Two-sample two-stage and purely sequential methodologies for tests of hypotheses with applications: comparing normal means when the two variances are unknown and unequal

In this paper, we develop appropriate sampling methodologies for testing hypotheses regarding the difference of mean values from two independent (or dependent) normal populations when their variances are unknown and unequal. We design two-stage and purely sequential testing methodologies of hypotheses for comparing the unknown means by determining the appropriate sample sizes while controlling both type-I and type-II error probabilities at or below preassigned levels α, β respectively. Such methodologies are constructed under both unequal and equal sample size designs. We prove that both two-stage and purely sequential testing strategies enjoy a number of practically appealing properties. Extensive sets of computer simulations and real data analyses empirically validate our theoretical findings.     

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.     

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.     

Critical Surface Tension of Non-crosslinking Linear Low Density Polyethylene-graft-acrylic Acid by the Contact Angle Method

Abstract

The contact angles θ of polar liquid on surface of non-crosslinking linear low density polyethylene-graft-acrylic acid (LLDPE-g-AA) were measured. The critical surface tension (γ_c) of LLDPE-g-AA films were evaluated by three different plots, the Zisman plot, the Young-Dupre-Good-Girifalco plot, and the log(1 + cos θ) versus log θ _L , plot. The θ _c of LLDPE-g-AA obtained by the 1 + cosθ versus θ _L ^?1/2 plot were higher than those obtained by other plots.     

A Numerical Analysis of Surface Diffusion in a Binary Adsorbed Film

Abstract

Based on the assumptions that the adsorption of a binary gas mixture may be described by Langmuir isotherms and that the rate of diffusion into the interior of the adsorbent is proportional to the gradient in chemical potential, equations are developed to describe the adsorption of a binary mixture by a slab and by a sphere. The equations describing the concentration changes of each component are coupled nonlinear second-order partial differentials and were solved numerically for the following boundary conditions (normalized concentrations θ of the components at the surface): θ_A:θ_B=0.05:0.90, 0.05:0.05, 0.475:0.475, and 0.875:0.095 and ratios of diffusivities, L _A/L _B=2, 10, and 200. Profiles of concentration against distance into a slab or sphere and also curves of total uptake against time (all in dimensionless form) were obtained. The distinctive feature of the results is that the component of higher diffusivity advances ahead of the second component and tends to attain temporary local concentrations much higher than the equilibrium values at the boundary. Consequently, the total uptake of the more-mobile component may pass through a maximum. These effects are most pronounced when the difference in diffusivities is large, when the equilibrium concentration of the component of higher diffusivity is small, and when the total equilibrium concentration of the two components is large. The calculations were applied to previously reported results of the adsorption of nitrogen-methane mixtures by zeolite A and were found to give reasonable correlations between the behavior of mixtures and that of the individual pure components. In the case of near saturation (θ_A+θ_B → 1 at the boundary), the fit with experiment is particularly sensitive to the exact value of (1 ? θ_A ? θ_B).     

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.     

Studies of linear polymer dimensions of poly(4-chlorostyrene) in various solvents around the θ temperatures by intrinsic viscosity

Intrinsic viscosities of poly(4-chlorostyrene) solutions in isopropyl-benzene and n-propylbenzene at temperatures above and below the theta (θ) temperatures of these solvents were measured using a capillary viscometer. The viscosity measurements were performed on three poly(4-chlorostyrene) samples having molecular weights (M?;_v) 1.75 × 10⁶, 6.5 × 10⁵ and 2.7 × 10⁵. A smooth and continuous contraction below the θ temperatures was observed for both solvents. The temperature dependence of [/eta] can be represented by a master curve in a plot of α³/eta|/tau|M^1/2 (g^1/2 mol^?1/2) versus |/tau|M^1/2 (g^1/2 mol^?1/2), where α/eta = [/eta(T)]/[/eta(θ)]^1/3 is the expansion factor and /tau = (T - θ)/T is the reduced temperature. A universal plot of reduced viscosity size (α/eta) versus reduced blob parameter (N/N_c) shows the achievement of collapsed state for T<θ. The prediction of thermal blob theory is also verified for T > θ. The temperature dependence of intrinsic viscosity, both below and above the θ temperature, exhibits similar behaviour to the temperature dependence of dipole moments for both of these solvents.     

A Confidence Interval That Is Consistent and More Informative Than a Sequential Equivalence Test for a Normal Mean

Abstract

This note shows how to construct a 1 ? β level symmetric confidence interval for a normal mean θ that is consistent with and more informative than a size β sequential equivalence test of H ₀ :|θ| > δ against the alternative H_a :|θ | ≤ δ, where δ>0 is a prespecified limit of equivalence. If the null hypothesis H ₀ is rejected and so equivalence can be declared by the test, then the confidence interval is contained in the interval [? δ, δ]. If the null hypothesis H₀ is not rejected and so equivalence cannot be declared by the test, then the confidence interval contains the interval [? δ, δ]. Therefore the test and the confidence interval are consistent in rejection or non-rejection of H ₀. But the confidence interval provides extra information on the magnitude of θ, and is therefore more informative than the test.     

The Rheology of Wetting By Polymer Melts

Theoretically, the rate of capillary penetration of a polymer melt into a slit, a model for a surface irregularity, has been shown to depend on γcosθ/η) where γ refers to the surface tension of the liquid, η its viscosity and θ a time-dependent contact angle. Analytical expressions relating the depth of penetration with time have been experimentally verified by observations of the penetration of molten polyethylene and poly-(ethylene-vinyl acetate) into aluminum channels. Values of η, calculated from the observed data, agree closely with independent determinations of this material parameter. A theoretical treatment has also been developed which describes the velocity of spreading of a liquid drop over a flat surface. Flow equations for the flow of free films were adapted for this purpose. The spreading velocity is predicted to depend on the product of three factors (1) a scaling factor, (γ/η1R_o), where R_o is the initial radius of curvature, (2) cosθ_∞. (l-cosθ/cosθ_∞) where θ_∞ refers to the equilibrium value of θ, and (3) geometric terms. After demonstrating that a drop of molten polymer may be treated as a spherical cap, the predicted dependence of spreading rate on drop size, cosθ_∞ (nature of the substrate) and the scaling factor was experimentally verified. Some discrepancies noted at long times and high temperatures are discussed.     

Stability of an inclined,pneumatically‐transported system of particles

A linear stability analysis is performed on the base‐state solution describing a pneumatically‐transported system inclined at an angle θ. One of two modes was found to be dominant, depending on the tilting angle: a low‐shear mode whose voidage eigenmode remains nearly symmetric from 0° < θ ≤ θ_c, and a “boundary layer” mode for θ > θ_c, marked by significant asymmetry throughout the domain and a high level of shear inside the thin regions adjacent to the walls. The critical angle θ_c increases with the amount of momentum and energy lost to particle–wall friction and collisions, respectively. The time‐evolution of the latter mode, superimposed onto the base‐state solution, reveals an alternating train of dense and dilute regions near the top of the domain, which agrees qualitatively with the development and propagation of bubbles in inclined systems observed in experiment. An analytic solution corresponding to a vertical system with a uniform base state was also derived to identify the instability mechanisms responsible for a class of non‐oscillatory transverse waves which exist in both vertical and tilted particle systems. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2248–2258, 2016     

Bayesian testing simple hypotheses on The drift of a wiener process With randomly relayed observations

We present a Byesian test of simple hypotheses on the drift of the observation process X_t = θt + W_twhere θis a bernouli random variable (w_t) is a standard Wiener process. sequential tests δ = (T, d) are used, where T is a (F^X _t)-stopping time, d is a (F^X _t+n)-measurable decision function, n is a postive random variable with a given distribution. The cost function V(π, T, d) of any sequential test θ = (T, d) is based on costs of wrong decisions and observation, respectively.     

Measurement of polar orientation in poled side-chain NLO polymers using infrared spectroscopy

Infrared spectroscopy has been used to provide an independent estimate of dipole orientation in poled films of a side-chain polymer based on poly(methyl methacrylate) with an oxynitrostilbene side group attached via an aliphatic spacer. In particular, we used the dichroism of the absorption band assigned to the NO₂ symmetric stretching vibration as a measure of the orientation of the nitrostilbene group. This yields 〈P₂(cos θ)〉, where θ is the angle between the transition dipole moment and the symmetry axis of the sample. We then used a Langevin-type model to calculate 〈cos θ〉 from 〈P₂(cos θ)〉 and concluded that values of 〈cos θ〉 as high as 0.5 are being achieved. We find, however, that 〈P₂(cos θ)〉 obtained by comparison of normal incidence measurements on unpoled and poled films does not agree with 〈P₂(cos θ)〉 obtained from tilted-film measurements on poled films. We propose that this discrepancy arises because the local field in a poled sample affects the polarizability of the molecules. To confirm this, we applied high fields to unpoled films in the infrared spectrometer at room temperature. The results clearly show a large reduction in absorbance with the applied field when the infrared electric field vector has a component perpendicular to the plane of the film. A major error therefore occurs as a consequence of the internal field in poled samples if 〈P₂(cos θ)〉 is derived from the tilted film measurements, whereas 〈P₂(cos θ)〉 derived from normal incidence measurements is essentially correct. © 1994 John Wiley & Sons, Inc.     

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.     

Multi-step Sequential and Accelerated Sequential Methodologies for a Replicable Linear Model

Abstract

We consider a sequence of observable p-dimensional independent and identically distributed N _p (A θ, Σ) observations X _l, X ₂, … where A _{p  ×  q} is a known matrix with rank q(1<q<p), θ _{q  × 1} consists of unknown regression parameters, and Σ is an unknown positive definite matrix. Chatterjee (Chatterjee, S.K. Multi-step sequential procedures for a replicable linear model with correlated variables. In Probability Statistics and Design of Experiments (Proceedings of the R. C. Bose Symposium 1988); Bahadur R.R., Ed.; Wiley Eastern: New Delhi, 1990; 217–226) gave a multi-step sequential methodology to construct a fixed-size confidence region for θ. This methodology used the largest characteristic root of an appropriate conditional dispersion matrix. We propose new multi-step sequential and accelerated methodologies, which are more efficient in lowering the oversampling rate substantially. Accelerated versions of these estimation techniques further reduce sampling operations and these preserve crucial properties associated with the original multi-step methodology of Chatterjee (1988). A substantial part of this investigation includes thorough comparisons of all available methodologies for moderate sample sizes. Some limited robustness issues are addressed. An illustration with realistic data on scores of college students on the college level examination program (CLEP) is also included.     

Surface Modification of Fluorinated Polymers by Microwave Plasmas

We developed a new plasma treating method, incorporating the use of microwaves generated by an electronic cooking range. Using this method, polytetrafluorethylene (PTFE) and a copolymer of tetrafluoroethylene and hexafluoropropylene (FEP) were treated. Dialkylphthalates (DAP) were used as the standard liquids of contact angle measurements for evaluation of the wetting properties of plasma treated polymers. The components of surface tension (γ_L) due to the dispersion force (γ^d _L) and the polar force (γ^P _L) of DAP were calculated by Fowkes' equation from the contact angles (θ) on polypropylene. After plasma treatment cos θ of several standard liquids on PTFE and FEP increased. The linear relationship between γ_L(1 + cos θ)/(γ^d _L)^½ and (γ^P _L/γ^P _L)^½ was verified. γ_s and γ^d _s and γ^d _s of the plasma treated PTFE and FEP also increased. From the results of ESCA analysis, it was found that a significant amount of oxygen was introduced to the polymer surface by the plasma treatment. Peel strengths of a pressure sensitive adhesive bonded to PTFE and FEP increased approximately two-to threefold if the plasma treatment was used prior to bonding.     

Two-stage and sequential estimation of parameter N of binomial distribution when p is known

In this article, we propose two-stage and purely sequential procedures to construct bounded width and prescribed proportional closeness confidence intervals for the unknown parameter N of B(N,p) distribution where the parameter p is assumed to be known. The exact distributions of the stopping variables and the estimators of N at stopping are derived for all cases. The coverage probabilities of the proposed interval estimator are computed exactly and are shown to be nearly the same as the prescribed level.     

Contribution of interferometry and mechanical parameters in evaluating molecular orientation for drawn polyester

Mechanical and structural parameters related to the optical properties of polyester (PET) (woollen type) fibres drawn at room temperature have been investigated. The changes in the strain were evaluated to obtain the molecular orientation factors 〈P₂(cos θ)〉 and 〈P₄(cos θ)〉. From the optical orientation, the values of f₂(θ), f₄(θ) and f₆(θ) orientation parameters were calculated. The structure and properties of oriented PET have been studied in the light of the rubber elasticity theory. The dielectric constant, magnetic susceptibility, number N′ of chains between crosslinks per unit volume, optical configuration parameter and the segment anisotropy, were among the calculated parameters. The results of the extension were used to calculate the shrinkage factor. Relationships between the calculated parameters and the draw ratios, together with micro-interferograms, are given for illustration. The present study throws light on how the applied stress changes the molecular orientation factors and the structural parameters. © 1999 Society of Chemical Industry     

Théorie et application de la distribution normale de θ1/3 aux temps de séjour des fluides dans les réservoirs agités en régime continu

Statistical models of residence time distributions in ideal stirred tanks in series are well approximated by a normal distribution of θ^1/2, θ^1/2 or θ according as the number of tanks is > 15, > 50 and very large respectively. For one stirred tank, the normal distribution of θ^1/3 approximates well the residence time distribution for both ideal and nonideal mixing. For ideal mixing, the mean x_m and standard deviation σ of the θ^1/2 distribution assume values of 8/9 and 1/3 respectively. For nonideal mixing, x_m is < 8/9 while σ is > 1/3. At constant fluid viscosities, when the rotational speed of the agitator increases, x_m increases gradually to reach its upper limiting value of 8/9 whilst σ decreases to reach its lower limiting value of 1/3. At fixed rotational speeds, if the viscosity increases, x_m decreases whilst σ increases. A simple mixing efficiency criteria derived from the normal distribution of θ^1/3 is proposed and is compatible with Danckwerts' segregation concept.