On a nonparametric estimation of P(X<Y<Z) in the presence of measurement errors

Let theta=P(X<Y<Z), where X similar to N(mu(X),sigma(2)), Z similar to N(mu(z),sigma(2)) and Y are independent random variables, with known constants mu(X), mu(Z) is an element of R, sigma>0. Suppose we observe a random sample Y-1 ', . . . Yn ' from the distribution of Y-1 '=Y+epsilon. Here epsilon is a random error and distributed with a known continuous distribution. Our aim is to estimate theta based on the sample as well as on the complete knowledge about the distributions of X, Z and epsilon. We propose a nonparametric estimator of theta by applying the Fourier deconvolution method. Our estimator is shown to be mean consistent with respect to the mean squared error and strongly consistent. Under some regularity assumptions on the densities of Y and epsilon, some error estimates of the proposed estimator are derived. A simulation study is also conducted to illustrate the effectiveness of our method.