Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-10T06:25:08.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations in a random record model with a nonidentically distributed initial record

Part of: Distribution theory Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Gadi Barlevy  and
H. N. Nagaraja
Gadi Barlevy*
Affiliation:
Federal Reserve Bank of Chicago
H. N. Nagaraja*
Affiliation:
The Ohio State University
*
∗ Postal address: Economic Research Department, Federal Reserve Bank of Chicago, 230 South La Salle Street, Chicago, IL 60604-1413, USA. Email address: gbarlevy@frbchi.org
∗∗ Postal address: Department of Statistics, The Ohio State University, 1958 Neil Avenue, Columbus, OH 43210-1247, USA. Email address: hnn@stat.ohio-state.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider a sequence, of random length M, of independent, continuous observations X i , 1 ≤ i ≤ M, where M is geometric, X 1 has cumulative distribution function (CDF) G, and X i , i ≥ 2, have CDF F. Let N be the number of upper records and let R n , n ≥ 1, be the nth record value. We show that N is independent of F if and only if G(x) = G 0(F(x)) for some CDF G 0, and that if E(|X 2|) is finite then so is E(|R n |), n ≥ 2, whenever N ≥ n or N = n. We prove that the distribution of N, along with appropriately chosen subsequences of E(R n ), characterize F and G and, along with subsequences of E(R n - R n-1), characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

Keywords

Moment sequencenumber of recordsrecord spacinggeometric distributionMüntz-Szász theoremTitchmarsh convolution theoremjob search model

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60G70: Extreme value theory; extremal processes 62G32: Statistics of extreme values; tail inference

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1119 - 1136
Copyright
© Applied Probability Trust 2006 

References

Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. John Wiley, New York.CrossRefGoogle Scholar
Barlevy, G. (2005). Identification of Job search models using record statistics. FRB Chicago mimeo.Google Scholar
Barlevy, G. and Nagaraja, H. N. (2006). Identification of Job search models with initial condition problems. FRB Chicago mimeo.CrossRefGoogle Scholar
Boas, R. P. (1954). Entire Functions. Academic Press, New York.Google Scholar
Borwein, P. and Erdélyi, T. (1995). Polynomials and Polynomial Inequalities. Springer, New York.Google Scholar
Bunge, J. and Nagaraja, H. N. (1991). The distribution of certain record statistics from a random number of observations. Stoch. Proc. Appl. 38, 167–183.Google Scholar
Burdett, K. and Mortensen, D. (1998). Wage differentials, employer size, and unemployment. Internat. Econom. Rev. 39, 257–273.Google Scholar
Eckstein, Z. and van den Berg, G. (2006). Empirical labor search: a survey. To appear in J. Econometrics.Google Scholar
Kamps, U. (1998). Characterizations of distributions by recurrence relations and identities for moments of order statistics. In Handbook of Statistics, Vol. 16, Order Statistics: Theory and Methods, eds Balakrishnan, N. and Rao, C. R., Elsevier, Amsterdam, pp. 291–311.Google Scholar
Müntz, C. H. (1914). Über den Approximationssatz von Weierstrass. In H. A. Schwarz Festschrift, Springer, Berlin, pp. 303–312.Google Scholar
Nagaraja, H. N. and Barlevy, G. (2003). Characterizations using record moments in a random record model and applications. J. Appl. Prob. 40, 826–833.Google Scholar
Nevzorov, V. B. (1986). Two characterizations using records. In Stability Problems for Stochastic Models (Lecture Notes Math. 1223), eds Kalashnikov, V. V., Penkov, B. and Zolotarev, V. M., Springer, Berlin, pp. 79–85.Google Scholar
Pfeifer, D. (1982). Characterizations of exponential distributions by independent nonstationary record increments. J. Appl. Prob. 19, 127–135. (Correction: 19 (1982), 906.)Google Scholar
Szász, O. (1916). Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen. Math. Ann. 77, 482–496.Google Scholar
Taylor, A. (1965). General Theory of Functions and Integration. Blaisdell, New York.Google Scholar
Titchmarsh, E. C. (1926). The zeros of certain integral functions. Proc. London Math. Soc. 2 25, 283–302.Google Scholar