ABSTRACT
Let {Xn , n ≥ 1} be a sequence of independent and identically distributed random variables with cumulative distribution function F(x) and the corresponding probability density function f(x). Set Yn = max(X 1,…, Xn ), n ≥ 1. We say Yj is an upper record value of {Xn , n ≥ 1} if Yj > Y j−1, j > 1. By definition, X 1 = Y 1 is an upper record value. Thus the upper record values in the sequence {Xn , n ≥ 1} are the successive maxima. For example, consider the weighing of objects on a scale missing it’s spring. An object is placed on this scale and its weight measured. The ‘needle’ indicates the correct value but does not return to zero when the object is removed. If various objects are placed on the scale, only the weights greater than the previous ones can be recorded. These recorded weights are the upper record value sequence. Suppose we consider a sequence of products which may fail under stress. We are interested to determine the minimum failure stress of the products sequentially. We test the first product until it fails with stress less than X 1 then we record its failure stress, otherwise we consider the next product. In general we will record stress Xm of the mth product if Xm < min(X 1,…, X m−1), m > 1. The recorded failure stresses are the lower record values. One can go from lower records to upper records by replacing the original sequence of random variables by {–Xi , i ≥ 1} or if P(Xi > 0) = 1 by {1/Xi, i ≥ 1). Unless mentioned otherwise, we will call the upper record values as record values. The indices at which the record values occur are given by the record times {U(n)}, n > 0, where U(n) = min{j | j > U(n – 1), Xj > X U(n−i) n > 1} and U/(1) = 1. The record times of the sequence {Xn, n > n) are the same as those for the sequence {F(Xn ), n ≥ 1}. Since F(X) has uniform distribution, it follows that the distribution of U(n), n ≥ 1, does not depend on F. Chandler (1952) introduced record values and record times. Feller (1966) gave some examples of record values with respect to gambling problems. Properties of record values of i.i.d. random variables have been extensively studied in the literature. See Ahsanullah (1988), Arnold and 280 Balakrishnan (1989), Arnold, Balakrishnan, and Nagaraja (1992), Nagaraja (1988) and Nevzorov (1988) for recent reviews.
