Khmaladze transformation

Khmaladze transformation

Consider empirical distribution function F_n based on asequence of i.i.d random variables, X_1,ldots, X_n.Suppose F is a hypothetical distribution function ofeach X_i. To test whether the choice of Fis correct or not, statisticians use the normalized difference,

: v_n(x)=sqrt{n} [F_n(x)-F(x)] .

This v_n, as a random process in x, is called empirical process. Various functional from v_n are used as test statistics. The change of the time 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 [0,1] if X_is indeed have distribution function F.

This fact, discovered and first utilized by Kolmogorov(1933), Wald and Wolfowitz(1936), Smirnov(1937), especially after Doob(1949) and Anderson and Darling(1952) led to the standard rule to choose test statistics from v_n: these are statistics psi(v_n,F) which possibly depends F, in such a way that there exists another statistic varphi(u_n) from the uniform empirical process, such that psi(v_n,F)=varphi(u_n). Examples are

: sup_x|v_n(x)|=sup_t|u_n(t)|,quad sup_xfrac{a(F(x))}=sup_tfrac {a(t)}


: int_{-infty}^{infty} v_n^2(x)d F(x)=int_{0}^{1} u_n^2(t),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 simple hypothesis, when a fixed F as a hypothesis is given. Much more often, one needs to verify parametric hypothesis when hypothetical F=F_{ heta_n}, depends on some parameter heta_n, which the hypothesis does not specify and which has to be estimated from the sample X_1,ldots,X_n itself.

Although the estimators hat heta_n, most commonly converge to true value of heta, it was discovered, Kac, Kiefer and Wolfowitz(1955) and Gikhman(1954), that the parametric, or estimated, empirical process

: hat v_n(x)=sqrt{n} [F_n(x)-F_{hat heta_n}(x)]

differs significantly from v_n and the time transformed process hat u_n(t)=hat v_n(x), t=F_{hat heta_n}(x) has the distribution, and the limit distribution as n oinfty, dependent on parametric form of F_{ heta} on hat heta_n and, in general, within one parametric family, on the value of heta.

From mid-50's to the late-80's, much work was done to clarify the situation and understand the nature of the process hat v_n.

In 1981, and then 1987 and 1993, E. V. Khmaladze suggested to replace the parametric empirical process hat v_n by its martingale part w_n only.

: hat v_n(x)-K_n(x)=w_n(x)

where K_n(x) is the compensator of 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_{hat heta_n}(x), as a function of both x and heta_n, the limit distribution of the time transformed process

: omega_n(t)=w_n(x), t=F_{hat heta_n}(x)

is that of standard Brownian motion on [0,1] , i.e., isagain standard and independent of the choice ofF_{hat heta_n}.

* The relationship between hat v_n and w_n and between their limits, is one to one, so that the statistical inference based on hat v_n or on w_n are equivalent, and in w_n, nothing is lost compared to hat v_n.

* The construction of innovation martingale w_n could be carried over to the case of vector-valued X_1,ldots,X_n, giving rise to the definition of the so-called scanning martingales in mathbb R^d.

For a long time the transformation was, although known, still not used. Later, the work of researchers like R. Koenker, W. Stute, J. Bai, H. Koul, A. Koening, ... and others made it popular in econometrics and other fields of statistics.

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