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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Khmaladze transformation ： ウィキペディア英語版 Khmaladze transformation In statistics, the Khmaladze transformation is a mathematical tool used in constructing convenient goodness of fit tests for hypothetical distribution functions. More precisely, suppose $X\_1,\backslash ldots,\; X\_n$ are i.i.d., possibly multi-dimensional, random observations generated from an unknown probability distribution. A classical problem in statistics is to decide how well a given hypothetical distribution function $F$, or a given hypothetical parametric family of distribution functions $\backslash$, fits the set of observations. The Khmaladze transformation allows us to construct goodness of fit tests with desirable properties. It is named after Estate V. Khmaladze. Consider the sequence of empirical distribution functions $F\_n$ based on a sequence of i.i.d random variables, $X\_1,\backslash ldots,\; X\_n$, as ''n'' increases. Suppose $F$ is the hypothetical distribution function of each $X\_i$. To test whether the choice of $F$ is correct or not, statisticians use the normalized difference, : $v\_n(x)=\backslash sqrt\; ().$ This $v\_n$, as a random process in $x$, is called the empirical process. Various functionals of $v\_n$ are used as test statistics. The change of the variable $v\_n(x)=u\_n(t)$, $t=F(x)$ transforms to the so-called uniform empirical process $u\_n$. The latter is an empirical processes based on independent random variables $U\_i=F(X\_i)$, which are uniformly distributed on $()$ if the $X\_i$s do indeed have distribution function $F$. This fact was discovered and first utilized by Kolmogorov (1933), Wald and Wolfowitz (1936) and Smirnov (1937) and, especially after Doob (1949) and Anderson and Darling (1952), it led to the standard rule to choose test statistics based on $v\_n$. That is, test statistics $\backslash psi(v\_n,F)$ are defined (which possibly depend on the $F$ being tested) in such a way that there exists another statistic $\backslash varphi(u\_n)$ derived from the uniform empirical process, such that $\backslash psi(v\_n,F)=\backslash varphi(u\_n)$. Examples are : $\backslash sup\_x|v\_n(x)|=\backslash sup\_t|u\_n(t)|,\backslash quad\; \backslash sup\_x\backslash frac=\backslash sup\_t\backslash frac$ and : $\backslash int\_^\backslash infty\; v\_n^2(x)\; \backslash ,\; dF(x)=\backslash int\_0^1\; u\_n^2(t)\backslash ,dt.$ For all such functionals, their null distribution (under the hypothetical $F$) does not depend on $F$, and can be calculated once and then used to test any $F$. However, it is only rarely that one needs to test a simple hypothesis, when a fixed $F$ as a hypothesis is given. Much more often, one needs to verify parametric hypotheses where the hypothetical $F=F\_$, depends on some parameters $\backslash theta\_n$, which the hypothesis does not specify and which have to be estimated from the sample $X\_1,\backslash ldots,X\_n$ itself. Although the estimators $\backslash hat\; \backslash theta\_n$, most commonly converge to true value of $\backslash theta$, it was discovered that the parametric,〔Gikhman (1954)〕 or estimated, empirical process : $\backslash hat\; v\_n(x)=\backslash sqrt\; ()$ differs significantly from $v\_n$ and that the transformed process $\backslash hat\; u\_n(t)=\backslash hat\; v\_n(x)$, $t=F\_(x)$ has a distribution for which the limit distribution, as $n\backslash to\backslash infty$, is dependent on the parametric form of $F\_$ and on the particular estimator $\backslash hat\backslash theta\_n$ and, in general, within one parametric family, on the value of $\backslash theta$. From mid-1950s to the late-1980s, much work was done to clarify the situation and understand the nature of the process $\backslash hat\; v\_n$. In 1981, and then 1987 and 1993, Khmaladze suggested to replace the parametric empirical process $\backslash hat\; v\_n$ by its martingale part $w\_n$ only. : $\backslash hat\; v\_n(x)-K\_n(x)=w\_n(x)$ where $K\_n(x)$ is the compensator of $\backslash hat\; v\_n(x)$. Then the following properties of $w\_n$ were established: * Although the form of $K\_n$, and therefore, of $w\_n$, depends on $F\_(x)$, as a function of both $x$ and $\backslash theta\_n$, the limit distribution of the time transformed process :: $\backslash omega\_n(t)=w\_n(x),\; t=F\_(x)$ : is that of standard Brownian motion on $()$, i.e., is again standard and independent of the choice of $F\_$. * The relationship between $\backslash hat\; v\_n$ and $w\_n$ and between their limits, is one to one, so that the statistical inference based on $\backslash hat\; v\_n$ or on $w\_n$ are equivalent, and in $w\_n$, nothing is lost compared to $\backslash hat\; v\_n$. * The construction of innovation martingale $w\_n$ could be carried over to the case of vector-valued $X\_1,\backslash ldots,X\_n$, giving rise to the definition of the so-called scanning martingales in $\backslash mathbb\; R^d$. For a long time the transformation was, although known, still not used. Later, the work of researchers like Koenker, Stute, Bai, Koul, Koening, and others made it popular in econometrics and other fields of statistics. ==See also== *Empirical process 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Khmaladze transformation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.