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Term(s):1995
Results:20
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Title:The silhouette, concentration functions, and ML-density estimation under order restrictions
Author(s):Polonik, Wolfgang;
Date issued:Dec 1995
http://nma.berkeley.edu/ark:/28722/bk000472r6h (PDF)
Abstract:Based on empirical Levy-type concentration functions a new graphical representation of the ML-density estimator under order restrictions is given. This representation generalizes the well-known representation of the Grenander estimator of a monotone density as the slope of the least concave majorant of the empirical distribution function. From the given representation it follows that a density estimator called silhouette which arises naturally out of the excess mass approach is the ML-density estimator under order restrictions. This fact brings in several new aspects to ML-density estimation under order restrictions. Especially, it provides new methods for deriving asymptotic results for ML-density estimators under order restrictions based on empirical process theory.
Pub info:Annals of Statistics, Vol.26, 1998, pp. 1857-1877
Keyword note:Polonik__Wolfgang
Report ID:445
Relevance:100

Title:Stopped Markov chains with stationary occupation times
Author(s):Evans, Steven N.; Pitman, Jim;
Date issued:Dec 1995
http://nma.berkeley.edu/ark:/28722/bk0000n2280 (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n229j (PostScript)
Abstract:Let $E$ be a finite set equipped with a group $G$ of bijective transformations and suppose that $X$ is an irreducible Markov chain on $E$ that is equivariant under the action of $G$. In particular, if $E = G$ with the corresponding transformations being left or right multiplication, then $X$ is a random walk on $G$. We show that when $X$ is started at a fixed point there is a stopping time $U$ such that the distribution of the random vector of pre-$U$ occupation times is invariant under the action of $G$. When $G$ acts transitively (that is, $E$ is a homogeneous space), any non-zero, finite expectation stopping time with this property can occur no earlier than the time $S$ of the first return to the starting point after all states have been visited. We obtain an expression for the joint Laplace transform of the pre-$S$ occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod $r$ the pre-$S$ occupation times do not generally have a group invariant distribution. This appears to contrast with the Brownian analog, as there is considerable support for the conjecture that the field of local times for Brownian motion on the circle prior to the counterpart of $S$ is stationary under circular shifts.
Pub info:Prob. Th. Rel. Fields. 106, 299-329, 1996
Keyword note:Evans__Steven_N Pitman__Jim
Report ID:444
Relevance:100

Title:Data Reduction and Statistical Inconsistency in Linear Inverse Problems
Author(s):Genovese, C. R.; Stark, P. B.;
Date issued:Oct 1995
Date modified:Revised February, 1996
Abstract:An estimator or confidence set is statistically consistent if, in a well defined sense, it converges in probability to the truth as the number of data grows. We give sufficient conditions for it to be impossible to find consistent estimators or confidence sets in some linear inverse problems. Several common approaches to statistical inference in geophysical inverse problems use the set of models that satisfy the data within a chi-squared measure of misfit to construct confidence sets and estimates. For example, the minimum-norm estimate of the unknown model is the model of smallest norm among those that map into a chi-squared ball around the data. We give weaker conditions under which the chi-square misfit approach yields inconsistent estimators and confidence sets. Both sets of conditions depend on a measure of the redundancy of the observations, with respect to an (\em a priori\/) constraint on the model. When the observations are sufficiently redundant, using a chi-square measure of misfit to selected averages of the data yields consistent confidence sets and minimum-norm estimates. Under still weaker conditions, one can find consistent estimates and confidence intervals for finite collections of linear functionals of the model. In an idealization of the problem of estimating the Gauss coefficients of the magnetic field at the core from satellite data, using a constraint on the energy stored in the field, suitable data averaging leads to consistent confidence intervals for finite collections of the Gauss coefficients.
Pub info:Physics of the Earth and Planetary Interiors, December, 1996.
Keyword note:Genovese__Christopher_Ralph Stark__Philip_B
Report ID:443
Relevance:100

Title:Coalescing Markov labelled partitions and a continuous sites genetics model
Author(s):Evans, S. N.;
Date issued:Oct 1995
http://nma.berkeley.edu/ark:/28722/bk0000n2214 (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n222p (PostScript)
Abstract:Let $Z$ be a Markov process with state-space $E$. For any finite set $S$ it is possible to associate with $Z$ a process $\zeta$ of coalescing partitions of $S$ with components labelled by elements of $E$ that evolve as copies of $Z$ between coalescences. Subject to very weak hypotheses on $Z$, there is a Feller process $X$ with state-space a certain space of probability measure valued functions on $E$. The process $X$ has its moments'' defined in terms of expectations for $\zeta$ in a manner suggested by various instances of martingale problem duality between coalescing Markov processes and voter model particle systems, systems of interacting Fisher-Wright and Fleming-Viot diffusions that arise in population genetics, and stochastic partial differential equations with Fisher-Wright noise that appear as rescaling limits of long-range voter models as well as in population genetics. Some sample path properties are examined in the special case where $Z$ is a symmetric stable process on $\bR$ with index $1 < \alpha \le 2$. In particular, we show that for fixed $t>0$ the essential range of the random probability measure valued function $X_t$ is almost surely a countable set of point masses.
Keyword note:Evans__Steven_N
Report ID:442
Relevance:100

Title:Cluster formation in a stepping stone model with continuous, hierarchically structured sites
Author(s):Evans, S. N.; Fleischmann, K.;
Date issued:Aug 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1x1k (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1x24 (PostScript)
Abstract:A stepping stone model with site space a continuous, hierarchical group is constructed via duality with a system of (delayed) coalescing stable'' L\'evy processes. This model can be understood as a continuum limit of discrete state-space, two allele, genetics models with hierarchically structured resampling and migration. The existence of a process rescaling limit on suitable large space and time scales is established and interpreted in terms of the dynamics of cluster formation. This paper was inspired by recent work of Klenke.
Keyword note:Evans__Steven_N Fleischmann__Klaus
Report ID:441
Relevance:100

Title:Mixing Property and Functional Central Limit Theorems for a Sieve
Author(s):Bickel, Peter J.; Bühlmann, Peter;
Date issued:Sep 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1r0f (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1r10 (PostScript)
Abstract:Bootstrap in Time Series We study a bootstrap method for stationary real-valued time series, which is based on the method of sieves. We restrict ourselves to autoregressive sieve bootstraps. Given a sample X_1,...,X_n from a linear process (X_t)_(t in Z), we approximate the underlying process by an autoregressive model with order p=p(n), where p(n) tends to infinity, p(n)=o(n) as the sample size n tends to infinity. Based on such a model a bootstrap process (X_t^*)_(t in Z) is constructed from which one can draw samples of any size. We give a novel result which says that with high probability, such a sieve bootstrap process (X_t^*)_(t in Z) satisfies a new type of mixing condition. This implies that many results for stationary, mixing sequences carry over to the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition.
Keyword note:Bickel__Peter_John Buhlmann__Peter
Report ID:440
Relevance:100

Title:Bias in Qualitative Measures of Concordance for Rodent Carcinogenicity Tests
Author(s):Lin, Tony; Gold, Lois Swirsky; Freedman, David;
Date issued:August 1995
http://nma.berkeley.edu/ark:/28722/bk000472j03 (PDF)
Abstract:According to current policy, chemicals are evaluated for possible cancer risk to humans at low dose by testing in bioassays, where high doses of the chemical are given to rodents. Thus, risk is extrapolated from high dose in rodents to low dose in humans. The accuracy of these extrapolations is generally unverifiable, since data on humans is limited. However, it is feasible to examine the accuracy of extrapolations from mice to rats. If mice and rats are similar in respect to carcinogenesis, this provides some evidence in favor of inter-species extrapolations; conversely, if mice and rats are different, this casts doubt on the validity of extrapolations from mice to humans. Once measure of inter-species agreement is concordance, the percentage of chemicals that are classified the same way as to carcinogenicity in mice and rats. Observed concordance in NCI/NTP bioassays is around 75%, which may seem on the low side -- because mice and rats are closely related species tested under the same experimental conditions. Theoretically, observed concondance could under-estimate true concordance, due to measurement error in bioassays. Thus, bias in concordance is of policy interest. Expanding on previous work by Piegorsh et. al. (1992), we show that the bias in observed concordance can be either positive or negative: an observed concordance of 75% can arise if the true concordance is anything between 20% and 100%. In particular, observed concordance can seriously overestimate true concordance. A variety of models more or less fit the data, with quite different implications for bias. Therefore, given our present state of knowledge, it seems quite unlikely that true concordance can be determined from bioassay data.
Keyword note:Lin__Tony_Hai Gold__Lois_S Freedman__David
Report ID:439
Relevance:100

Title:Some Conditional Expectations Given an Average of a Stationary or Self-similar Process
Author(s):Pitman, J.; Yor, Marc;
Date issued:August 1995
Keyword note:Pitman__Jim Yor__Marc
Report ID:438
Relevance:100

Title:Polynomial splines and their tensor products in extended linear modeling
Author(s):Stone, Charles J.; Hansen, Mark; Kooperberg, Charles; Truong, Young K.;
Date issued:August 1995
Keyword note:Stone__Charles Hansen__Mark_Henry Kooperberg__Charles_Louis Truong__Young_Kinh-Nhue
Report ID:437
Relevance:100

Title:Random discrete distributions derived from self-similar random sets
Author(s):Pitman, Jim; Yor, Marc;
Date issued:Aug 1995
Abstract:A model is proposed for a decreasing sequence of random variables $(V_1, V_2, \cdots)$ with $\sum_n V_n = 1$, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let $V_n$ be the length of the $n$th longest component interval of $[0,1]\backslash Z$, where $Z$ is an a.s. non-empty random closed of $(0,\infty)$ of Lebesgue measure $0$, and $Z$ is self-similar, i.e. $cZ$ has the same distribution as $Z$ for every $c > 0$. Then for $0 \le a < b \le 1$ the expected number of $n$'s such that $V_n \in (a,b)$ equals $\int_a^b v^(-1) F(dv)$ where the structural distribution $F$ is identical to the distribution of $1 - \sup ( Z \cap [0,1] )$. Then $F(dv) = f(v)dv$ where $(1-v) f(v)$ is a decreasing function of $v$, and every such probability distribution $F$ on $[0,1]$ can arise from this construction.
Pub info:EJP Vol 1 (1996) Paper 4
Keyword note:Pitman__Jim Yor__Marc
Report ID:436
Relevance:100

Title:The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
Author(s):Pitman, Jim; Yor, Marc;
Date issued:Aug 1995
http://nma.berkeley.edu/ark:/28722/bk0000n211m (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n2125 (PostScript)
Abstract:The two-parameter Poisson-Dirichlet distribution, denoted \PD$(\alpha,\theta)$, is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter $\theta$, introduced by Kingman, is \PD$(0,\theta)$. Known properties of \PD$(0,\theta)$, including the Markov chain description due to Vershik-Shmidt-Ignatov, are generalized to the two-parameter case. The size-biased random permutation of \PD$(\alpha,\theta)$ is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For $0 < \alpha < 1$, \PD$(\alpha,0)$ is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index $\alpha$. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is \PD$(1/2,0)$, and the corresponding distribution for Brownian bridge is \PD$(1/2,1/2)$. The \PD$(\alpha,0)$ and \PD$(\alpha,\alpha)$ distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semi-stable Markov process whose zero set is the range of a stable subordinator of index $\alpha$.
Pub info:Annals of Probability 25, pages 855-900 (1997)
Keyword note:Pitman__Jim Yor__Marc
Report ID:433
Relevance:100

Title:A New Duality Relation for Random Walks
Author(s):Dette, Holger; Pitman, Jim; Studden, William;
Date issued:Jul 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1x47 (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1x5s (PostScript)
Abstract:For the random walk on the nonnegative integers with reflecting barrier it is shown that the right tails of the probability of the first return from state $0$ to state $0$ are simple transition probabilities of a dual''random walk which is obtained from the original process by interchanging the one step probabilities. A combinatorical and analytical proof are presented and extensions and relations to other concepts of duality in the literature are discussed.
Keyword note:Dette__Holger Pitman__Jim Studden__William
Report ID:432
Relevance:100

Title:Sieve Bootstrap for Time Series
Author(s):Bühlmann, Peter;
Date issued:Apr 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1q7s (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1q8b (PostScript)
Abstract:We study a bootstrap method which is based on the method of sieves. A linear process is approximated by a sequence of autoregressive processes of order p=p(n), where p(n) tends to infinity, but with a smaller rate than n, as the sample size n increases. For given data, we then estimate such an AR(p(n)) model and generate a bootstrap sample by resampling from the residuals. This sieve bootstrap enjoys a nice nonparametric property. We show its consistency for a class of nonlinear estimators and compare the procedure with the blockwise bootstrap, which has been proposed by K\"{u}nsch (1989). In particular, the sieve bootstrap variance of the mean is shown to have a better rate of convergence if the dependence between separated values of the underlying process decreases sufficiently fast with growing separation. Finally a simulation study helps illustrating the advantages and disadvantages of the sieve compared to the blockwise bootstrap.
Keyword note:Buhlmann__Peter
Report ID:431
Relevance:100

Title:Large deviations from the McKean-Vlasov limit for super-Brownian morion with mean-field interaction
Author(s):Overbeck, L.;
Date issued:Jan 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1w0h (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1w12 (PostScript)
Abstract:A large deviation principle is proved for the mean-field measure of a sequence of n-type super-Brownian motions with a mean-field dependent immigration. The rate-function is identified as a pertubation of the rate function for a non-interacting case.
Keyword note:Overbeck__L
Report ID:430
Relevance:100

Title:Non-linear superprocesses
Author(s):Overbeck, L.;
Date issued:Jan 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1v7v (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1v8d (PostScript)
Abstract:Non-linear martingale problems in the McKean-Vlasov sense for superprocesses are studied. The stochastic calculus on historical trees is used in order to show that there is a unique solution of the non-linear martingale problems under Lipschitz conditions on the coefficients.
Keyword note:Overbeck__L
Report ID:429
Relevance:100

Title:Superprocesses and McKean-Vlasov equations with creation of mass
Author(s):Overbeck, L.;
Date issued:Jan 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1v46 (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1v5r (PostScript)
Abstract:Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of superprocesses. The solutions can be approximated by a sequence of non-interacting superprocesses or by the mean-field of multitype superprocesses with mean-field interaction. The latter approximation is associated with a propagation of chaos statement for weakly interacting multitype superprocesses.
Keyword note:Overbeck__L
Report ID:428
Relevance:100

Title:Kac's moment formula for additive functionals of a Markov process
Author(s):Pitman, Jim;
Date issued:Jul 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1z0j (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1z13 (PostScript)
Abstract:Mark Kac introduced a method for calculating the distribution of the integral $A_v = \int_0^T v(X_t) dt$ for a function $v$ of a Markov process $(X_t, t \ge 0 )$, and suitable random $T$, which yields the Feynman-Kac formula for the moment-generating function of $A_v$. This paper reviews Kac's method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of $A_v$ which may be stated as follows. For any random time $T$ such that the killed process $(X_t, 0 \le t <T)$ is Markov with some substochastic semi-group $K_t(x,dy) = P_x(X_t \in dy, T > t)$, and any non-negative measurable $v$, for $X$ with initial distribution $\lambda$ the $n$th moment of $A_v$ is $P_\lambda A_v^n = n! \lambda (G M_v) ^n 1$ where $G = \int_(0)^\infty K_t dt$ is the Green's operator of the killed process, $M_v$ is the operator of multiplication by $v$, and $1$ is the function that is identically 1.
Keyword note:Pitman__Jim
Report ID:427
Relevance:100

Title:Shrinkage Estimators, Skorokhod's Problem, and Stochastic Integration by Parts
Author(s):Evans, S. N.; Stark, P. B.;
Date issued:Mar 1995
Abstract:For a broad class of error distributions that includes the spherically symmetric ones, we give a short proof that the usual estimator of the mean in a $d$ - dimensional shift model is inadmissible under quadratic loss when $d \ge 3$. Our proof involves representing the error distribution as that of a stopped Brownian motion, and using elementary stochastic analysis to obtain a generalisation of an integration by parts lemma due to Stein in the Gaussian case.
Pub info:Annals of Statistics, 24, 809-815, 1996.
Keyword note:Evans__Steven_N Stark__Philip_B
Report ID:426
Relevance:100

Title:Cyclically Stationary Brownian Local Time Processes
Author(s):Pitman, Jim;
Date issued:Jun 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1x7w (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1x8f (PostScript)
Abstract:Local time processes parameterized by a circle, defined by the occupation density up to time $T$ of Brownian motion with constant drift on the circle, are studied for various random times $T$. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray-Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For $T$ the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar L\'evy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams' decomposition of It\^o's law of Brownian excursions.
Pub info:Prob. Th. Rel. Fields. 106 299-329, 1996
Keyword note:Pitman__Jim
Report ID:425
Relevance:100

Title:Moving-Average Representation of Autoregressive Approximations
Author(s):Bühlmann, Peter;
Date issued:Jan 1995
http://nma.berkeley.edu/ark:/28722/bk0000n1r79 (PDF)
http://nma.berkeley.edu/ark:/28722/bk0000n1r8v (PostScript)
Abstract:We study the properties of an infinite MA-representation of an autoregressive approximation for a stationary, real-valued process. In doing so we give an extension of Wiener's Theorem in the deterministic approximation set-up. When dealing with data, we can use this new key result to obtain insight into the structure of infinite MA-representations of fitted autoregressive models where the order increases with the sample size. In particular, we give a uniform bound for estimating the moving-average coefficients via autoregressive approximation being uniform over all integers.
Pub info:Stochastic Processes and their Applications, Vol. 60 (1995) 331-342
Keyword note:Buhlmann__Peter
Report ID:423
Relevance:100