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As later on I give explications of some concepts of mathematical statistics I want to call the reader's atention to a distinction essential for the understanding of the character of mathematical statistics, and to anticipate objections which might be raised against those explications. When we speak of the concepts and theorems of mathematical statistics—similarly of any mathematical theory—it is necessary to distinguish concepts and theorems of pure theory from their interpretations occurring in the applications of the theory. The abstract object called in the Euclidean geometry “triangle” is not identical with a part of a slip of paper confined by three intersecting liens drawn with a pencil and a ruler to which we apply the theorems of geometry in our practical activity. A similar or perhaps an even clearer distinction applies to mathematical statistics. Let us consider it by discussing an example. In the textbooks of mathematical statistics we may encounter the following definition of “random variable”, based on that first formulated byKolmogorov. “A real single-valued functionX(e) defined on a setE of elementary eventse is called a random variable, if the counter-imageA of any intervalI of the set of real numbers of the form (−∞,x) belongs to the Borel's field of setsZ of random events formed of subsets of the setE” (M. Fisz:Rachunek prawdopodobieństwa i statystyka matematyczna, Warswawa 1954, p. 26.). Random variable as defined above is an abstract mathematical construction. It is possible to analyze various mathematical properties of this construction without entering into the application of the concept of random variable to extramathematical reality. In the present paper we are interested, however, in practical rules of inference supplied by mathematical statistics; we shall appeal therefore not to the abstract concept of random variable but to that which occurs in the applications of mathematical statistics. I think that in accordance with the practical usage we can define random variable as a measurable propertyC of elements of a populationP assuming in this population definite values (or values out of definite sets of values) with definite probabilities. These probabilities are defined for practical purposes as frequencies of elements ofP possessing definite values of the propertyC (or the ones out of definite sets of values) or the (hypothetical) limits of these frequencies. What has been said above on the concept of random variable appliesmutatis mutandis to other concepts of mathematical statistics introduced below. Being fully aware of the preciseness of many concepts and theorems appearing in mathematical statistics as an abstract theory I shall introduce here statistical concepts and theorems in the sense they have in the applications of the abstract theory of statistics, as the aim of this paper is not to analyze the mathematical foundations of statistics but to compare the practical rules of statistical inference with those of traditional logic of induction and deduction.
By sample characteristic I mean any functions n =f(x 1,...x n ) ofn values of the investigated propertyC assumed byn examined elements belonging to the sampleS drawn from the populationP Sample characteristics are e.g. distribution parameters of the examined measurable property in the sample. Comp.M. Fisz, op. cit.Rachunek prawdopodobieństwa i statystyka matematyczna, Warszawa 1954, p. 187.
When testing a hypothesish which ascribes a definite value to an unknown distribution parameter of the examined propertyC inP we usually make some assumptions concerning the form of the distribution function of the propertyC inP and eventually assumptions concerning the values of other parameters of this distribution. It is only by the formulated hypothesis in conjunction with these assumptions that the distribution function ofC inP is uniquely determined. For the present discussion the question as to what are these assumptions based upon is inessential. It should only be noted that they are not questioned in the course of testing the hypothesish.
This definition is based upon that ofJ. Neyman (First Course in Probability and Statistics, N. York 1950, p. 258) with the only difference that Neyman's definition is not in terms of rejecting or accepting a hypothesis (or of continuing investigations) but in terms of undertaking some actions, as Neyman identifies the acceptance or the rejection of a statistical hypothesis with undertaking an action: A discussion of the relation between the acceptamce of a hypothesis and the undertaking of an action would go beyond the limits of this paper. It seems, however, that it is possible to speak of accepting (rejecting) a hypothesis without identifying either acceptance or rejection with undertaking an action, as examples may be given of such cases of testing statistical hypotheses which do not result (directly at least) in undertaking any action. It should be noted, moreover, that there are tests which do not allow to accept the tested hypothesis on the sole ground that the examined sample characteristic has assumed a value from outside the critical region, and prescribe to continue investigations. We will, however, consider exclusively such tests which prescribe either to accept or to reject the tested hypothesis and we will draw the reader's attention to the conditions these tests should satisfy.
The meaning of “follow” as used here must not be identified with “follows logically” or “is implied” in the sense of material implication. The meaning in which I use this expression here is close to the meaning of the so called “strict implication”:q follows fromp if it is impossible thatp be andq not be the case. Despite various objections raised against this concept, any analysis of inferences actually applied in science seems impossible without it. The relation of following (in the sense explained above) is denoted in this paper by the symbol “→”. Schemas in which this symbol appears are therefore not equisignificant to the respective schemas of the propositional calculus, where “→” denotes material implication; notwithstanding this difference I use for the schemas formulated by means of the symbol “→” names usually applied to the corresponding schemas of the propositional calculus.
The rule (schema) postulating the rejection ofh ifs n εK remains unaffected.
If, moreover, such a test is the uniformly most powerful one, these probabilities are the least possible, i.e. they are smaller than in case any critical regionK 1, other than the regionK indicated by the test, would be chosen.
For the sake of simplicity I assume that the number of alternative hypotheses is finite.
I want to emphasize that in the induction by elimination we a i m at constructing an inference falling under this schema; the methods of eliminative induction usually presented in the textbooks of logic are not exactly substitutions of this schema, but they might be if additional assumptions were supplied.
The difference between testing a statistical hypothesis and estimating a parameter is in the fact that while in the former case a hypothesis ascribing a definite value to the parameter is formulated before drawing the sample, in the latter case we try to guess the value of the parameter on the ground of observed sample values without first formulating anya priori hypothesis.
For the sake of simplicity it is assumed here that the number of values the parameter μ may assume is finite and thus that it is a noncontinuous random variable, if it is a random variable at all. Analogous argument may easily be presented on the assumption of continuous variation of μ and in terms of density probability functions and integrals instead of frequency functions and sums. This simplified model is, however, sufficient for the discussion of logical problems, undertaken in this paper.
The summation sign in the last quotient applies to all the valuesx the parameter μ may assume.
The name “principle of maximum likelihood” is due to the fact that the probability of obtaining a given sample (given a hypothesis concerning the distribution of the examined property in the population) is called likelihood (of the hypothesis relatively to this sample).
Comp.M. Fisz op. cit.Rachunek prawdopodobieństwa i statystyka matematyczna, Warszawa 1954, p. 274 orH. Cramer:Mathematical Methods of Statics, p. 499.
Comp.H. Cramer, op. cit.Mathematical Methods of Statics, pp. 512–13;M. Fisz, op. cit.Rachunek prawdopodobieństwa i statystyka matematyczna, Warszawa 1954, pp. 275–77.
Brief reflection suffices to notice that if μ is a parameter of a population which does not belong to a population of populations in which the parameter assumes various values, (iii) is a meaningless expression. It states namely the probability that a definite numerical value belongs to a definite set of numbers.
We assume here that rule (14′) is applied on a high level of confidence.
It should also be added that the general theory of statistical decision functions was not discussed here.
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On the relation of statistical inference to traditional induction and deduction. Stud Logica 7, 243–264 (1958). https://doi.org/10.1007/BF02548940
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DOI: https://doi.org/10.1007/BF02548940