Abstract
As already mentioned, the procedures discussed in Chapter II for the testing of an hypothesis possess without doubt a certain intuitiveness and are rather convincing. However, we have already pointed out that it is desirable to develop a general test theory which depends on few basic assumptions. In particular, we have given no clear definition of the notion of a “test”. We also want to develop criteria for deciding when one of two tests can be viewed as the “better” one.
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References
Neyman, J. and E. S. Pearson, Biometrika 20 A, 175–240 and 263–294 (1928). Philos. Trans. Roy. Soc. London, Ser. A, 231, 289-337 (1933).
A standard reference for the theory of tests and confidence regions, which treats numerous details, is the book by Lehmann I.e. Not.11: Testing Statistical Hypotheses, J. Wiley, New York 1959.
We will also write E(φ;γ) for E(φ;Py).
This means that one exploits the given level of significance “as far as possible”.
We will sometimes write P(M,γ) for P γ(M).
This terminology is due to the idea that the random experiment which delivers the sample (x1,...,xn) is the result of n trials each of which has the same probability distribution with parameter a. The correct value of a is unknown and the null hypothesis, which is to be tested, assumes that a=a0. See also II, p. 127.
The case α = 0 is trivial and need not be considered.
Strictly speaking, the assumptions of Theorem 12.2 of I are not fulfilled everywhere since the Jacobian vanishes for r=0, ϑ=0 and ϑ = π. However, it is easy to see that the exceptional sets have measure 0.
For the evaluation of this integral see for example N. Hofreiter and W. Grobner, Integraltafel, Zweiter Teil: Bestimmte Integrale, 2. Aufl. Springer-Verlag, Wien 1961.
This can be also be written as \( \mathop \Sigma \limits_{j = 0}^\infty \frac{1}{{j!}}{\left( {\frac{{|a{|^2}}}{2}} \right)^j}{e^{ - |a{|^2}/2}}\frac{{{z^{(2j + n - 2)/2}}}}{{{2^{(2j + n)/2}}\Gamma \left( {j + \frac{n}{2}} \right)}}{e^{ - z/2}}. \) But for \( z \ge 0,\frac{{{2^{(2j + n - 2)/2}}}}{{{2^{(2j + n)/2}}\Gamma \left( {j + \frac{n}{2}} \right)}}{e^{ - z/2}} \) is the density of a χ2-distribution with 2j+n degrees of freedom.
The first version of this fundamental theorem is in E. S. Pearson, Statist. Res. Mem. Univ. London 1,1–37 (1936).
If k =-∞ define v(M k) = v(Mk+).
k = ∞ requires (also for IIIF) a trivial special argument.
See Dantzig and A. Wald loc. cit. 11.
See L. Schmetterer, Sankhya 25, 207–210 (1963). A much deeper result has been given by W. Sendler, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18, 183-196 (1971).
This means that-g is convex.
Without explicitly giving non-trivial tests, one can also argue as follows: From \( \mathop {\lim }\limits_{\alpha \to 0 + 0} \,g(\alpha )/\alpha = 1 \) we have from Theorem 3.4, g(α)=α,0⩽ α ⩽ 1. Hence, from the definition of \( \int\limits_R {\phi {f_1}d\mu \le \int\limits_R {\phi {f_0}d\mu } } \) for each test φ∈Φα and O ⩽ α ⩽ 1. For the test φ = cE we thus have μ(E)=0 which contradicts the assumption.
Practically speaking, p⩽p0, resp., p⩾p1 is a more reasonable requirement but we want to consider only simple hypotheses here.
The first systematic investigation of the connection between linear programs and test theory is in E. W. Barankin, Univ. California Publ. Statist. 1, 161–214 (1949–1953).
See for instance S. Vajda, Theory of Games and Linear Programming, John Wiley, New York 1956.
We follow essentially a paper by O. Krafft und H. Witting, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7, 289–302 (1967).
To avoid trivial complications we now assume 0 < α < l.
Necessary and sufficient conditions for the existence of product measurable densities can be found in J. Pfanzagl, Sankhya, Ser. A. 31, 13–18 (1969).
Φα is defined on p. 178.
See p. 185ff.
Essentially, the following considerations represent only an illustration of the uniqueness claim of Theorem 3.1.
P. R. Halmos and L. J. Savage, Ann. Math. Statist. 20, 225–241 (1949).
To justify this conclusion also in the case α = 0 and c = l, one must define 0·∞ = 0.
From this it naturally does not necessarily follow that the set of corresponding probability measures Pr is convex.
J. Pfanzagl, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1, 109–115 (1963).
4 ⫅ B v-a.e. means that the set of elements of A which do not belong to B form a v-null set. A = B v-a.e. means v(A-B) + v(B-A) = 0.
Here, and occasionally later, we will supress the fact that certain relations hold only &λ-a.e.
See J. Pfanzagl, Sankhya Ser. A 30, 147–156 (1968).
In practical work, however, alternatives which are “too close” to the null hypothesis are likewise uninteresting.
H. Scheffe, Ann. Math. Statist. 18, 434–438 (1947).
This terminology is not restricted to test problems. It will be used analogously for confidence regions (see IV) and theory of estimation (see also V). See also VII. I 2.
The choice of endpoint for this interval is not important, oo can be replaced by an arbitrary real number <γ0.
This μ-null set may depend on γ.
We define here 0· ∞ =0 and-∞ · 0 =-·.
One can also allow l1 = ± ·, l2 = ± ·.
J. Neyman and E. S. Pearson, Statist. Res. Mem. Univ. London 2, 25–57 (1938).
See St. L. Isaacson, Ann. Math. Statist. 22, 217–234 (1951).
See p. 60.
For a more precise terminology see 122.
More precisely, this means that for each A∈S0 there is a B∈S0 (1) such that Py((A-B)⋃(B-A))=0 for all γ∈Γ and likewise when the roles of S0 and S0 (1) are interchanged.
In the statistical literature the existence of a sufficient transformation for a set of probability measures over \( ({R_n}{_n}) \) is often proved.
Thus for all real a the inverse image of (-∞, α) under fy belongs to S0 up to a γ-null set.
\( {E_\lambda }({f_\lambda }|{S_0}) \) denotes the conditional expectation w.r.t. the measure γ.
We have shown this only for the case T(R) = Q. However, see I23.
See J. Neyman, Giorn. Ital. Attuari 6, 320–334 (1953) as well as P.R. Halmos and L.J. Savage, l.e.27.
Actually, I, Theorem 18.3 yields this only for n = 1, but’1, Theorem 18.3 can easily be extended to the case where the function f named there has range Rn with n >1.
The assumption that the densities are >0 in all of R1 is made only for convenience. It is enough, for example, that the fy be >0 for all γ∈Γ in a fixed open interval and vanish for all γ outside of this fixed interval.
Essentially due to E. B. Dynkin, Uspehi Mat. Nauk 6,68–90 (1951). See also B. O. Koopman, Trans. Amer. Math. Soc. 39, 399–409 (1936).
E. W. Barankin and M. Katz, Sankhya 21, 217–246 (1959) and E.W. Barankin and A. P. Maitra, Sankhya 25, 217–244(1963).
J. L. Denny, Proc. Nat. Acad. Sci. U.S.A. 57, 1184–1187 (1967). Ann. Math. Statist. 41, 401–411 (1970). See also O. Barndorff-Nielsen and Karl Pedersen, Math. Scand. 22,197–202 (1968).
D. L. Burkholder, Ann. Math. Statist. 32, 1191–1200 (1961) and Ann. Math. Statistics 33, 596–599 (1962). See also T. S. Pitcher, loc. cit. VII7.
The importance of this definition in mathematical statistics was clearly presented for the first time in E.L. Lehmann and H. Scheffe, Sankhya 10, 305–340 (1950).
See H. Steinhaus—L. Kaczmarz: Theorie der Orthogonalreihen, Monografje Matematyczne VI, Warschau 1935.
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An application of Holder’s inequality shows that these expectations always exist.
Introduced by J. Neyman and E.S. Pearson, Philos. Trans. Roy. Soc. London l.c.1.
The first known example of this is in W. Feller, Statist. Res. Mem. Univ. London 2, 117–125 (1938). See also H. Kellerer, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1, 240–246 (1963).
E.L. Lehmann and H. Scheffe, l.c.57.
For an analysis see Lehmann, l.c.2, 134ff. and especially G. Noelle, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11, 208–229 (1969).
G. B. Dantzig, Ann. Math. Statist. 11, 186–192 (1940) proved that there exists no (non-trivial) test for the mean of a normal distribution with given sample size whose power function is independent of a. fürther examples for similar tests are also found in VI.
W.U. Behrens, Landwirtschaftliche Jahrbucher 48, 807–837 (1929).
Ju. V. Linnik, Statistical problems with nuisance parameters, Translations of Mathematical Monographs Vol. 20, Amer, Math. Soc, Providence, R.L, 1968.
Due to J. Neyman and E. S. Pearson, Biometrika, l.c.1.
See 5.
See V, Lemma 3.2.
Such a discussion is given by P. Hoel, Ann. Math. Statist. 16, 362–368 (1945).
For details see H. Scheffe, The Analysis of Variance, John Wiley & Sons-Chapman & Hall, New York-London 1959.
Frequent use is made of such decompositions in the analysis of variance. For the underlying algebraic relations, see H. B. Mann, Ann. Math. Statist. 31, 1–15 (1960).
For a detailed analysis of this model see A.N. Kolmogorov, Proc. Second All-Union Congress Math. Statistics, Sept. 27–Oct. 2, 1948, Acad. Sci. Uzbekistan Soviet Socialist. Republic, Tashkent 1949, 240–268.
K. Pearson, Philos. Mag. 50, Ser. 5, 157–175 (1900).
For the grouping problem see H. B. Mann and A. Wald, Ann. Math. Statist. 13, 306–317 (1942) and H. Witting, Arch. Math. 10, 468-479 (1959).
For this and fürther important results see H. Cramer, l.c. I58. The first formulation of such results is in R. A. Fisher, J. Roy. Statist. Soc. 85, 87–94 (1922). Also see W. G. Cochran, Ann. Math. Statist. 23, 315-345 (1952).
A. Wald, Trans. Amer. Math. Soc. 54, 462–482 (1943).
Strictly speaking, gγ is for the time being not at all defined for γ∈T; only is defined.
G is thus a homomorphic image of G.
The group is then called transitive.
The notion of an invariant test is viewed somewhat more generally in this theorem: φ is called invariant if there exists a μ-null set M such that φ(gx) = φ(x)for all g∈G and each x∈R-M.
See for example A. Weil, L’integration dans les groupes topologiques et ses applications,Actualites scientifiques et industrielles 869–1145, Hermann & Cie, 2nd ed., Paris 1953.
See e.g. E. L. Lehmann, l.c.2, 335. Also O. Wesler, Ann. Math. Statist. 30,1–20 (1959).
See for example J. Neyman and E. Scott, Econometrica 16,1–32 (1948).
E(n)(φn;γ) naturally means \( \int\limits_{{R^{(n)}}} {{\phi _n}} dP_\gamma ^{(n)}. \).
See A. Berger, Ann. Math. Statist. 22, 289–293 (1951) and Ch. Kraft, Univ. California Publ. Statist. 2, 125–141 (1953–1958).
S. Kakutani, Ann. of Math. II. Ser. 49, 214–224 (1948).
See for details A. Wald, l.c.78.
See J. L. Hodges Jr. and E. L. Lehmann, Proc. Fourth Berkeley Sympos. Math. Statist, and Prob. Vol. I, pp. 307–317, Univ. California Press, Berkeley Calif, 1961.
Note that \( g_n^{(1)} \) is somewhat differently defined as \( g_n^{(2)}. \).
If m = 1, this condition is omitted.
See J.G. Pitman, Lecture Notes on Nonparametric Inference. Columbia University, New York 1949. See also G.E. Noether, Ann. Math. Statist. 26, 64–68 (1955).
See, however, the developments on p. 243. Also J. L. Hodges Jr. and E. L. Lehmann, Ann. Math. Statist. 27, 324–335 (1956).
R.R. Bahadur, Ann. Math. Statist. 31, 276–295 (1960).
The fundamental paper is A. Wald. Ann. Math. Statist. 16, 117–186 (1945). Also see A. Wald, Sequential Analysis, John Wiley & Sons-Chapman & Hall, New York-London 1947. See also G.A. Barnard, Suppl. J. Roy. Statist. Soc. 8, 1–21 (1946).
For details see A. Wald and J. Wolfowitz, Ann. Math. Statist. 19, 326–339 (1948).
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Schmetterer, L. (1974). Introduction to the Theory of Hypothesis Testing. In: Introduction to Mathematical Statistics. Die Grundlehren der mathematischen Wissenschaften, vol 202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65542-5_5
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