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Distribution of Record Values

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Distribution of Record Values Recent Developments in Record Values M. Ahsanullah Rider University Lawrenceville, NJ,USA In this talk the record values of continuous and discrete random variables will be discussed. 1. 2 Consider the following sequence of observations 6,10,5,6,11, 11,14,4,18,4, 1, … 6,10,11,14,18, … -upper record values 6,5,4,1, … -lower record values 6,10,11,11,14.18, - upper weak records 6,5,4,4.1, … - lower weak records Suppose we consider 5 more observations. 6,10,5,6,11,11,14,4,18,4,1, ,8,0,21,21,0,… 6,10,11,14,18, 21,.. upper record values 6,5,4,1, 0,.. lower record values 6,10,11,11,14.18,21,21, upper weak records 6,5,4,4.1, 0,0,.. - lower weak records Thus the past records are in the data. 1.0. INTRODUCTION AND EXAMPLES OF RECORD VALUES In 1952 Chandler defined so called Record Times and Record Values and a groundwork for a mathematical Theory of Records. For six decades beginning his pioneering work there appeared about 500 papers and some monographs devoted to different aspects of the theory of records. Records are very popular because they arise naturally in many fields of studies such as climatology, sports, medicine, traffic, industry and so on. The Records as such are memories of their time. 4 Suppose we consider the weighing of objects on a scale missing its spring. An object is placed on the scale and its weight is measured. The 'needle' indicated the correct weight 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. Let us consider a sequence of products that 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 X1 then we record its failure stress, otherwise we consider the next product. In general we will record stress Xm of th the m product if Xm < min (X1,..., Xm-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 {Xj} by {-Xj j >1} or if P(Xj > 0) = 1 by { 1/Xj, i > 1}, j=1,2,… Chandler (1952) proved the interesting result that for any given distribution of the random variables the expected value of the inter record time is infinite. Feller 6 (1952) gave some examples of record values with respect to gambling problems. 1.1. DEFINITION OF RECORD VALUES AND RECORD TIMES Suppose that X1, X2, .... is a sequence of independent and identically distributed random variables with cumulative distribution function F(x). Let Yn = max (min){X1,X2,...,Xn} for n > 1. We say Xj is an upper(lower) record value of {Xn, n >1}, if Yj > (<)Yj-1, j > 1. By definition X1 is an upper as well as a lower record value. One can transform the upper records to lower records by replacing the original sequence of {Xj} by {-Xj , j > 1} or (if P(Xi > 0)=1 for all i) by {1/Xi, i >1}; the lower record values of this sequence will correspond to the upper record values of the original sequence. The indices at which the upper record values occur are given by the record times {U(n)}, n > 0, where U(n) = min{j|j>U(n-1),Xj >XU(n-1), n >1} and U(1) =1. The record times of the sequence {Xn n >1} are the same as those for the sequence {F(Xn), n > 1}. 8 Since F(X) has an uniform distribution, it follows that the distribution of U(n) , n > 1 does not depend on F. We will denote L(n) as the indices where the lower record values occur. By our assumption U(1) = L(1) = 1. The distribution of L(n) also does not depend on F. The kth record values XU(n,k) are a natural extension of records XU(n) . It is interesting that distributions of the kth records can be expressed via distributions of the classical record values. Really, together with a sequence of independent random variables X1,X2,… having a common distribution function F, let us consider one more sequence Y1=min{X1,X2,…,Xk}, Y2= min{Xk+1,Xk+2,…,X2k }, …. 10 Let now XU(n,k) be the kth record value based on X’s and Y(n) be the classical records based on the sequence Y1, Y2,…. Then for any fixed k=2,3,… and any n=1,2,… the vector (XU(1,k) ,XU(2,k) ,…,XU(n.k) has the same distribution as the vector (YU(1) ,YU(2) ,…,YU(n). 1.2. THE EXACT DISTRIBUTION OF RECORD VALUES (CONTINUOUS RVS) Many properties of the record value sequence can be expressed in terms of R(x), where R(x) = - ln Fx(), 0 < F (x) < 1 and Fx() = 1 - F(x). Here 'ln' is used for the natural logarithm. If we define Fn(x) as the distribution function of XU(n) for n > 1, then we have 12 F1(x) = P[XU(1) < x] = F(x) (1.2.1) F2(x) = P[XU(2) < x ] x y i1 = (F(u)) dF(u)dF(y) i1 x y dF(u) dF(y) 1 F(u) x R(y)dF(y) (1.2.2) If F(x) has a density f(x), then the probability density function (pdf) of XU(2) is f2(x) = R(x) f(x) (1.2.3) It can similarly be shown that the pdf Fn(x) of XU(n) is Fn(x) = P( XU(n) < x ) x un u2 f (un1) f (u1) f (un )dun dun1. du1. 1 F(un1) 1 F(u1) n1 x R (u) dF(u), x . (n) (1.2.4) This can be expressed as n1 R(x) u F (x) = eu du , n (n) x n1 j (R(x)) Fn (x)1Fn (x)F (x) j0 ( j 1) n1 j (R(x)) eR( x) j0 ( j 1) 14 The pdf fn(x) of XU(n) is R n1 (x) fn(x) = f (x), x . (n) (1.2.5) The joint pdf f(x1,x2, ... , xn) of the n record values XU(1), XU(2), ..., XU(n)) is given by f (x1, x2 ,....,xn ) r(x1 )r(x2 )....r(xn1 )f (xn ) for x1 x2...xn1xn , (1.2.6) d f (x) where r(x) = R(x) , dx 1 F(x) 0 < F(x) < 1. The function r(x) is known as hazard rate. The joint pdf of XU(i) and XU(j) is f(xi,xj) (R(x ))i1 (R(x ) R(x )) ji1 = i r(x ) j i f (x ) (i) i ( j i) j (1.2.7) for xi x j . Thus E(Vk) = k and Var(Vk) = k. The conditional pdf of XU(j)| XU(i) = xi if( xj | XU(i) = xi ) = fij (xi , x j ) fi (xi ) ji1 (R(x j ) R(xi )) f (x j ) = ( j i) 1 F(xi ) 16 (1.2.8) for xij x . fik,im (x, y |XU(i) z) 1 1 [R(y) R(x)]mk1[R(X ) R(z)]k1 (m k) (k) f (y)r(x) . F(z) for - < z < x < y < . The marginal pdf of the nth lower record value can be derived by using the same procedure as that of the nth upper record value. Let H(u) = -ln F(u), 0< F(u) < 1 and d h(u) H (u), then du P(XL(n) < x ) = n1 x {H (u)} dF(u) (n) (1.2.9) and the corresponding the pdf f(n) can be written as (H (x))n1 f(n)(x) = f (x). (n) (1.2.10) The joint pdf of XL(1), XL(2), ... , XL(m) can be written as f (1),(2),...,(m) (x1,x2 ,...,xm )h(x1 )h(x2 )...h(xm1 )f (xm ) xm xm1 ...x1 18 = 0, otherwise. (1.2.11) The joint pdf of XL(r) and XL(s) is f(r),(s) (x,y) = (H (x))r 1[H (y) H (x)]sr 1 h(x)f (y) (r) (s r) for s > r and - < y < x < . (1.2.12) Proceeding as in the case of upper record values, we can obtain the conditional pdfs of the lower record values. Example 1.2.1. Let us consider the exponential distribution with pdf f(x) as f(x)=e-x ,0x and the cumulative distribution function (cdf) F(x) as F(x) = 1- e-x, 0 < x <∞. Then R(x) = x and n1 x x fn(x) = e , x 0 (n) =0, otherwise. The joint pdf of XU(m) and XU(n) , n> m is fm,n(x,y) xm1 = (y x)nm1e y (m)(nm) 0 xy = 0, otherwise. 20 The conditional pdf of XU(n) | XU(m) = x) is f(y|XU(m)=x) (y x)nm1 = e( yx) (n m) = 0, otherwise. Thus the conditional distribution of XU(n) - XU(m) given XU(m) is the same as the unconditional distribution of XU(n-m) for n > m. Example 1.2.2. Suppose that the random variable X has the Gumbel distribution with pdf f(x) = x ex e e ,x . Let F(n) and f(n) be the cdf and pdf of XL(n).
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