A refined continuity correction for the negative binomial distribution and asymptotics of the median. (English) Zbl 07734492
Summary: In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a \(\text{Negative Binomial}(r,p)\) random variable jittered by a \(\text{Uniform}(0,1)\), which answers a problem left open in Coeurjolly and Trépanier (Metrika 83(7):837-851, 2020). This is used to construct a simple, robust and consistent estimator of the parameter \(p\), when \(r > 0\) is known. The case where \(r\) is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.
MSC:
| 62E20 | Asymptotic distribution theory in statistics |
| 62F12 | Asymptotic properties of parametric estimators |
| 62F35 | Robustness and adaptive procedures (parametric inference) |
| 62E15 | Exact distribution theory in statistics |
| 60F15 | Strong limit theorems |
| 62B15 | Theory of statistical experiments |
Keywords:
local limit theorem; continuity correction; quantile coupling; negative binomial distribution; Gaussian approximation; median; comparison of experiments; Le Cam distance; total variationReferences:
| [1] | Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC. http://www.ams.org/mathscinet-getitem?mr=MR0167642 · Zbl 0171.38503 |
| [2] | Adell, JA; Alzer, H., Inequalities for the median of the gamma distribution, J Comput Appl Math, 232, 2, 481-495 (2009) · Zbl 1185.26034 · doi:10.1016/j.cam.2009.06.024 |
| [3] | Adell, JA; Jodrá, P., The median of the Poisson distribution, Metrika, 61, 3, 337-346 (2005) · Zbl 1079.62014 · doi:10.1007/s001840400350 |
| [4] | Adell, JA; Jodrá, P., Sharp estimates for the median of the \(\Gamma (n+1,1)\) distribution, Stat Probab Lett, 71, 2, 185-191 (2005) · Zbl 1074.60011 · doi:10.1016/j.spl.2004.10.025 |
| [5] | Adell, JA; Jodrá, P., On a Ramanujan equation connected with the median of the gamma distribution, Trans Am Math Soc, 360, 7, 3631-3644 (2008) · Zbl 1151.41021 · doi:10.1090/S0002-9947-07-04411-X |
| [6] | Alm, SE, Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence, Bernoulli, 9, 2, 351-371 (2003) · Zbl 1015.62007 |
| [7] | Alzer, H., Proof of the Chen-Rubin conjecture, Proc R Soc Edinb Sect A, 135, 4, 677-688 (2005) · Zbl 1077.62005 · doi:10.1017/S0308210500004066 |
| [8] | Alzer, H., A convexity property of the median of the gamma distribution, Stat Probab Lett, 76, 14, 1510-1513 (2006) · Zbl 1096.60010 · doi:10.1016/j.spl.2006.03.011 |
| [9] | Berg, C.; Pedersen, HL, The Chen-Rubin conjecture in a continuous setting, Methods Appl Anal, 13, 1, 63-88 (2006) · Zbl 1107.60006 · doi:10.4310/MAA.2006.v13.n1.a4 |
| [10] | Berg, C.; Pedersen, HL, Convexity of the median in the gamma distribution, Ark Mat, 46, 1, 1-6 (2008) · Zbl 05319828 · doi:10.1007/s11512-006-0037-2 |
| [11] | Brown, LD; Carter, AV; Low, MG; Zhang, C-H, Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift, Ann Stat, 32, 5, 2074-2097 (2004) · Zbl 1062.62083 · doi:10.1214/009053604000000012 |
| [12] | Carter, AV, Deficiency distance between multinomial and multivariate normal experiments. Dedicated to the memory of Lucien Le Cam, Ann Stat, 30, 3, 708-730 (2002) · Zbl 1029.62005 · doi:10.1214/aos/1028674839 |
| [13] | Chen, C-P, The median of gamma distribution and a related Ramanujan sequence, Ramanujan J, 44, 1, 75-88 (2017) · Zbl 1379.41011 · doi:10.1007/s11139-016-9796-y |
| [14] | Chen, J.; Rubin, H., Bounds for the difference between median and mean of gamma and Poisson distributions, Stat Probab Lett, 4, 6, 281-283 (1986) · Zbl 0596.62015 · doi:10.1016/0167-7152(86)90044-1 |
| [15] | Choi, KP, On the medians of gamma distributions and an equation of Ramanujan, Proc Am Math Soc, 121, 1, 245-251 (1994) · Zbl 0803.62007 · doi:10.1090/S0002-9939-1994-1195477-8 |
| [16] | Coeurjolly, J-F; Trépanier, JR, The median of a jittered Poisson distribution, Metrika, 83, 7, 837-851 (2020) · Zbl 1475.62144 · doi:10.1007/s00184-020-00765-3 |
| [17] | Cramér, H., On the composition of elementary errors, Scand Actuar J, 1928, 1, 13-74 (1928) · JFM 54.0557.02 · doi:10.1080/03461238.1928.10416862 |
| [18] | Cramér H (1937) Random variables and probability distributions. In: Cambridge Tracts in Mathematics and Mathematical Physics, vol. 36, 1st ed. Cambridge Univ. Press · JFM 63.1080.01 |
| [19] | Cressie, N., A finely tuned continuity correction, Ann Inst Stat Math, 30, 3, 435-442 (1978) · Zbl 0445.62033 · doi:10.1007/BF02480234 |
| [20] | Esseen, C-G, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math, 77, 1-125 (1945) · Zbl 0060.28705 · doi:10.1007/BF02392223 |
| [21] | Göb, R., Bounds for median and \(50\) percentage point of binomial and negative binomial distribution, Metrika, 41, 1, 43-54 (1994) · Zbl 0791.62017 · doi:10.1007/BF01895303 |
| [22] | Govindarajulu, Z., Normal approximations to the classical discrete distributions, Sankhyā Ser A, 27, 143-172 (1965) · Zbl 0202.48401 |
| [23] | Groeneveld, RA; Meeden, G., The mode, median, and mean inequality, Am Stat, 31, 3, 120-121 (1977) |
| [24] | Hamza, K., The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions, Stat Probab Lett, 23, 1, 21-25 (1995) · Zbl 0819.60017 · doi:10.1016/0167-7152(94)00090-U |
| [25] | Jodrá, P., Computing the asymptotic expansion of the median of the Erlang distribution, Math Model Anal, 17, 2, 281-292 (2012) · Zbl 1242.41033 · doi:10.3846/13926292.2012.664571 |
| [26] | Kolassa JE (1994) Series approximation methods in statistics. Lecture Notes in Statistics, vol 88. Springer-, New York. http://www.ams.org/mathscinet-getitem?mr=MR1295242 · Zbl 0797.62008 |
| [27] | Lyon, RF, On closed-form tight bounds and approximations for the median of a gamma distribution, PLOS ONE, 1, 18 (2021) · doi:10.1371/journal.pone.0251626 |
| [28] | Mariucci, E., Le Cam theory on the comparison of statistical models, Grad J Math, 1, 2, 81-91 (2016) · Zbl 1497.62029 |
| [29] | Nussbaum, M., Asymptotic equivalence of density estimation and Gaussian white noise, Ann Stat, 24, 6, 2399-2430 (1996) · Zbl 0867.62035 · doi:10.1214/aos/1032181160 |
| [30] | Ouimet, F., On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators, J Math Anal Appl, 499, 1 (2021) · Zbl 1462.62772 · doi:10.1016/j.jmaa.2021.125033 |
| [31] | Ouimet, F., A precise local limit theorem for the multinomial distribution and some applications, J Stat Plann Inference, 215, 218-233 (2021) · Zbl 1474.62036 · doi:10.1016/j.jspi.2021.03.006 |
| [32] | Patil, GP, On the evaluation of the negative binomial distribution with examples, Technometrics, 2, 501-505 (1960) · Zbl 0096.13403 · doi:10.1080/00401706.1960.10489915 |
| [33] | Payton, ME; Young, LJ; Young, JH, Bounds for the difference between median and mean of beta and negative binomial distributions, Metrika, 36, 6, 347-354 (1989) · Zbl 0704.60016 · doi:10.1007/BF02614111 |
| [34] | Pearson, K., On the applications of the double Bessel function \(K_{\tau_1,\tau_2}(x)\) to statistical problems, Biometrika, 25, 1-2, 158-178 (1933) · Zbl 0007.07105 · doi:10.1093/biomet/25.1-2.158 |
| [35] | Pinelis, I., Monotonicity properties of the gamma family of distributions, Stat Probab Lett, 171 (2021) · Zbl 1461.62033 · doi:10.1016/j.spl.2020.109027 |
| [36] | Teicher, H., An inequality on Poisson probabilities, Ann Math Stat, 26, 147-149 (1955) · Zbl 0064.38203 · doi:10.1214/aoms/1177728608 |
| [37] | van de Ven, R.; Weber, NC, Bounds for the median of the negative binomial distribution, Metrika, 40, 3-4, 185-189 (1993) · Zbl 0769.62014 |
| [38] | van der Vaart AW (1998) Asymptotic statistics. Cambridge Series in Statistical and Probabilistic Mathematics, vol 3. Cambridge University Press, Cambridge. http://www.ams.org/mathscinet-getitem?mr=MR1652247 · Zbl 0910.62001 |
| [39] | You, X., Approximation of the median of the gamma distribution, J Number Theory, 174, 487-493 (2017) · Zbl 1355.41022 · doi:10.1016/j.jnt.2016.11.019 |
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