Correlation of ranges of correlated deviates
| Autor: | T. E. KURTZ, R. F. LINK, J. W. TUKEY, D. L. WALLACE, null E.S.P. |
|---|---|
| Rok vydání: | 1966 |
| Předmět: |
Statistics and Probability
education.field_of_study Applied Mathematics General Mathematics Population Multivariate normal distribution Agricultural and Biological Sciences (miscellaneous) Sampling variance Fractional power Standard deviation Normal distribution Correlation Statistics Range (statistics) Statistics Probability and Uncertainty General Agricultural and Biological Sciences education Mathematics |
| Zdroj: | Biometrika. 53:191-197 |
| ISSN: | 1464-3510 0006-3444 |
| DOI: | 10.1093/biomet/53.1-2.191 |
| Popis: | In samples from independent normal distributions with a common variance, the mean of the sample ranges is a useful short-cut estimate of (a multiple of) the standard deviation. (Lord, 1947; Patnaik, 1950). If the distributions sampled are not independent, the mean range is still a useful estimate, but to assess its sampling variance requires knowledge of the correlation between ranges of the component variables in samples from a multivariate normal population. The latter situation arises when ranges are used to estimate variability in a randomized block experiment (Hartley, 1950; Kurtz, Link, Tukey & Wallace, 1965 a, b). The results of this paper were developed for such applications, as detailed in the references cited. The problem may be specialized to the following: If (Yi, Z1), ..., (Ynw Z.) are independent samples from a bivariate normal distribution with unit variances and correlation p, if v is the range of (Yl, * * *, Yn) and w the range of (z1, Zn), what is the value of pw(n, p), the correlation between v and w? The problem was first attacked by Hartley (1950). The limiting values of pw(n,p) are immediately seen to be pw(n, O) = 0, pw(n, ? 1) = 1, and to these Hartley added the values of pw(n,-0 2) for n = 2, 3, ..., 9, obtained by complex and arduous numerical quadratures. Hartley then obtained the intermediate values of pw(n,p) that he required by fractional power interpolation between the three values, O-0,-02, 1, of p. The transformation yi -? yi but zi zi shows that |
| Databáze: | OpenAIRE |
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