Kimberly K. J. Kinateder

Clustering Functional Data

The problem of clustering functional data is addressed. Results on principal points (cluster means for probability distributions) are given for functional Gaussian distributions. Examples and simulations are provided to illustrate results.

A new approach to the busy period of the M/M/1 queue

In this paper, we provide a new approach to the computation of the Laplace transform of the length of the busy period of the M/M/1 queue with constrained workload (finite dam), without the use of complex analysis.

The expected wet period of finite dam with exponential inputs

We use martingale methods to obtain an explicit formula for the expected wet period of the finite dam of capacity V, where the amounts of inputs are i.i.d exponential random variables and the output rate is one, when the reservoir is not empty. As a consequence, we obtain an explicit formula for the expected hitting time of either 0 or V and a new expression for the distribution of the number of overflows during the wet period, both without the use of complex analysis.

Hypersurfaces in R-D and the Variance of Exit Times for Brownian Motion

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

Variational principles for average exit time moments for diffusions in Euclidean space

Let D be a smoothly bounded domain in Euclidean space and let Xt be a diffusion in Euclidean space. For a class of diffusions, we develop variational principles which characterize the average of the moments of the exit time from D of a particle driven by Xt, where the average is taken over all starting points in D.

Exact confidence intervals in analysis of nonorthogonal saturated designs

SYNOPTIC ABSTRACT Let m×1 = (i) ~ Nm (θ, Σ) have an m-variate normal distribution, where Σ = A′Aσ2, A′A is a known, nondiagonal positive definite matrix, and σ is unknown. The objective is to construct an exact confidence interval for each effect θi, the ith component of θ. For a saturated design, there are no error degrees of freedom from which to compute an independent estimator of the error variance component σ2. However, under effect sparsity, the smaller effect estimates can be used to provide comparable information with which to construct confidence intervals for the effects. Voss (1999) provided a method for the construction of exact individual confidence intervals for each effect θi in the analysis of orthogonal saturated factorial experiments, for which the covariance matrix Σ is diagonal. We extend his results to the case of saturated designs, for which Σ is not diagonal. Such nonorthogonality can be planned, in order to keep the number of observations small, or it may be the unplanned consequence of lost observations. In the latter case, A is a random matrix, so our results are conditional.

A statistical test for the hypothesis of Gaussian random function

ABSTRACT A Gaussian random function is a functional version of the normal distribution. This paper proposes a statistical hypothesis test to test whether or not a random function is a Gaussian random function. A parameter that is equal to 0 under Gaussian random function is considered, and its unbiased estimator is given. The asymptotic distribution of the estimator is studied, which is used for constructing a test statistic and discussing its asymptotic power. The performance of the proposed test is investigated through several numerical simulations. An illustrative example is also presented.

Understanding average Brownian exit time

Let W denote Brownian motion starting from the origin. The idea of this paper is give a computation of the expected exit time E[tau][a,b] from an interval [a,b], where a

Analysis of Nearly Saturated Designs Using Composite Variance Estimators

SYNOPTIC ABSTRACT Langsrud and Naes (1998) proposed forward-selection and backward-elimination strategies for the analysis of nearly saturated designs using composite variance estimators. Their variance estimators combine an estimator that is a function of the smaller sums of squares of the effect estimators (assuming effect sparsity) with an independent variance estimator based on the available error degrees of freedom. However, exact control of error rates for their stepwise methods remains an open problem. We investigate procedures that likewise use composite variance estimates but also provide exact control of error rates.

Random Fluctuations of Convex Domains and Lattice Points

In this paper, we examine a random version of the lattice point problem. Let 7H denote the class of all homogeneous functions in C2 (Rn) of degree one, positive away from the origin. Let qX be a random element of X-, defined on probability space (Q, .F, P), and define F.,, W() = J e-i(x,0)dx {x: b(P(WX)<11 for w E Q. We prove that, if E (I F.1)) < C[(] 2 , where [(] = 1 + 1, then E(N?)(t) = Vtn + O(tn-2+ n+) where V = E(I{x: (D(.,x) < 1}1), the expected volume. That is, on average, N(Dt)= Vn+(t-2+ 2 Nb (t) = Vtn+O(tn 2+ n+1 ). We give explicit examples in which the Gaussian curvature of {x: (D(w, x) < 1} is small with high probability, and the estimate N~1 (t) = Vtn + O(tn 2? n+1) holds nevertheless.