Optimal Stopping for I.I.D. Random Variables Based on the Sequential Information of the Location of Relative Records Only

Abstract

Let X _j , j = 1,…, n be independent and identically distributed random variables. Like in the classical secretary problem, the optimal stopper only observes Y _j  = 1, if X _j is a (relative) record, and Y _j  = 0, otherwise. The actual X _j values are not revealed. The goal is to maximize the expected X value at which one stops. We show that the optimal number of observations one should skip before considering stopping depends heavily on the underlying distribution.     

Pseudo-prophet Inequalities in Average-Optimal Stopping

Abstract

This note considers the average-optimal expected return of two players observing independent random variables X ₁, … , X _n , whose distributions are generated at random. One player, the pseudo prophet, knows the distributions prior to observing the random variables. The other player, the gambler, has no such foresight. Sharp difference and ratio comparisons of the two players' optimal expected returns are given. The key step in the proof is a reduction to a classical prophet inequality for i.i.d. random variables proved by Hill and Kertz (Hill, T.P.; Kertz, R.P. Comparisons of stop rule and supremum expectations of i.i.d. random variables. Ann. Probab. 1982, 10 (2), 336–345).     

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     

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.     

DRYING OF CARROTS IN A FLUIDIZED BED. I. EFFECTS OF DRYING CONDITIONS AND MODELLING

ABSTRACT

Drying curves were determined in a mechanically agitated fluidized bed dryer, at temperatures between 70°C and 160°C, air velocities between 1.1 m/s and 2.2 m/s and stirring rates between 30 rpm and 70 rpm for batch drying of 3 kg lots of carrot slices, measuring the moisture content and shrinking of the particles in time. This was complemented by a study of the rate and degree of swelling of dried carrot particles in water between 20 and 75°C. Drying kinetics were modeled by Fick's second law, for which an optimal agreement with the experimental data was obtained when the effective diffusivity (D _e ) was determined by a correlation based on the air velocity (v), the air temperature (T) and the dimensional moisture content of the carrot particles (X/X _o ). Loss of carotenes is minimized when dehydration is carried out at about 130°C with a drying time below 12 min.     

Polymer-Solid Interface Connectivity and Adhesion: Design of a Bio-Based Pressure Sensitive Adhesive

Adhesion at polymer-solid interfaces was explored for a new bio-based pressure sensitive adhesive (PSA) in terms of sticker groups, φ_X, on the polymer phase, receptor groups, φ_Y, on the solid surface, and the bond strength of the sticker-receptor X-Y acid-base interaction, χ. The polymer-solid interface restructuring models of Gong and Lee et al. were extended with new percolation models of entanglements and interface strength to determine the optimal sticker group concentration, φ?_X. For the general case where φ_Y and χ are constant, it is predicted that when φ_X < φ?_X, that the critical peel energy behaves as G_1c ～ φ_X/φ?_X and the locus of failure is adhesive between the polymer and the solid. However, when φ_X > φ?_X, failure occurs cohesively in a polymer-polymer interface adjacent to the solid and the strength decreases as G_1c ～ φ?_X/φ_X. The switch from adhesive to cohesive failure can be understood in terms of the changes in the chain conformations of the adhered chains and their decreasing interpenetration, X_i, with the bulk chains, via X_i ～ 1/r, where r = χφ_Xφ_Y. The optimal value of φ_X which maximizes the adhesion and determines the mode of failure is given by φ?_X ≈ 0.129/C_∝, and for typical values of the characteristic ratio C_∝ in the range 7–20, φ?_X ≈ 1% mole fraction, corresponding to about 2 sticker groups per entanglement molecular weight, M_e. This result was verified for a bio-based PSA synthesized from an acrylated high oleic fatty acid, which was copolymerized with maleic anhydride as the sticker group. The observed behavior is counterintuitive to the current wisdom for the effect of acid-based interactions on adhesion, where the strength is expected to increase with the number of X-Y contacts. The surprisingly low value of φ?_X ≈ 1% sticker groups which maximizes the adhesion strength can now be readily calculated using the percolation model of entanglements and fracture.     

Esquentian of the mean by three stage procedure

Let X₁, X₂... be i.i.d. observations with mean μ and variance σ², and suppose that given a sample of size n one wishes to estimate μ by X_n, subject to the loss function L_n= A²(X_n-μ)²+n. When σ² is unknown, the optimal sample size n₀ for minimizing the risk R_n = E{L_n} cannotbe found. To resolve this problem, Robbins (1959) proposed a sequential procedure whenX_i is normally distributed, and then the asymptotic properties have been extensively studied. In this paper, we apply Hall's three stage procedure to the problem without specifying the distribution of X_i and derive the second order asymptotic expansions of the expected sample size and the risk.     

Distributional Properties of CUSUM Stopping Times

Abstract

Let S _n be the partial sum of i.i.d. random variables X ₁,X ₂,…, and let N be the usual CUSUM stopping time based on S _n . Under suitable conditions we determine ψ(α,β) = E exp(α S _N  ? β N), where β > 0 and α is a suitable number. The given formula can be used to study the distributional properties of N, S _N , and S _N  ? h. Because the CUSUM based on maxima is reducible to N, the formula can be used to obtain the distributional properties of the maximal process as well. Several examples are discussed, and certain applications are shown in the so called trading securities. The formulas can also be used to study the distributional properties of a symmetric two-sided CUSUM.     

Solubility of hydrogen sulphide,dimethyl ether,methyl chloride and sulphur dioxide in liquids. The prediction of solubility of all gases

The solubilities of hydrogen sulphide, dimethyl ether, methyl chloride, and sulphur dioxide expressed as the mole ratio, X_gas, at different temperatures and 1 bar^a are given for a number of liquid compounds of types commonly used in operational chemistry. Representative curves showing the variations of X_gas, and N_gas (mole fraction), with pressure, p_gas, are discussed with reference to Henry's law and Raoult's law. With the exception of hydrogen halides and ammonia, the X_gas (at a stated t° and p_gas) for water is much less than for any other liquid reported as examined. The primary factor determining the X_gas value at 1 bar is the boiling point of the gas (He to SO₂) at 1 bar; the Raoult's law X_gas value emerges from this, and places the gases in the same order as the overlapping ranges of real X_gas values. The actual X_gas value within a particular range for a chosen gas is, for a chosen liquid, seemingly determined by acid-base functions as defined herein. The prediction of the X_gas value is discussed in these terms.     

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.     

Optimization of nonlinear process based on sequential extreme learning machine

In this paper, a new approach to the optimal control with constraints is proposed to achieve a desired end product quality for nonlinear processes based on new kernel extreme learning machine (KELM). The contributions of the paper are as follows: (1) In existing ILC algorithm, the model was built only between manipulated input variables U and output variables Y without considering the state variables. However, the states variables X_state are important in the industrial processes, which are usually constrained. In this paper, the variables are divided into state variables X_state, manipulated input variables U and output Y in the process of modeling. Then ΔU can be obtained by batch-to-batch iterative learning control separately. Kernel algorithm is added to ELM. (2) Constraints of state variables X_state and the input variables U are considered in the current version. PSO is used to solve the optimization problem. (3) Kernel trick is introduced to improve accuracy of ELM modeling. New KELM algorithm is proposed in the current version. The input trajectory for the next batch is accommodated by searching for the optimal value through the error feedback at a minimum cost. The particle swarm optimization algorithm is used to search for the optimal value based on the iterative learning control (ILC). The proposed approach has been shown to be effective and feasible by applying bulk polymerization of the styrene batch process and fused magnesium furnace.     

Effective Two-Stage Estimation for a Linear Function of High-Dimensional Gaussian Means

We consider fixed-size estimation for a linear function of mean vectors from π _i : N _p (μ _i , Σ _i ), i = 1,…, k. The goal of inference is to construct a fixed-span confidence region with required accuracy. We find a new sample-size formulation and propose a two-stage estimation methodology to give the fixed-span confidence region satisfying the probability requirement approximately. We show that the proposed methodology greatly reduces the sample size to enjoy the asymptotic first-order consistency when the dimensionality p is extremely high. With the help of simulation studies, we show that the proposed methodology is still efficient even when p is moderate. We give an actual example to illustrate how it should be done by using the proposed methodology in the inference.     

Wastewater Treatment by Chemically Activated Carbons from Giant Reed: Effect of the Activation Atmosphere on Properties and Adsorptive Behavior

Abstract

Uptake of nickel and benzene from dilute single‐solute solutions, mimicking wastewater with low concentrations of heavy metals or volatile organic compounds, was examined using activated carbons with similar good surface properties (BET surface area of ≈1100 m²/g). They were developed through H₃PO₄ acid activation of giant reed (Arundo donax L.) under flowing air or N₂. The carbons obtained in air proved more effective to capture Ni(II) ions under pre‐established equilibrium conditions. Inversely, the N₂‐derived carbons exhibited a better ability for benzene adsorption. The behavior was related to the smaller total content of acidic/polar surface oxygen functionalities of the carbons developed under N₂ (1.9 meq/g), compared to that of the air‐derived ones (3.3 meq/g). Two‐, three‐parameter models described properly the isotherms, predicting similar maximum adsorption capacities (X_m ) for the investigated systems. The X_m parameter in the Langmuir's model was 0.44 mmol/g for the adsorption of Ni(II) ions on the air‐derived carbons, and 0.45 mmol/g for benzene adsorption on those obtained in N₂. Present results highlight the relevance of the surface chemistry developed upon activation to optimize the performance of activated carbons for wastewater treatment according to the pollutants' nature.     

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.     

On a Type of Probability Stopping Rule for Toxicity Study

Abstract

In early phase cancer clinical trials where toxicity events follow independent and identical Bernoulli distributions indexed by patients, the Bayesian stopping rule has been used for continuous monitoring of toxicity along with an affordable maximum sample size (N). This article studies some properties of an heuristic procedure where the trial will stop at the first time that the posterior probability that the toxicity rate (p) is greater than a threshold (η) is greater than certain probability threshold (τ). Specifically, we study the pattern formed by stopping times and regions, recursive stopping probability computation, and toxicity rate estimation. Some relevant theoretical results are given. The presented results are potentially useful for guiding toxicity clinical trial designs.     

Formula and process optimization of controlled‐release microcapsules prepared using a coordination assembly and the response surface methodology

A model chlorpyrifos microcapsule was prepared using coordination assembly between Fe³⁺ and tannic acid (TA). The influence of independent variables such as the dropping rate of TA (X₁) and Fe³⁺ (X₃), the concentration of TA (X₂) and Fe³⁺ (X₄), and the reaction temperature (X₅) on the encapsulation efficiency (R₁) and release characteristics (R₂) of the microcapsule had been investigated, based on a central composite design with five factors and five levels. The results showed that the main factors influencing R₁ and R₂ were X₄ and X₂, then the interaction between X₂ and X₄, followed by X₅ and X₃. The optimal formula mainly based on higher R₁ and lower R₂ were determined and then tested. The optimized conditions led to an encapsulation efficiency and cumulative release proportion of 97.12% ± 0.72% and 40.07% ± 0.53%, along with the average relative errors of predicted values being 1.78% and ?1.60%, respectively. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016 , 132, 42865.