Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-06T05:24:38.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Normality of a Class of Nonparametric Statistics

Published online by Cambridge University Press:  11 February 2009

Munsup Seoh  and
Madan L. Puri
Munsup Seoh
Affiliation:
Department of Mathematics and Statistics, Wright State University
Madan L. Puri
Affiliation:
Department of Mathematics, Indiana University
Get access Rights & Permissions [Opens in a new window]

Abstract

Asymptotic normality is established for a class of statistics which includes as special cases weighted sum of independent and identically distributed (i.i.d.) random variables, unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear function of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

Type
Research Article
Information
Econometric Theory , Volume 3 , Issue 3 , June 1987 , pp. 313 - 347
Copyright
Copyright © Cambridge University Press 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Albers, W.One sample rank tests under autoregressive dependence. Annals of Statistics 6 (1978): 836–845.CrossRefGoogle Scholar
2.Bergström, H. & Puri, M.L.. Convergence and remainder terms in linear rank statistic. Annals of Statistics 5 (1977): 671–680.CrossRefGoogle Scholar
3.Bhattacharya, P.K.Convergence of sample paths of normalized sums of induced order statistics. Annals of Statistics 2 (1974): 1034–1039.CrossRefGoogle Scholar
4.Bhattacharya, P.K.An invariance principle in regression analysis. Annals of Statistics 4 (1976) 621–624.CrossRefGoogle Scholar
5.Boos, D.D.A differential for L-statistics. Annals of Statistics 7 (1979): 955–959.CrossRefGoogle Scholar
6.Chernoff, H., Gastwirth, J.L. & Johns, M.V.. Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. Annals of Mathematical Statistics 38 (1967): 52–72.CrossRefGoogle Scholar
7.Chung, K.L.A course in probability theory. New York: Academic Press, 1974.Google Scholar
8.David, H.A.Order statistics. New York: John Wiley & Sons, 1970.Google Scholar
9.Denker, M. & Rösler, U.. A note on the asymptotic normality of rank statistics. In Révész, P. (ed), Probability and statistics. Colloquia Mathematica Societatis Janos Bolyai, 36 (1982): 405–425.Google Scholar
10.Elbadawi, I., Gallant, A.R., & Souza, G.. An elasticity can be estimated consistently with out a priori knowledge of functional form. Econometrica 51 (1983): 1731–1751.CrossRefGoogle Scholar
11.Govindarajulu, Z.Asymptotic normality of linear combinations of functions of order statistics, II. Proceedings of the National Academy of Sciences, U.S.A. 59 (1968): 713–719.CrossRefGoogle ScholarPubMed
12.Hájek, J.Some extensions of the Wald-Wolfowitz-Noether theorem. Annals of Mathe matical Statistics 32 (1961): 506–523.CrossRefGoogle Scholar
13.Hájek, J.Asymptotically most powerful rank order tests. Annals of Mathematical Statistics 33 (1962): 1129–1147.CrossRefGoogle Scholar
14.Hájek, J.Asymptotic normality of simple linear rank statistics under alternatives. Annals of Mathematical Statistics 39 (1968): 325–346.CrossRefGoogle Scholar
15.Hájek, J. & Šidàk, Z.. Theory of rank statistics. New York: Academic Press, 1967.Google Scholar
16.Hallin, M., Ingenbleek, J.-F., & Puri, M.L.. Linear serial rank tests for randomness against ARMA alternatives. Annals of Statistics 13 (1985a): 1156–1181.CrossRefGoogle Scholar
17.Hallin, M., Ingenbleek, J.-F., & Puri, M.L.. Tests de rangs linéar pour une hypothèse de bruit blanc. C.R. Acad. Sc. Paris 301, Série 1 (1985b): 49–52.Google Scholar
18.Hallin, M., Ingenbleek, J.-F., & Puri, M.L.. Tests de rangs quadratiques pour une hypothèse de bruit blanc. C.R. Acad. Sc. Paris 301, Série 1 (1985): 935–938.Google Scholar
19.Hallin, M., Ingenbleek, J.-F., & Puri, M.L.. Tests de rangs localement optimaux pour une hypothèse de bruit blanc multivariè. C.R. Acad. Sc. Paris 303, Série 1 (1986): 901–904.Google Scholar
20.Hallin, M., Ingenbleek, J.-F., & Puri, M.L.. Linear and quadratic serial rank tests for ran domness against serial dependence. Journal of Time Series Analysis 8 (1987): to appear.CrossRefGoogle Scholar
21.Hoeffding, W.On the centering of a simple linear rank statistic. Annals of Statistics 1 (1973): 54–66.CrossRefGoogle Scholar
22.Huškov´, M.Asymptotic distribution of simple linear rank statistics for testing symmetry. Zeitschrift für Wahrscheinlichkeitstheorie 14 (1970): 308–322.CrossRefGoogle Scholar
23.Lehmann, E. L.Some concepts of dependence. Annals of Mathematical Statistics 37 (1966): 1137–1153.CrossRefGoogle Scholar
24.Manski, C.F.Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3 (1975): 205–228.CrossRefGoogle Scholar
25.Mason, D.M.Asymptotic normality of linear combinations of order statistics with a smooth score function. Annals of Statistics 9 (1981): 899–908.CrossRefGoogle Scholar
26.Moore, D.S.An elementary proof of asymptotic normality of linear functions of order statistics. Annals of Mathematical Statistics 19 (1968): 263–265.CrossRefGoogle Scholar
27.Pakes, A. & Pollard, D.. The asymptotics of simulation estimators. Unpublished manuscript (1986).Google Scholar
28.Puri, M.L. & Ralescu, S.S.. Centering of signed rank statistic with continuous score-generating function. Teoria Veroyatnostei i ee Primeneniya 29 (1984): 580–584.Google Scholar
29.Pyke, R. & Shorack, G.R.. Weak convergence of a two-sample empirical process and a new approach to Chernoff-Savage theorems. Annals of Mathematical Statistics 39 (1968a): 755–771.CrossRefGoogle Scholar
30.Pyke, R. & Shorack, G.R.. Weak convergence and a Chernoff-Savage theorem for random sample sizes. Annals of Mathematical Statistics 39 (1968b): 1675–1685.CrossRefGoogle Scholar
31.Robinson, P.M.Nonparametric methods in specification. Supplement to the Economic Journal 96: (1986) 134–141.CrossRefGoogle Scholar
32.Seoh, M., Ralescu, S.S., & Puri, M.L.. Cramér type large deviations for generalized rank statistics. Annals of Probability 13 (1985): 115–125.CrossRefGoogle Scholar
33.Shorack, G.R.Asymptotic normality of linear combinations of functions of order statis tics. Annals of Mathematical Statistics 40 (1969): 2041–2050.CrossRefGoogle Scholar
34.Shorack, G.R.Functions of order statistics. Annals of Mathematical Statistics 43 (1972): 412–427.CrossRefGoogle Scholar
35.Singh, R.S. & Ullah, A.. Nonparametric time-series estimation of joint DGP, conditional DGP, and vector autoregression. Econometric Theory 1 (1985): 27–52.CrossRefGoogle Scholar
36.Stigler, S.M.Linear functions of order statistics. Annals of Mathematical Statistics 40 (1969): 770–788.CrossRefGoogle Scholar
37.Stigler, S.M.The asymptotic distribution of the trimmed mean. Annals of Statistics 1 (1973): 472–477.CrossRefGoogle Scholar
38.Stigler, S.M.Linear functions of order statistics with smooth weight functions. Annals of Statistics 2 (1974): 676–693.CrossRefGoogle Scholar
39.Stigler, S.M.Correction to linear functions of order statistics with smooth weight functions. Annals of Statistics 7 (1979): 466.CrossRefGoogle Scholar
40.Yang, S.S.Linear functions of concomitants of order statistics with application to nonpara metric estimation of a regression function. Journal of the American Statistical Association 76 (1981): 658–662.CrossRefGoogle Scholar