Nonparametric change-point detection for two populations

Suppose F and G are unknow continuous distributions and one can observe sequential a series of independent random vectors (X₁,Y₁),(X₂,Y₂),...such that (X_i,Y_i)'s initially have distribution F×F and at some unknow time their distribution may become F×G. Namely,a change in the distribution of the Y observations may occur for some reason, while the X observation maintain their distribution. We coinsider the case where the X observation maintain their distribution We consider the case where the post-change distribution is a Lehmann alternative of F, i.e.., G = F^δ for some δ > 0. The problem is to detect the change as soon as possible after its occurrence, subject to constriant on the rate of false alarms Let Aⁿ _k denote the likehood ratio of the ranks of the combined data(X₁,...,X_n,Y₁,...,Y_n) for the test of no change versus na change to a Lehmann alternative at k+1 in the Y sequence. We consider the nonparmetric Shiryaev-Roberts stopping rule based on Aⁿ _k and compute its average run length to dectection by decoupling method     

sequential estimation of the hgarginal density function for a strongly mixing process

Suppose that {X_n} is a strongly mixing process with unknow marginal density f(x) and that we estimate f(x) by a kernel estimator [fcirc]_n(x|h_n)and want to achive the MISE no larger than some preassigned postive number w. However,the appropriate sample size n^*depends on a functional of the unknow density function. Therefore some sequential procedure is required and we adopt a fully sequential procedure. In this paper we investigate the asymptotic properties of the procedure and show that the producure is asymptotically efficient in a certain sense as w→0. The results are almost the same in the i.i.d. setting. our result extend a class of models to which the methodology can be applied. For example economic variable,experiments on a single subject in which obervation are not indepent, and so on.     

Limit theorems on occupation times for perturbed random walks

Let F be a strongly non-lattice distribution function with mean zero variance one, and a finite third moment.Let X₁, X₂,… be i.i.d. with common distribution function F. Let S_n = X₁ + X₂ +…+ X_n and ξ_n be random variables independent of X_n+1, X_n+2,…for n = 1,2,....The main results reported here are two limit theorems (Theorem 3 and Theorem 11) on occupation times for a type perturbed random walks Z_n = S_n + ξ_n when ξ_n's are slowly changing (also called slowly varying in this paper). A stone type local limit theorem, an Edgeworth expansion and an invariance principle are used in the proofs.     

Flow property analysis of irregular powders

Accurate yield locus analysis using computer techniques can now be carried out on simple and complex irregular powders, as well as on regular powders, by performing a least squares regression on the equation n log(τ_i/τ_L) = log [(T_s + σ_i/σ_L)/(T_s + 1)]This equation reduces any number of yield loci having a common shear index value, n, to a single straight line of slope n provided that all individual shear stress,τ_i, and normal stress, σ_i points are divided by their respective steadystate stress values τ_L and σ_L, and that the correct value for specific tension, T_s = T/σ_L is entered for each locus. Improved values for both n and T_s are found by successive reiteration of the above regression and by subsequent regeneration of individual yield locus values for cohesion, C, and tension, T, from a solution of the equation.

Simple irregular powders which show a constant n, but a variable angle of static internal friction, ψ_s, are easily handled by this technique. Complex irregular powders, in which both n and ψ_s vary, present greater difficulty, but can be analysed by taking a large number of points and operating on individual loci in pairs or one at a time. Examples are given of industrial powders showing both simple and complex irregularity, and particular reference is made to the effect of variable friction characteristics on overall flowability, ζ_i.     

The maximum of a sequence with prior information

Let X₁...,X_n denote a random sample of size n form distribution function F. The X's are observed one at a time in sequence; the problem is to stop at a maximum of the n observations, where recall is not permitted and where the loss is 0 if a maximum is selected and 1 if not. Here it is assumed that the distribution function is from a Dirichlet process with known parameter. Sequential stopping rules are developed and the probabilities of maximal selection computed. robustness of the procedure is investigated in an example.     

Fixed-width confidence intervals for a function of normal parameters

This paper considers sequantial producers to construct fixed-width confidence intervals for some function θ of mean μ and variance σ² of normal distribution.Consideration is devoted to θ=exp( μ + σ²/2 ) and θ=μ/σ.Nonlinear renewal theory is used to drive asymptotic expansion of expectation of the stopping time and the estimate as the width of confidence interval decreases to zero.An improvement of the coverage probability is also discussed.     

Adaptive sequential confidence intervals for A location Parameter

Let θ + X₁, θ + X₂... be a random sample and X_i's have a common distribution function F, where F has an absolutely continuous density f that is symmetric about origin. For a given confidence coefficient, a fully sequential procedure is proposed for constructing a fixed-width confidence interval for the location parameter θ. The result is based on Stone's (1975) adaptive maximum likelihood estimator. under mild conditions the procedure is shown to be asymptotically consistent and fully efficient.     

Second-Order Efficiency for Two-Stage Estimation of a Linear Function of Normal Mean Vectors when Covariance Matrices Have Some Structures

Abstract

We consider two-stage estimation for a fixed-span confidence region about a linear function of mean vectors from π _i : N _p (μ _i , Σ _i ), i = 1,…,k (≥2), when Σ _i 's have some structures. The purpose of this article is to investigate asymptotic efficiency of the estimation up to the second order in terms of the sample size. An adjustment of the design constant and a proper choice of the initial sample size appearing in the two-stage estimation are proposed to have asymptotic second-order efficiency. Some simulations are carried out to see moderate sample size performances of the proposed two-stage estimation. An example is given for a demonstration.     

Fixed width confidence intervals based on bootstrap procedures

Let T_n be an estimate of the parameter θ and for a fixed d let P_n* be the bootstrap estimate of the probability that θ is included in the interval (T_n - d,T_n +d). Let N* be the first n such that P^* _n ≥ 1-α with 0<α<1. If T_n is the sample mean, then the asymptotic behaviour of N* as d↓0 is the same as that of the well known Chow Robbins procedure. Monte Carlo calculations show that these procedures are also very similar non asymptotically. If T_n is the median, the asymptotic behaviour of N* is the same as that of the procedure of Geertsema but Monte Carlo calculations in this case favours the bootstrap procedure.     

On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher’s “Nile” example

Fisher’s “Nile” example is a classic that involves a bivariate random variable (X,Y) having a joint probability density function given by f(x,y;𝜃) = exp(?𝜃x?𝜃^?1y), 0<x,y<∞, where 𝜃>0 is a single unknown parameter. We develop (i) fixed-width and (ii) fixed-accuracy confidence intervals for 𝜃 with a preassigned confidence coe?cient. In problem (i), we develop a purely sequential estimation strategy along with its asymptotic properties. In problem (ii), we determine that a fixed-sample-size estimation strategy will su?ce and yet the requisite sample size would have to be found. We have done that both exactly as well as approximately and we report that for all practical purposes the approximations nearly provide the exact sample size whether it is small, moderate, or large. The last problem we address is bounded-accuracy fixed-sample-size estimation of P_𝜃{X>Y}. All theoretical properties are adequately validated by large-scale simulations.     

Programmed polymer light-scattering data

An ALGOL computer program has been devised to manipulate light-scattering data from the Brice-Phoenix photometer. The input consist of experimental values of the galvanometer deflections and filter factors used for each concentration c and angle of measurement θ. These are transformed to the appropriate variables in the fundamental equation including the particle scattering factor, viz: c/Qθ = (W/K^*)M?_w^?1[1 + (16/3) × π²n₁²λ〈S²〉 sin² (θ/2)] + (W/K^*)2A₂c + (W/K^*)3A₃c² in which Qθ is a corrected from of the Rayleigh ratio and (W/K^*) is a composite constant term for the instrument and polymer–solvent system. By writing X?_ij for the variable c/Q_θ at θ_i and c_j, a function X is found by least squares to fit X?_ij, thus X = l + m sin (θ/2) + nc_j + bc_j². The equations arising from minimizing Σ_i^K=1 Σ_j^L=1 (X_ij ? X?_ij)² are solved by the computer to yield the best-fitting coefficients l, m, n, and b. These can then be related simply to the molecular weight, root-mean-square radius of gyration, second and third virial coefficients, respectively. The final portion of the program is designed to check the fit of these coefficients. It yields a table of the differences between all experimental c/Qθ values and the coressponding ones obtained by inserting the derived l, m, n, and b into the fundamental equation. The procedure has been tested satisfactorily by using a well-standardized sample of polystyrene in toluene at 30°C. and a wavelength of 436 mμ.     

The Softness Parameters of Ions. Prediction of the Occurrence of Miscibility Gaps in Molten Salt Mixtures

Softness parameters σ_M for cations and σ_X for anions, have been calculated as dimensionless quantities for approx. 90 cations and 18 anions. They are given by σ_M = [σ_A (M^m+) - σ_A(H⁺)]/σ_A(H⁺) and σ_X = [σ_B(X^a?) - σ_B(OH^?)]/σ_A(H⁺) where σ_A = [σI_i(M) + ΔH⁰_h(M^m+)]/m and σ_B = [-E_a(X) + ΔH⁰_h(X^a?)]/a are Ahrland's parameters. The new normalized and comparative (to the test ions H⁺ and OH^?) softness parameters are positive for soft ions and negative for hard ones. These parameters, obtained independently, are used with a four-coefficient equation to calculate coordinate bond energies for metal halides with acceptable accuracy. Considerations of the average coordination in reciprocal molten salt mixture lead to an expression for the metathesis energy change as proportional to the product of the differences in softness parameters of the two cations and the two anions. An empirical one-coefficient equation involving the softness parameters is proposed to deal with next-nearest-neighbor interactions in binary common-ion molten salt mixtures. These relationships are then used with Blander and Topol's equation to predict the occurrence of irascibility gaps in uni-univalent reciprocal salt mixtures. The gaps found in other systems are also discussed in terms of the softness of the constituent ions.     

Sequential estimation of parameters in one-truncation parameter families under type ii censoring

In this paper, we consider one-truncation parameter family of distribution of the form f(x;θ)=q(θ)h(x),c<x<θ) We look at the theoretical properties of the stopping rule and the sequential estimation of q(θ) of this family under the assumptions that the sample is type II censored so that one observes only X_1:n,X_2:n,…X_r:n,1<r<n.     

A partial sequential t-test

Several authors have proposed sequential test procedures for testing one sided hypotheses about the mean μ of a normal distribution with unknown variance σ². Some of these procedures have concerned themselves with tests for units of (μ/σ). I this work, a sequential test procedure is introduced for hypoth-eses concerned with μ itself. In the present procedure the transformed observa-tions follow a central t-distribution as distinct from the noncentral t-distribution used in pnvious tests. Insights into properties of the operating characteristic and the average sample number functions are found for the present test; specifically, approximations to the theoretical operating characteristic function are derived and the sensitivity of the average sample number function to changes in some of the test parameters is investigated.     

Combined Absorption and Self-Decomposition of Ozone in Aqueous Solutions with Interfacial Resistance

A theoretical analysis is performed employing the film model for the isothermal absorption and self-decomposition of ozone in aqueous solutions with interfacial resistance, which is inversely proportional to the interfacial mass transfer coefficient k_s. A closed-form solution has been obtained. The effects of system parameters on the ozone mass transfer rate are examined. These parameters include the interfacial resistance (1/k_s), the acidic and basic self-decomposition reaction rate parameters (M_m ^0.5, M_n ^0.5.; M_m = [2D_Ak_mC_Ai ^m-1/(m+1)]/(k_L ⁰)², M_n=(2D_Ak_nC_Ai ^n-1/(n+1))/(k_L ⁰)², the reaction orders (m,n), the pH value of solution, and the liquid-phase mass transfer coefficient (k_L ⁰). The results indicate that the reduction effect of the interfacial resistance on the absorption rate is most significant for the situation with the larger values of M_m and M_n as well as with higher pH values. Also, for any particular finite value of k_L ⁰/k_s, the reduction effect encountered is greater for a gas liquid contactor with a lower k_L ⁰. The reduction effect should be avoided in order to maintain a higher mass transfer rate of ozone in aqueous solution. This analysis is of importance for the efficient use of ozone in water/wastewater treatment processes in the presence of interfacial resistance substances such as surface active agents. For some known special cases (for example, cases with no interfacial resistance), the present solution reduces to the previous works of other investigators.