Testing the hypothesis that a point is Poisson

1977 ◽  
Vol 9 (4) ◽  
pp. 724-746 ◽  
Author(s):  
Robert B. Davies
Keyword(s):  
Point Process ◽  
Infinite Series ◽  
Partial Sums ◽  
Test Statistic ◽  
Optimal Test ◽  
One Dimensional

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

Testing the hypothesis that a point is Poisson

1977 ◽  
Vol 9 (04) ◽  
pp. 724-746
Author(s):  
Robert B. Davies
Keyword(s):  
Point Process ◽  
Infinite Series ◽  
Partial Sums ◽  
Test Statistic ◽  
Optimal Test ◽  
One Dimensional

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

Random Walk in Random Scenery

2019 ◽  
pp. 382-404
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Random Walk ◽  
Central Limit ◽  
Functional Form ◽  
Stationary Sequence ◽  
Limit Behavior ◽  
Partial Sums ◽  
Determinantal Process ◽  
One Dimensional ◽  
The Central Limit Theorem ◽  
Random Scenery

In this chapter we investigate the question of central limit behavior and its functional form for the partial sums associated with a centered L2-stationary sequence of real-valued random variables (usually called the random scenery) sampled by a recurrent one-dimensional strongly aperiodic random walk. This question is handled under various conditions dependent on the random scenery. In particular, we assume that the random scenery either satisfies an asymptotic negative dependence condition, or is a function of a determinantal process and a Gaussian sequence, or satisfies a mild projective criterion. We first show that study of central limit behavior for such random walks in random scenery can be handled with results related to linear statistics developed in Chapter 12, provided the random walk has good properties. We then look extensively at the properties of a recurrent one-dimensional strongly aperiodic random walk. The functional form of the central limit theorem is also investigated.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

Possible and Impossible Series

2015 ◽  
Vol 108 (7) ◽  
pp. 560
Author(s):  
Mark MacLean
Keyword(s):  
Infinite Series ◽  
Partial Sums ◽  
Its Sequence

A lesson helps students discern possible relationships between an infinite series, its sequence of terms, and the sequence of partial sums.

scholarly journals On the statistical power of Baarda’s outlier test and some alternative

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
R. Lehmann ◽  
A. Voß-Böhme
Keyword(s):  
Model Test ◽  
Statistical Power ◽  
Local Model ◽  
Test Statistics ◽  
Test Statistic ◽  
Optimal Test ◽  
Outlier Test ◽  
Uniformly Most Powerful ◽  
Ump Test

AbstractBaarda’s outlier test is one of the best established theories in geodetic practice. The optimal test statistic of the local model test for a single outlier is known as the normalized residual. Also other model disturbances can be detected and identified with this test. It enjoys the property of being a uniformly most powerful invariant (UMPI) test, but is not a uniformly most powerful (UMP) test. In this contribution we will prove that in the class of test statistics following a common central or non-central χ

Stochastic comparisons for queueing models via random sums and intervals

1992 ◽  
Vol 24 (4) ◽  
pp. 960-985 ◽  
Author(s):  
Alain Jean-Marie ◽  
Zhen Liu
Keyword(s):  
Point Process ◽  
Response Times ◽  
Waiting Times ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Partial Sums ◽  
Polling Systems ◽  
Convex Ordering ◽  
Random Intervals ◽  
Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

scholarly journals Improved Differentially Private Analysis of Variance

2019 ◽  
Vol 2019 (3) ◽  
pp. 310-330 ◽  
Author(s):  
Marika Swanberg ◽  
Ira Globus-Harris ◽  
Iris Griffith ◽  
Anna Ritz ◽  
Adam Groce ◽  
...  
Keyword(s):  
Analysis Of Variance ◽  
Statistical Power ◽  
Differential Privacy ◽  
Hypothesis Test ◽  
Test Statistic ◽  
Reference Distribution ◽  
Optimal Test ◽  
The Public ◽  
Order Of Magnitude ◽  
Public Setting

Abstract Hypothesis testing is one of the most common types of data analysis and forms the backbone of scientific research in many disciplines. Analysis of variance (ANOVA) in particular is used to detect dependence between a categorical and a numerical variable. Here we show how one can carry out this hypothesis test under the restrictions of differential privacy. We show that the F -statistic, the optimal test statistic in the public setting, is no longer optimal in the private setting, and we develop a new test statistic F1 with much higher statistical power. We show how to rigorously compute a reference distribution for the F1 statistic and give an algorithm that outputs accurate p-values. We implement our test and experimentally optimize several parameters. We then compare our test to the only previous work on private ANOVA testing, using the same effect size as that work. We see an order of magnitude improvement, with our test requiring only 7% as much data to detect the effect.

Stochastic comparisons for queueing models via random sums and intervals

1992 ◽  
Vol 24 (04) ◽  
pp. 960-985 ◽  
Author(s):  
Alain Jean-Marie ◽  
Zhen Liu
Keyword(s):  
Point Process ◽  
Response Times ◽  
Waiting Times ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Partial Sums ◽  
Polling Systems ◽  
Convex Ordering ◽  
Random Intervals ◽  
Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered. These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

On Absolute Summability by Riesz and Generalized Cesàro Means. I

1970 ◽  
Vol 22 (2) ◽  
pp. 202-208 ◽  
Author(s):  
H.-H. Körle
Keyword(s):  
Infinite Series ◽  
Absolute Summability ◽  
Partial Sums ◽  
Cesàro Means ◽  
Riesz Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Image Position ◽  
Generalized Cesàro Means

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series(1)with complex terms an. Throughout, we will assume that(2)and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means(of the partial sums of (1)) tend to s as x → ∞.

Sign in / Sign up

Export Citation Format

Share Document