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Sur l'extension de certaines evaluations statistiques au cas de petits echantillons

Maurice Fréchet
Revue de l'Institut International de Statistique / Review of the International Statistical Institute
Vol. 11, No. 3/4 (1943), pp. 182-205
DOI: 10.2307/1401114
Stable URL: http://www.jstor.org/stable/1401114
Page Count: 24
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Sur l'extension de certaines evaluations statistiques au cas de petits echantillons
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Abstract

Writer hopes that the reader will find that some of the ideas or of the theorems of this memoir are new to him. However, as its contents were first intended as a chapter of a course on mathematical statistics, this paper must be judged from the didactic or pedagogical, as well as from the scientific point of view. It is devoted to a critical study and to an extension of some of the methods of estimation of parameters of a priori adopted probability laws, when a small sample has been drawn. More precisely: Let X1,......Xn be a system of n random variables, and assume the elementary probability f (x. θ) dx of X to be know, leaving only undetermined the value of a parameter θ. The problem is to estimate θ by means of n observed values x1,....xn of X1,....,Xn. The estimation taken as an approximate value of θ shall be one function t=H(x1,....,xn) of the known values x1,....,xn. Different methods have been proposed for the purpose of fixing which function H shall be chosen. First, writer deals with what is sometimes called the best unbiassed estimate of the parameter. H will be called an "unbiassed" estimate of θ, if the average value of H(x1..xn) is equal to θ. It is called the "best" unbiassed estimate, if the variance of H(x1,. ... xn) - i.e. the average of [H(x1,....,xn)-Θ]2 - is minimum. Writer makes clear that these definitions - though natural enough - are conventional and arbitrary ones. Most of the previous writers had applied this definition in the hypothesis of n being great and X satisfying the laplacian (so called normal) probability law. n arbitrary: Writer showed in a course of lectures in 1939, how to extend their formula to the general case by using Schwarz' inequality. He first proved that $\sqrt{n}\sigma _{H}\geq {\textstyle\frac{1}{\sigma _{A}}}$, where σ 2v is written for the variance of v and A=1/f(x,Θ)∂/∂ Θf(x,Θ). Later on, he happened to hear that Doob had previously obtained the same result by the same method. n great: When, instead of supposing the average value of H to be θ, it is only assumed that this is so at the limit when n → ∞. writer shows that the lowest limit of $\sqrt{n}\sigma _{H}\ \text{is}\geq {\textstyle\frac{1}{\sigma _{A}}}$. Then, reverting to the first case, the question is to find, whether σ H may be equal to its lower bound ${\textstyle\frac{1}{\sigma _{A}\sqrt{n}}}$. Writer points out that, generally, this does not happen. For example, when f(x, θ) = g(x - θ), the only case in which a function H may be found such that $\sigma _{H}={\textstyle\frac{1}{\sigma _{A}\sqrt{n}}}$ is that one when X is a laplacian (so called normal) variable. Writer shows that a necessary and sufficient condition that one of the σ H be equal to ${\textstyle\frac{1}{\sigma _{A}\sqrt{n}}}$ is that functions h(x),l(x),μ(θ) exist, such that f(x)=eμ (Θ)[h(x)-Θ]+μ (Θ)+l(x). (We will then say that f(x) is a "distinguished function"). Then the minimizing function H is H(x1,....,xn)=1/n[h(x1)+....+h(xn)]: so that the estimate of θ is t=1/nΣ ih(xi) In addition: (σ H)2=1/n μ ′′Θ. When f is known, but h is not, t is the unique root of Σ i∂/∂ tL f(x i,t)=0. Writing Tn=H(x1,....,xn) and $\xi _{n}={\textstyle\frac{T_{n}-\Theta}{\sigma _{A}}}\sqrt{n},\xi _{n}$ tends to be a laplacian variable. Then writer raises the question: which change of parameter t = a (τ) - whence f(x, a(τ)) = φ(x, τ) - is such that φ(x, τ) is a distinguished function. And he finds that in this case, functions ϱ (θ), χ(θ), k(x), m(x) must exist, such that f(x,Θ)=eϱ (Θ )k(x)+λ(Θ)+m(x) (x, θ, being in such a case a "quasi" distinguished function. It happens that G. Darmois previously came to the same class of functions by solving a different problem, mentioned in the present memoir p. 195. In the second part, a comparison is made of the previous method with the method of the most probable value and with the method of maximum likelihood. The first one is based on the Bayes-Laplace formula. It is pointed out that the main objection to the use of this formula (that generally we do not know the a priori probability of θ) is balanced by the advantage of our knowing exactly the hypotheses on which the solution is based and our therefore also knowing the weakness of the solution, a weakness which is just as real, but not so apparent in the other methods. It is then shown that the best unbiassed estimate and the most probable value (when all values of θ are assumed to be equally probable) coïncide when f(x, θ) is a distinguished function (and even when it is a quasi-distinguished function, provided ϱ Θ does not change sign). As for the method of maximum likelihood, it is pointed out that it is founded on a plausible, but arbitrary principle which can only be justified by the successes it will achieve in those cases, where verification is possible.

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