Section (fiber bundle)

Abstract: The problem of testing outlying observations, although an old one, is of considerable importance in applied statistics. Many and various types of significance tests have been proposed by statisticians interested in this field of application. In this connection, we bring out in the Histrical Comments notable advances toward a clear formulation of the problem and important points which should be considered in attempting a complete solution. In Section 4 we state some of the situations the experimental statistician will very likely encounter in practice, these considerations being based on experience. For testing the significance of the largest observation in a sample of size $n$ from a normal population, we propose the statistic $\frac{S^2_n}{S^2} = \frac{\sum^{n-1}_{i=1} (x_i - \bar x_n)^2}{\sum^n_{i=1} (x_i - \bar x)^2}$ where $x_1 \leq x_2 \leq \cdots \leq x_n, \bar x_n = \frac{1}{n - 1} \sum^{n-1}_{i=1} x_i$ and $\bar x = \frac{1}{n}\sum^{n}_{i=1} x_i.$ A similar statistic, $S^2_1/S^2$, can be used for testing whether the smallest observation is too low. It turns out that $\frac{S^2_n}{S^2} = 1 - \frac{1}{n - 1} \big(\frac{x_n - \bar x}{s}\big)^2 = 1 - \frac{1}{n - 1} T^2_n,$ where $s^2 = \frac{1}{n}\sigma(x_i - \bar x)^2,$ and $T_n$ is the studentized extreme deviation already suggested by E. Pearson and C. Chandra Sekar [1] for testing the significance of the largest observation. Based on previous work by W. R. Thompson [12], Pearson and Chandra Sekar were able to obtain certain percentage points of $T_n$ without deriving the exact distribution of $T_n$. The exact distribution of $S^2_n/S^2$ (or $T_n$) is apparently derived for the first time by the present author. For testing whether the two largest observations are too large we propose the statistic $\frac{S^2_{n-1,n}}{S^2} = \frac{\sum^{n-2}_{i=1} (x_i - \bar x_{n-1,n})^2}{\sum^n_{i=1} (x_i - \bar x)^2},\quad\bar x_{n-1,n} = \frac{1}{n - 2} \sum^{n-2}_{i=1} x_i$ and a similar statistic, $S^2_{1,2}/S^2$, can be used to test the significance of the two smallest observations. The probability distributions of the above sample statistics $S^2 = \sum^n_{i=1} (x_i - \bar x)^2 \text{where} \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ $S^2_n = \sum^{n-1}_{i=1} (x_i - \bar x_n)^2 \text{where} \bar x_n = \frac{1}{n-1} \sum^{n-1}_{i=1} x_i$ $S^2_1 = \sum^n_{i=2} (x_i - \bar x_1)^2 \text{where} \bar x_1 = \frac{1}{n-1} \sum^n_{i=2} x_i$ are derived for a normal parent and tables of appropriate percentage points are given in this paper (Table I and Table V). Although the efficiencies of the above tests have not been completely investigated under various models for outlying observations, it is apparent that the proposed sample criteria have considerable intuitive appeal. In deriving the distributions of the sample statistics for testing the largest (or smallest) or the two largest (or two smallest) observations, it was first necessary to derive the distribution of the difference between the extreme observation and the sample mean in terms of the population $\sigma$. This probability$X_1 \leq x_2 \leq x_3 \cdots \leq x_n$ $s^2 = \frac{1}{n} \sum^n_{i=1} (x_i - \bar x)^2 \quad \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ distribution was apparently derived first by A. T. McKay [11] who employed the method of characteristic functions. The author was not aware of the work of McKay when the simplified derivation for the distribution of $\frac{x_n - \bar x}{\sigma}$ outlined in Section 5 below was worked out by him in the spring of 1945, McKay's result being called to his attention by C. C. Craig. It has been noted also that K. R. Nair [20] worked out independently and published the same derivation of the distribution of the extreme minus the mean arrived at by the present author--see Biometrika, Vol. 35, May, 1948. We nevertheless include part of this derivation in Section 5 below as it was basic to the work in connection with the derivations given in Sections 8 and 9. Our table is considerably more extensive than Nair's table of the probability integral of the extreme deviation from the sample mean in normal samples, since Nair's table runs from $n = 2$ to $n = 9,$ whereas our Table II is for $n = 2$ to $n = 25$. The present work is concluded with some examples.

Abstract: I. Introduction.- 1.1. General Remarks.- 1.2. The Robbins-Monro Process.- 1.3. A "Continuous" Process Version of Section 2.- 1.4. Regulation of a Dynamical System a simple example.- 1.5. Function Minimization: The Kiefer-Wolfowitz Procedure.- 1.6. Constrained Problems.- 1.7. An Economics Example.- II. Convergence w.p.1 for Unconstrained Systems.- 2.1. Preliminaries and Motivation.- 2.2. The Robbins-Monro and Kiefer-Wolfowitz Algorithms: Conditions and Discussion.- 2.3. Convergence Proofs for RM and KW-like Procedures.- 2.3.1. A Basic RM-like Procedure.- 2.3.2. One Dimensional RM and Accelerated RM Procedures.- 2.3.3. A Continuous Parameter RM Procedure.- 2.3.4. The Basic Kiefer-Wolfowitz Procedure.- 2.3.5. Random Directions KW Methods.- 2.4. A General Robbins-Monro Process: "Exogenous Noise".- 2.4.1. The Case of Bounded h(*,*).- 2.4.2. Unbounded h(*,*): Exogenous Noise.- 2.5. A General RM Process State Dependent Noise.- 2.5.1. Extensions and Localizations of Theorem 2.5.2.- 2.6. Some Applications.- 2.7. Mensov-Rademacher Estimates.- III. Weak Convergence of Probability Measures.- IV. Weak Convergence for Unconstrained Systems.- 4.1. Conditions and General Discussion.- 4.2. The Robbins-Monro and Kiefer-Wolfowitz Procedures.- 4.2.1. The Basic Robbins-Monro Procedure.- 4.2.2. The One-Dimensional Robbins-Monro Procedure.- 4.2.3. The Kiefer-Wolfowitz Procedure.- 4.2.4. A Case Where the Limit Satisfies a Generalized ODE.- 4.2.5. A Continuous Parameter KW Procedure.- 4.3. A General Robbins-Monro Process: Exogenous Noise.- 4.4. A General RM Process: State Dependent Noise.- 4.5. The Identification Problem.- 4.6. A Counter-Example to Tightness.- 4.7. Boundedness of {Xn} and Tightness of {Xn(*)}.- V. Convergence w.p.1 For Constrained Systems.- 5.1. A Penalty-Multiplier Algorithm for Equality Constraints.- 5.1.1. A Basic RM-like Algorithm, Conditions and Discussion.- 5.1.2. The Noise Condition, Discussion and Generalization.- 5.1.3. Boundedness of {Xn}.- 5.1.4. Proof of the Main Theorem.- 5.1.5. Constrained Function Minimization and Other Extensions.- 5.2. A Lagrangian Method for Inequality Constraints.- 5.2.1. The Algorithm and Conditions.- 5.2.2. The Convergence Theorem 18.- 5.2.3. A Non-Convergent but Useful Algorithm.- 5.2.4. An Application to the Identification Problem.- 5.3. A Projection Algorithm.- 5.4. A Penalty-Multiplier Method for Inequality Constraints.- VI. Weak Convergence: Constrained Systems.- 6.1. A Multiplier Type Algorithm for Equality Constraints.- 6.1.1. Boundedness of {Xn}.- 6.1.2. The Noise Condition, Discussion.- 6.1.3. The Convergence Theorem.- 6.2. The Lagrangian Method.- 6.3. A Projection Algorithm.- 6.4. A Penalty-Multiplier Algorithm for Inequality Constraints.- VII. Rates of Convergence.- 7.1. The Problem Formulation.- 7.2. Conditions and Discussions.- 7.3. Rates of Convergence for Case 1, the KW Algorithm.- 7.4. Discussion of Rates of Convergence for Two KW Algorithms.

Abstract: The present status of high-pressure research with the diamond anvil cell (DAC) is reviewed in this article, mainly from an experimental aspect. After a brief description of the different types of DAC's that are currently in vogue, the techniques used in conjunction with the DAC in modern high-pressure research are presented. These include techniques for low- and high-temperature studies, x-ray diffractometry, spectroscopy with the DAC, and other measurements. Results on selected materials, with a view to illustrating the physics behind high-pressure phenomena, are presented and discussed. These include metal-semiconductor transitions, electronic transitions, phonons and high-pressure lattice dynamics, and phase transitions. A whole section is devoted to the behavior of condensed gases, principally ${\mathrm{H}}_{2}$, ${\mathrm{D}}_{2}$, ${\mathrm{O}}_{2}$, ${\mathrm{N}}_{2}$, and rare-gas solids. The concluding section briefly deals with speculations on ultra-high-pressure research with the DAC in the future.

Abstract: On September 14, 2015, the Laser Interferometer Gravitational-wave Observatory (LIGO) detected a gravitational-wave transient (GW150914); we characterise the properties of the source and its parameters. The data around the time of the event were analysed coherently across the LIGO network using a suite of accurate waveform models that describe gravitational waves from a compact binary system in general relativity. GW150914 was produced by a nearly equal mass binary black hole of $36^{+5}_{-4} M_\odot$ and $29^{+4}_{-4} M_\odot$ (for each parameter we report the median value and the range of the 90% credible interval). The dimensionless spin magnitude of the more massive black hole is bound to be $0.7$ (at 90% probability). The luminosity distance to the source is $410^{+160}_{-180}$ Mpc, corresponding to a redshift $0.09^{+0.03}_{-0.04}$ assuming standard cosmology. The source location is constrained to an annulus section of $590$ deg$^2$, primarily in the southern hemisphere. The binary merges into a black hole of $62^{+4}_{-4} M_\odot$ and spin $0.67^{+0.05}_{-0.07}$. This black hole is significantly more massive than any other known in the stellar-mass regime.