Geodesic
Estimating distributions from samples with random size
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2023), pp. 5-24
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The article is concerned with the estimating problem of a distribution function and the limiting behavior of the distance between the empirical and theoretical laws, namely, integrated square errors and Smirnov and Kolmogorov statistics by the samples with random size. We suppose that this random size and the initial sample are independent random variables and this random variable has the generalized negative binomial distribution. We find limiting distributions for integrated square errors of kernel distribution function estimators by the samples with random size. It is shown that for samples with random size the limiting distribution of the Smirnov and Kolmogorov statistics has more heavier tails than the Weibull and Kolmogorov distribution function for samples with the fixed size. We propose an asymptotic expansion approach to naturally balance the asymptotic distribution and random sample size. The problem of sequential estimation of the shift parameter of the uniform distribution is considered. The negative binomial distribution (samples size $\nu $) arises naturally here from a statistical experiment of performing a series of independent trials.
Keywords: sample with random size, empirical distribution function, Kolmogorov statistics, sequential estimation, generalized negative binomial distribution. 
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M. S. Tikhov. Estimating distributions from samples with random size. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2023), pp. 5-24. http://geodesic.mathdoc.fr/item/VTPMK_2023_4_a0/

[1] Wald A., “Sequential Test of Statistical Hypotheses”, The Annals of Mathematical Statistics, 16:2 (1945), 117–186 | DOI | MR | Zbl

[2] Linnik Yu. V., Romanovsky I. V., “Some new results in sequential estimation theory”, Berkeley Symposium on Mathematical Statistics and Probability, 1972, 85–96 | MR | Zbl

[3] Tihov M. S., “On optimal sequential estimation procedures for non-quadratic loss functions”, Theory of Probability and its Applications, 23:1 (1978), 132–138 | DOI | MR | Zbl

[4] Tikhov M. S., “Sequential estimation of the uniform distribution shift parameter for censored type II samples”, Notes of LOMI scientific seminars, 136 (1984), 183–192 (in Russian) | MR | Zbl

[5] Renyi A., “On the central limit theorem for the sum of a random number of independent random variables”, Acta Mathematica Academiae Scientiarum Hungaricae, 11 (1963), 97–102 | DOI | MR

[6] Blum J. R., Hanson D. L., Rosenblatt J. I., “On the central limit theorem for the sum of a random number of independent random variables”, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1 (1963), 389–393 | DOI | MR | Zbl

[7] Gnedenko B. V., Stomatovich S., Shukri A., “About the distribution of the median”, Bulletin of the Moscow University. Series 1: Mathematics. Mechanics, 1984, no. 2, 59–63 (in Russian) | Zbl

[8] Bening V. E., “On the power of criteria in the case of samples of random size”, Herald of Tver State University. Series: Applied Mathematics, 2018, no. 4, 5–25 (in Russian) | DOI

[9] Bening V. E., Korolev V. Y., “On an application of the Student distribution in the theory of probability and mathematical statistics”, Theory of Probability and its Applications, 49:3 (2005), 377–391 | DOI | MR | Zbl

[10] Slepov N. A., “Convergence rate of random geometric sum distributions to the Laplace law”, Theory of Probability and its Applications, 66:1 (2021), 121–141 | DOI | DOI | MR | Zbl

[11] Toda A. A., Weak limit of the geometric sum of independent but not identically distributed random variables, 2011 | DOI

[12] Christoph G., Ulyanov V., “Second Order Chebyshev-Edgeworth-Type Approximations for Statistics Based on Random Size Samples”, Mathematics, 11 (2023), 1848 | DOI

[13] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and series. Elementary functions, Vol. 1: Elementarnye funktsii, Fizmatlit Publ., Moscow, 2002, 632 pp. (in Russian) | MR

[14] Tikhov M. S., “Statistical estimation based on interval censored data”, Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, Statistics for Industry and Technology, eds. N. Balakrishnan, M.S. Nikulin, M. Mesbah, N. Limnios, Birkhauser, Boston, MA, 2004, 211–218 | DOI | MR

[15] Tikhov M. S., “Asimptoticheskie raspredeleniya summiruemykh kvadratichnykh uklonenij otsenok funktsii raspredeleniya v zavisimosti «doza-effekt»”, Review of Applied and Industrial Mathematics journal, 16:5 (2009), 772–786 (in Russian)

[16] Okumura H., Naito K., “Non-parametric kernel regression for multinomial data”, Journal Multivariate Analysis, 97 (2006), 2009–2022 | DOI | MR | Zbl

[17] Tikhov M. S., “Negative $\lambda$-binomial regression in dose-effect relationship”, Herald of Tver State University. Series: Applied Mathematics, 2022, no. 4, 53–75 (in Russian) | DOI | MR

[18] Gradshtejn I. S., Ryzhik I. M., Table of Integrals, Series, and Products, Fizmatlit Publ., Moscow, 1963, 1100 pp. (in Russian) | MR

[19] Kolmogorov A. N., “On the empirical definition of the law of distribution”, Probability theory and mathematical statistics, Nauka Publ., Moscow, 1986, 134–141 (in Russian) | MR

[20] Gikhman I. I., Skorokhod A. V., Introduction to the theory of random processes, Nauka Publ., Moscow, 1965, 356 pp. (in Russian) | MR

[21] Sarhan A., Greenberg B., “Estimation of location and scale parameter for the rectangular population fron censored samples”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 21:2 (1959), 356–363 | Zbl

[22] Kendall M., Styuart A., Statistical conclusions and connections, Nauka Publ., Moscow, 1973, 900 pp. (in Russian)

[23] David H., Nagaraja H., Order Statistics, Wiley, 2003, 475 pp. | MR | Zbl

[24] Bejtman G., Erdeji A., Vysshie transtsendentnye funktsii, Nauka Publ., Moscow, 1973, 296 pp. (in Russian)

[25] Smirnov N. V., “Approximate laws of distribution of random variables from empirical data”, Achievements of mathematical sciences, 1:10 (1944), 179–206 (in Russian)

[26] Korolyuk V. S., “Asymptotic expansions for the criteria of agreement by A.N.Kolmogorov and N.V.Smirnov”, Izvestia of the USSR Academy of Sciences. Mathematical series, 19:2 (1955), 103–124 (in Russian) | MR | Zbl

[27] Li-Tsyan Ch., “On the exact distribution of N.V.Smirnov's statistics and its asymptotic decomposition”, Mathematics, 4:2 (1960), 121–134 (in Russian)

[28] Bol'shev L. N., “Asymptotically Pearson's Transformations”, Theory of Probability and its Applications, 8:2 (1963), 121–146 | DOI | Zbl

[29] Hoel P. G., “On the chi-square distribution for small samples”, The Annals of Mathematical Statistics, 9:3 (1938), 158–165 | DOI | Zbl

[30] Kagan A. M., “Theory of estimation for families with shift, scale and exponential parameters”, Proceedings of the Steklov Mathematical Institute, 104 (1968), 19–87 (in Russian) | Zbl