Let \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) be independent and identically distributed random vectors. Denote by \(Y_{[i:n]}\) the Y-variate corresponding to the ith order statistic \(X_{i:n}\) of \(X_ 1,...,X_ n\). In this paper statistics of the form \[ T_ n=n^{- 1}\sum^{n}_{i=1}J(i/n)h(X_{i:n},Y_{[i:n]}) \] are considered. The function J: (0,1)\(\to {\mathbb{R}}\) is bounded and smooth, h is some real- valued function of two arguments. The quantities \(Y_{[i:n]}\) are called the concomitants of \(X_{i:n}\). One may also write \[ T_ n=T(F_ n)=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}J(F_ n(x))h(x,y)dF_ n(x,y), \] where \(F_ n\) denotes the empirical distribution function. The author establishes in this paper the asymptotic normality of \(n^{1/2}(T(F_ n)-T(F))\), using the concept of a stochastic Gateaux differential as his main tool. In certain applications random variables are assumed to be collected from a finite population, according to some sampling design. A finite population of size N is a collection of N units which are labelled from 1 to N. The population is identified with a set of labels \(U=\{1,2,...,N\}\). A sequence of populations \(U_ t=\{1,2,...,N_ t\}\) such that \(N_ t\to \infty\) as \(t\to \infty\) is considered. For fixed t with the jth unit, \(j\in U_ t\), some vector \((X_ j,Y_ j)\) is associated, which can be viewed as a result of measuring unit j. A sample is a subset of \(U_ t\). For fixed t and fixed sample size \(n_ t\) the sample may be defined as the set \(s_ t=\{j_ i,...,j_{n_ t}|\) \(j_ i\in U_ t\}\). A sampling experiment will give a sample \(s_ t\) according to a probability distribution \(P(s_ t)\), \(s_ t\subset U_ t\). The sampling plan is specified in terms of probability distributions \(P(s_ t)\), \(s_ t\subset U_ t\). Note that the size of the sample \(s_ t\) is denoted by \(n_ t.\) The author establishes in this finite population set up the asymptotic normality of \(n_ t^{1/2}(T(F^*_{n_ t})-T(F_{N_ t}))\), where \(F^*_{N_ t}\) denotes a suitable weighted empirical distribution function and \(F_{N_ t}^ a \)finite population distribution function. An application in which income inequality in a finite poulation must be estimated, using a sampling design of a certain type, is also briefly discussed.