Nonparametric inference on the difference of location parameters of correlated variables from fragmentary samples (Q1101160)

Let X, Y be two dependent random variables with marginal cdf's \(F_ 1(x)\), \(F_ 2(y)\), respectively, such that \(F_ 2(x)=F_ 1(x-\theta)\). \(\theta\) is to be estimated from a random sample \(X_ 1,...,X_{n+m}\), \(Y_ 1,...,Y_{n+\ell}\) where only the first n observations can be paired properly. Two classes of estimators of \(\theta\) are studied. The first class is that of L-estimators \(\theta\) \(*=\sum^{k}_{i=1}c_ i\xi_{p_ i}\), where the quantiles \(\xi_{p_ i}\) are from the empirical cdf's \[ \hat G(t) = M^{-1}[\alpha \sum^{n+m}_{1}\sum^{n+\ell}_{1}I(Y_ j- X_ i\leq t)\quad +\quad \beta \sum^{n}_{1}I(Y_ i+Y_ j-X_ i- X_ j\leq 2t)], \] where I(\(\cdot)\) is the indicator function, and \(\alpha\), \(\beta\) are suitable constants. It is shown that for suitable \(c_ i\), \(p_ j\), \(\alpha\), \(\beta\) the estimators \(\theta\) * are consistent. The second group is the class of M-estimators, i.e. solutions \({\hat \theta}\) of \[ \alpha \sum^{n+m}_{1}\sum^{n+\ell}_{1}\Psi (Y_ i-X_ i-c)\quad +\quad \beta \sum^{n}_{1}\Psi (2^{-1}(Y_ i+Y_ j-X_ i-X_ j)-c) = 0. \] Sufficient conditions are given to ensure consistency of \({\hat \theta}\). Related approaches of constructing confidence intervals are also studied.