# Introduction to Non-parametric Theories

Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 202)

## Abstract

In the previous chapters we have frequently made use of the assumption that the set Γ of parameters is a (open or closed) subset of R n, n⩽1. In addition, in connection with essential results, we have imposed continuity and differentiability requirements on the likelihood function. For some time, so-called non-parametric methods have received considerable attention. Their beginnings, however, lie far in the past. Recent progress is due to Anglo-american, Dutch and Soviet statisticians. The term “non-parametric” is rather unfortunately chosen and it is not easy to give a satisfactory definition of this notion. Roughly speaking, one can call a test or method of estimation nonparametric when no assumptions such as those above are made o n Γ 2 .

## Keywords

Null Hypothesis Probability Measure Order Statistic Stochastic Approximation Empirical Distribution Function
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## References

1. 1.
See the report of D. van Dantzig and J. Hemelrijk, Bull. Inst. Internat. Statist. 34,239–267 (1954).
2. 2.
See E. Ruist, Ark. Mat. 3, 133–163 (1955). A detailed analysis of the notion “nonparametric”is in M. G. Kendall and R. M. Sundrum, Rev. Inst. Internat. Statist. 21,124–134 (1953). Also see III, p. 199.
3. 3.
See III, p. 235. Lemma 1.1 can be viewed as an illustration of the arguments carried through there.Google Scholar
4. 4.
For a =-∞ or b = ∞, the definition of F is to be trivially modified.Google Scholar
5. 5.
More precisely, (1.14) holds, without continuity assumptions on f, only up to a null-set.Google Scholar
6. 6.
We refer especially in this connection to W. Hoeffding, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability 1951, Vol. I, pp. 83–92, University of California Press, Berkeley and Los Angeles.Google Scholar
7. 7.
D.A.S. Fraser, Canad. J. Math. 6, 42–45 (1954). Also see P.R. Halmos, Ann. Math. Statist. 17, 34–43 (1946) An interesting general result on sufficient transformations connected with the theory of invariance is in T. S. Pitcher, Trans. Amer. Math. Soc. 85, 166–173 (1957).
8. 8.
This is obviously the case for P∈Bm.Google Scholar
9. 9.
One also writes n(β, δ) = (x(β)) for this.Google Scholar
10. 10.
For the literature see foremost the detailed report of S. S. Wilks, Bull. Amer. Math. Soc. 54, 6–50 (1948).
11. 11.
See A. Renyi, Acta Math. Acad. Sci. Hungar. 4, 191–231 (1953) and G. Hajos and A. Renyi, Acta Math. Acad. Sci. Hungar. 5, 1–6 (1954).
12. 12.
Naturally, the density vanishes elsewhere.Google Scholar
13. 13.
The literature on limit theorems for order statistics is quite large. We mention: N.V. Smirnov, Trudy Mat. Inst. Steklov 1949.Google Scholar
14. 14.
See L. Le Cam, Proceedings of the 3 r d Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. I, pp. 129–156, University of California Press, Berkeley and Los Angeles.Google Scholar
15. 15.
The first proof is due to A. N. Kolmogorov, Giorn. 1st. Ital. Attuari 4,83–91 (1933). Other proofs are in W. Feller, Ann. Math. Statist. 19, 177–189 (1948) and J.L. Doob, Ann. Math. Statist. 20, 393–403 (1949), augmented by M. D. Donsker, Ann. Math. Statist. 22,277–281 (1952).
16. 16.
Z. W. Birnbaum and F.H. Tingey, Ann. Math. Statist. 22, 592–596 (1951). E. Hlawka has applied this result in an investigation of order statistics from a quite different viewpoint: see E. Hlawka, Math. Ann. 150, 259–267 (1963).
17. 17.
For the meaning of un, see9.Google Scholar
18. 18.
This proof is due to Gnedenko and Koroljuk: B. V. Gnedenko and V. S. Koroljuk, Dokl. Akad. Nauk SSSR, n. Ser. 80, 525–528 (1951). See also V.S. Koroljuk, Izv. Akad. Nauk SSSR, Ser. Mat. 19, 81–96 (1955).
19. 19.
We mention B. V. Gnedenko, Math. Nachr. 12, 29–63 (1954) and Darling’s instructive report: D. A. Darling, Ann. Math. Statist. 28, 823–839 (1957).
20. 20.
This theorem is due originally to Smirnov: N. V. Smirnov, Mat. Sb., n. Ser. 6, 3–26 (1939), who makes the additional assumption that $$\mathop {\lim }\limits_{m,n \to \infty } \,m/n = c \ne 0$$. The version given here was proved by I.I. Gichman, Dokl. Akad. Nauk SSSR, n. Ser. 82, 837–840 (1952).
21. 21.
H. Robbins, Proceedings of the 3r d Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, pp. 157–163, University of California Press, Berkeley and Los Angeles. See also M. V. Johns jr. Ann. Math. Statist. 28, 649–669 (1957).Google Scholar
22. 22.
We also abbreviate these distribution functions by $$\mathop \Pi \limits_{i = 1}^n {\psi _i} \circ F.$$ Google Scholar
23. 23.
We naturally assume that the set of null hypotheses and the set of alternatives are disjoint.Google Scholar
24. 24.
F. Wilcoxon, Biometrics 1, 80–83 (1945). H. B. Mann and D. R. Whitney, Ann. Math. Statist. 18,50–60 (1947).
25. 25.
B.L. van der Waerden, Math. Ann. 126, 93–107 (1953).Google Scholar
26. 26.
M. E. Terry, Ann. Math. Statist. 23, 346–366 (1952).
27. 27.
Obviously, this and the following expectations are taken w.r.t. (F, G).Google Scholar
28. 28.
E.L. Lehmann, Ann. Math. Statist. 22, 165–179 (1951).
29. 29.
A. Wald and J. Wolfowitz, Ann. Math. Statist. 11, 147–162 (1940).
30. 30.
See W. Feller, 61 ff. I.e. I1.Google Scholar
31. 31.
At most one separation bar can be placed in each space.Google Scholar
32. 32.
For the sign test see W. J. Dixon and A. M. Mood, J. Amer. Statist. Assoc. 41, 557–566 (1946) and E. Ruist, l.c.2.
33. 33.
J. Hemelrijk, Indag. Math. 12, 340–350 (1950). Also see E. Ruist, l.c.2. For another general test of symmetry see C. van Eeden and A. Benard, Indag. Math. 19, 381–408 (1957).Google Scholar
34. 34.
That is: k2(x) isthe largest integer contained in $$\frac{{p(x) + n(x) + 1}}{2}.$$ Google Scholar
35. 35.
H. Hornich, Mh. Math. Phys. 50,142–150 (1941). Z. W. Birnbaum and H. S. Zuckerman, Ann. Math. Statist. 15, 328–329 (1944) have couched his formulation (within the framework of risk theory) in the language of probability theory.
36. 36.
See H. Uzawa, Ann. Math. Statist. 31, 685–702 (1960). Uzawa’s results include those of E.L. Lehmann, Ann. Math. Statist. 24, 23–43 (1953).
37. 37.
For Fy∈Cm, Fy -1 is simply the inverse map.Google Scholar
38. 38.
The uniform distribution over (0,1) thus also belongs to 9).Google Scholar
39. 39.
These statements follow at once from Fejer–s theorem on the behavior of the arithmetic mean of a Fourier series. See A. Zygmund, Trigonometric series. 2nd ed., Vol. I, Cambridge University Press, New York 1959, 89.Google Scholar
40. 40.
This inequality, valid for arbitrary trigonometric polynomials, is due to S. N. Bernstein. See N.K. Bary, A treatise on trigonometric series. Vol. I, Pergamon Press Book. The Macmillan Co., New York 1964, 35.Google Scholar
41. 41.
See, for example, B. E. Netto, Lehrbuch der Combinatorik, Teubner, Leipzig 1901, 252.Google Scholar
42. 42.
W. Hoeffding, Ann. Math. Statist. 19, 293–325 (1948).
43. 43.
For generalizations see E.L. Lehmann, l.c.28. A systematic treatment of the asymptotic theory of rank-tests is given in Hajek and Sidak l.c.1.Google Scholar
44. 44.
We believe this theorem was first proved by H. R. van der Vaart, Nederl. Akad. Wetensch., Proc. Ser. A, 53, 494–520 (1956).Google Scholar
45. 45.
We mention J.L. Hodges and E.L. Lehmann, l.c.III90, H. Chernoff and I.R. Savage, Ann. Math. Statist. 29, 972–994 (1958), M. Dwass, Ann. Math. Statist. 27, 352–374 (1956). See also F. C. Andrews, Ann. Math. Statist. 25, 724–736 (1954).
46. 46.
D. van Dantzig, Indag. Math. 13, 1–8 (1951).Google Scholar
47. 47.
The basic reference is H. Robbins and S. Monro, Ann. Math. Statist. 22,400–407 (1951).
48. 48.
The theorem and its proof in this form are due to J. Wolfowitz, Ann. Math. Statist. 23, 457–461 (1952).
49. 49.
This naturally means that $$\sum\limits_{n = 1}^\infty {a_n^2}$$ converges.Google Scholar
50. 50.
See L. Schmetterer, Osterreich. Ing.-Arch. 7,111–117 (1953); K. L. Chung, Ann. Math. Statist. 25, 463–483 (1954); A. Dvoretzky, Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability1954-1955, Vol. I, pp. 39–55, University of California Press, Berkeley and Los Angeles (1956).Google Scholar