.J.N. and Rao, P.V. (1974). The convergence of Bhattacharyya bounds. Biometrika 61 137-142. MR0381097
    • (1974) Biometrika , vol.61 , pp. 137-142
    • Blight, B.J.N.1    Rao, P.V.2
  • 10
    • 0005387906 scopus 로고    scopus 로고
    • On Gaussian and Bernoulli covariance representations
    • Bobkov, S.G., Götze, F. and Houdré, C. (2001). On Gaussian and Bernoulli covariance representations. Bernoulli 7 439-451. MR1836739 (Pubitemid 33793049)
    • (2001) Bernoulli , vol.7 , Issue.3 , pp. 439-451
    • Bobkov, S.G.1    Gotze, F.2    Houdre, C.3
  • 11
    • 0036455622 scopus 로고    scopus 로고
    • An application of a density transform and the local limit theorem. Teor. Veroyatnost. i Primenen. 46 803-810; translation in
    • MR1971837
    • Cacoullos, T., Papadatos, N. and Papathanasiou, V. (2001). An application of a density transform and the local limit theorem. Teor. Veroyatnost. i Primenen. 46 803-810; translation in Theory Probab. Appl. 46 (2003) 699-707. MR1971837
    • (2001) Theory Probab. Appl. , vol.46 , Issue.2003 , pp. 699-707
    • Cacoullos, T.1    Papadatos, N.2    Papathanasiou, V.3
  • 12
    • 38249042346 scopus 로고
    • Bounds for the variance of functions of random variables by orthogonal polynomials and Bhattacharyya bounds
    • MR0822720
    • Cacoullos, T. and Papathanasiou, V. (1986). Bounds for the variance of functions of random variables by orthogonal polynomials and Bhattacharyya bounds. Statist. Probab. Lett. 4 21-23. MR0822720
    • (1986) Statist. Probab. Lett. , vol.4 , pp. 21-23
    • Cacoullos, T.1    Papathanasiou, V.2
  • 13
    • 38249022748 scopus 로고
    • Characterizations of distributions by variance bounds
    • MR1001133
    • Cacoullos, T. and Papathanasiou, V. (1989). Characterizations of distributions by variance bounds. Statist. Probab. Lett. 7 351-356. MR1001133
    • (1989) Statist. Probab. Lett. , vol.7 , pp. 351-356
    • Cacoullos, T.1    Papathanasiou, V.2
  • 14
    • 0000985130 scopus 로고
    • Variational inequalities with examples and an application to the central limit theorem
    • MR1303658
    • Cacoullos, T., Papathanasiou, V. and Utev, S.A. (1994). Variational inequalities with examples and an application to the central limit theorem. Ann. Probab. 22 1607-1618. MR1303658
    • (1994) Ann. Probab. , vol.22 , pp. 1607-1618
    • Cacoullos, T.1    Papathanasiou, V.2    Utev, S.A.3
  • 15
    • 0001067389 scopus 로고
    • Poisson approximation for dependent trials
    • MR0428387
    • Chen, L.H.Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534-545. MR0428387
    • (1975) Ann. Probab. , vol.3 , pp. 534-545
    • Chen, L.H.Y.1
  • 16
    • 0001136294 scopus 로고
    • A note on an inequality involving the normal distribution
    • MR0614640
    • Chernoff, H. (1981). A note on an inequality involving the normal distribution. Ann. Probab. 9 533-535. MR0614640
    • (1981) Ann. Probab. , vol.9 , pp. 533-535
    • Chernoff, H.1
  • 17
    • 84972535617 scopus 로고
    • Closed form summation for classical distributions: Variations on a theme of De Moivre
    • MR1144242
    • Diaconis, P. and Zabell, S. (1991). Closed form summation for classical distributions: Variations on a theme of De Moivre. Statist. Sci. 6 284-302. MR1144242
    • (1991) Statist. Sci. , vol.6 , pp. 284-302
    • Diaconis, P.1    Zabell, S.2
  • 18
    • 0031260681 scopus 로고    scopus 로고
    • Stein's method and the zero-bias transformation with application to simple random sampling
    • MR1484792
    • Goldstein, L. and Reinert, G. (1997). Stein's method and the zero-bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935-952. MR1484792
    • (1997) Ann. Appl. Probab. , vol.7 , pp. 935-952
    • Goldstein, L.1    Reinert, G.2
  • 19
    • 16244412903 scopus 로고    scopus 로고
    • Distributional transformations, orthogonal polynomials, and Stein characterizations
    • DOI 10.1007/s10959-004-2602-6
    • Goldstein, L. and Reinert, G. (2005). Distributional transformations, orthogonal polynomials, and Stein characterizations. J. Theoret. Probab. 18 237-260. MR2132278 (Pubitemid 40465394)
    • (2005) Journal of Theoretical Probability , vol.18 , Issue.1 , pp. 237-260
    • Goldstein, L.1    Reinert, G.2
  • 20
    • 0002321209 scopus 로고
    • Variance inequalities for functions of Gaussian variables
    • MR1308667
    • Houdré, C. and Kagan, A. (1995). Variance inequalities for functions of Gaussian variables. J. Theoret. Probab. 8 23-30. MR1308667
    • (1995) J. Theoret. Probab. , vol.8 , pp. 23-30
    • Houdré, C.1    Kagan, A.2
  • 21
    • 0010773289 scopus 로고
    • Covariance identities and inequalities for functionals on Wiener and Poisson spaces
    • MR1330776
    • Houdré, C. and Pérez-Abreu, V. (1995). Covariance identities and inequalities for functionals on Wiener and Poisson spaces. Ann. Probab. 23 400-419. MR1330776
    • (1995) Ann. Probab. , vol.23 , pp. 400-419
    • Houdré, C.1    Pérez-Abreu, V.2
  • 22
    • 0010994548 scopus 로고
    • A note on variance bounds for a function of a Pearson variate
    • MR1257861
    • Johnson, R.W. (1993). A note on variance bounds for a function of a Pearson variate. Statist. Decisions 11 273-278. MR1257861
    • (1993) Statist. Decisions , vol.11 , pp. 273-278
    • Johnson, R.W.1
  • 23
    • 0000319146 scopus 로고
    • On characterizations of distributions by mean absolute deviation and variance bounds
    • MR1128869
    • Korwar, R.M. (1991). On characterizations of distributions by mean absolute deviation and variance bounds. Ann. Inst. Statist. Math. 43 287-295. MR1128869
    • (1991) Ann. Inst. Statist. Math. , vol.43 , pp. 287-295
    • Korwar, R.M.1
  • 24
    • 12244291299 scopus 로고    scopus 로고
    • Generalized
    • Pearson distributions and related characterization problems, MR1954042
    • Lefévre, C., Papathanasiou, V. and Utev, S.A. (2002). Generalized Pearson distributions and related characterization problems. Ann. Inst. Statist. Math. 54 731-742. MR1954042
    • (2002) Ann. Inst. Statist. Math. , vol.54 , pp. 731-742
    • Lefévre, C.1    Papathanasiou, V.2    Utev, S.A.3
  • 25
    • 0347998575 scopus 로고    scopus 로고
    • Estimation based on the winzorized mean in the geometric distribution
    • MR1820824
    • López-Blázquez, F. and Salamanca-Miño, B. (2000). Estimation based on the winzorized mean in the geometric distribution. Statistics 35 81-95. MR1820824
    • (2000) Statistics , vol.35 , pp. 81-95
    • López-Blázquez, F.1    Salamanca-Mino, B.2
  • 26
    • 84962022819 scopus 로고    scopus 로고
    • Unified variance bounds and a Stein-type identity
    • (C.A. Charalambides, M.V. Koutras and N. Balakrishnan, eds.). New York: Chapman & Hall/CRC
    • Papadatos, N. and Papathanasiou, V. (2001). Unified variance bounds and a Stein-type identity. In Probability and Statistical Models With Applications (C.A. Charalambides, M.V. Koutras and N. Balakrishnan, eds.) 87-100. New York: Chapman & Hall/CRC.
    • (2001) Probability and Statistical Models With Applications , pp. 87-100
    • Papadatos, N.1    Papathanasiou, V.2
  • 27
    • 0036754022 scopus 로고    scopus 로고
    • Poisson approximation for a sum of dependent indicators: An alternative approach
    • DOI 10.1239/aap/1033662168
    • Papadatos, N. and Papathanasiou, V. (2002). Poisson approximation for a sum of dependent indicators: An alternative approach. Adv. in Appl. Probab. 34 609-625. MR1929600 (Pubitemid 35297880)
    • (2002) Advances in Applied Probability , vol.34 , Issue.3 , pp. 609-625
    • Papadatos, N.1    Papathanasiou, V.2
  • 28
    • 12244268590 scopus 로고
    • A characterization of the Pearson system of distributions and the associated orthogonal polynomials
    • MR1341214
    • Papathanasiou, V. (1995). A characterization of the Pearson system of distributions and the associated orthogonal polynomials. Ann. Inst. Statist. Math. 47 171-176. MR1341214
    • (1995) Ann. Inst. Statist. Math. , vol.47 , pp. 171-176
    • Papathanasiou, V.1
  • 29
    • 0042910559 scopus 로고    scopus 로고
    • Extended covariance identities and inequalities
    • DOI 10.1016/S0167-7152(01)00140-7, PII S0167715201001407
    • Privault, N. (2001). Extended covariance identities and inequalities. Statist. Probab. Lett. 55 247-255. MR1867528 (Pubitemid 33619137)
    • (2001) Statistics and Probability Letters , vol.55 , Issue.3 , pp. 247-255
    • Privault, N.1
  • 30
    • 0001270801 scopus 로고
    • Sur le problème des moments et le théorème de Parseval correspondant (in French)
    • Riesz, M. (1923). Sur le problème des moments et le théorème de Parseval correspondant (in French). Acta Litt. Ac. Sci. (Szeged) 1 209-225.
    • (1923) Acta Litt. Ac. Sci. (Szeged) , vol.1 , pp. 209-225
    • Riesz, M.1
  • 31
    • 0035863994 scopus 로고    scopus 로고
    • Orthogonal Polynomials in Stein's Method
    • DOI 10.1006/jmaa.2000.7159, PII S0022247X00971592
    • Schoutens, W. (2001). Orthogonal polynomials in Stein's method. J. Math. Anal. Appl. 253 515-531. MR1808151 (Pubitemid 33382562)
    • (2001) Journal of Mathematical Analysis and Applications , vol.253 , Issue.2 , pp. 515-531
    • Schoutens, W.1
  • 32
    • 0002314745 scopus 로고
    • On the variance of estimates
    • MR0029141
    • Seth, G.R. (1949). On the variance of estimates. Ann. Math. Statist. 20 1-27. MR0029141
    • (1949) Ann. Math. Statist. , vol.20 , pp. 1-27
    • Seth, G.R.1
  • 33
    • 0000457248 scopus 로고
    • A bound for the error in the normal approximation to the distribution of a sum of dependent random variables
    • Berkeley, CA: Univ. California Press. MR0402873
    • Stein, C.M. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 583-602. Berkeley, CA: Univ. California Press. MR0402873
    • (1972) Proc. Sixth Berkeley Symp. Math. Statist. Probab. , vol.2 , pp. 583-602
    • Stein, C.M.1
  • 34
    • 0000169918 scopus 로고
    • Estimation of the mean of a multivariate normal distribution
    • MR0630098
    • Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135-1151. MR0630098
    • (1981) Ann. Statist. , vol.9 , pp. 1135-1151
    • Stein, C.1


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