~:~.."/¢m ELSEVIER Statistics & Probability Letters 31 (1996) 7-12 STATISTICS& PROBABILITY LEr~RS On the estimation of the unknown sample size from the number of records J.L. Moreno Rebollo *, I. Barranco Chamorro, F. L6pez Blfizquez, T. G6mez G6mez Departamento de Estadistica e I. 0., Facultad de Matemdtticas, Universidad de Sevilla, Tarfia, sin. 41012 Sevilla, Spain Received June 1995; revised February 1996 Abstract Suppose we have observed only the number of records in a random sample of unknown size n from a continuous distribution. Based on this data, we propose an unbiased estimator of n and derive expressions for its maximum likelihood estimator and an estimator based on the method of moments. Keywords: Lower records; Estimation; Sample size 1. Introduction As the distribution theory for the number of records is the same whether it is an upper or lower record, without loss of generality, we will restrict our attention to lower records. Let Y1, Ye ..... Y~ be a random sample with common continuous distribution function F. The lower record times are defined recurrently by L(1)=I, L(k)=min{j:j>L(k-1),Yj < YL(k-1)}, k~>2, the corresponding lower records are the random variables Ark = YL(k), k >>- 1. Let us also define the r.v. N (n) • number of lower records among Y1, Y2 ..... Yn. We assume that an unknown number of the observations Yi's are missing, but in spite of such circumstance we still keep the value of all the lower records in the sample. In this situation, the sample size, n, is unknown, but note that by the previous assumptions the value of N (n) is known. In this paper, we will introduce three estimators of n, exploiting some properties of N (n). Firstly, we will study the distribution of N (n) and its properties. In Section 3, we will obtain an unbiased estimator and characterize the maximum likelihood estimator (m.l.e.) of n. We also introduce an estimator suggested by the method of moments. * Corresponding author. E-mail: moreno@cica.es. 0167-7152/96/$12.00 (E) 1996 Elsevier Science B.V. All rights reserved PH SO 167-71 52( 96)00006-5