The bounded normal mean problem has important applications in nonparametric function estimation. It is to estimate the mean of a normal distribution whose mean is restricted to a bounded interval. The minimax risk for such a problem is generally unknown. It is shown in Donoho, Liu and MacGibbon(1990) that the linear minimax risk provides a good approximation to the minimax risk. We show in this...

Let X be a random variable from a normal distribution with unknown mean θ and known variance σ2. In many practical situations, θ is known in advance to lie in an interval, say [−m,m], for some m > 0. As the usual estimator of θ, i.e., X under the LINEX loss function is inadmissible, finding some competitors for X becomes worthwhile. The only study in the literature considered the problem of min...

Consider the problem of estimating the mean of a single normal random variable when the mean is known to be bounded. We establish the minimax aane estimator under zero-one loss and discuss minimal xed-length aane conndence intervals. Moreover, the minimal length of arbitrary xed-size conndence intervals is described and the maximal loss of eeciency caused by the restriction to aane estimators i...

Let u = u(q) satisfy a hyperbolic equation with impulsive input: ∂ t u(x, t)−4u(x, t) + q(x)u(x, t) = δ(x1)δ(t) and let u|t<0 = 0. Then we consider an inverse problem of determining q(x), x ∈ Ω from data u(q)|ST and (∂u(q)/∂ν) |ST . Here Ω ⊂ {(x1, . . . , xn) ∈ R|x1 > 0}, n ≥ 2, is a bounded domain, ST = {(x, t); x ∈ ∂Ω, x1 < t < T + x1}, ν = ν(x) is the unit outward normal vector to ∂Ω at x ∈ ...

Let X be a compact subset of the complex plane with property that every relatively open has positive area and let $$A_0(X)$$ denote space VMO functions are analytic on X. is said to admit bounded point derivation order t at $$x_0 \in \partial X$$ if there exists constant C such $$|f^{(t)}(x_0)|\le \Vert f\Vert _{{\text {BMO}}}$$ for all in $${\text {VMO}}(X)$$ In this paper, we give necessa...