AN IMPROVEMENT TO THE SSF MODEL FOR PREDICTION OF LIGHT PARTICLE DISPERSION

This study investigates the predicted motion of small, light particles using the stochastic separated flow models proposed by Shuen et al. (1983 Shuen , J. S. , Chen , L. D. , and Faeth , G. M. ( 1983 ). Evaluation of a stochastic model of particle dispersion in a turbulent round jet , AIChE J. , 29 , 167 – 170 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and Lightstone and Raithby (1998 Lightstone , M. F. and Raithby , G. D. ( 1998 ). A stochastic model of particle dispersion in a turbulent gaseous environment , Combust. Flame , 113 , 424 – 441 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Model predictions are compared to the benchmark experiment of Snyder and Lumley (1971 Snyder , W. H. and Lumley , J. L. ( 1971 ). Some measurements of particle velocity autocorrelation functions in a turbulent flow , J. Fluid Mech. , 48 , 41 – 71 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) for particles released from a point source in grid-generated turbulence. It is found that the predictions of particle velocity variance and dispersion exhibit nonphysical waves. This behavior is explored in order to understand its mathematical basis arising from the model formulation. A modification to the model is presented. The modified model is successful in removing the characteristic waves.     

Partial Sequential Tests for Generalized Behrens-Fisher Problem

Abstract

In this article we study the problem of finding a locally most powerful sequentially planned test in a one-sided sequentially planned testing problem for stochastic processes of the exponential class with continuous time parameter, thereby applying the theory of optimal sampling in continuous time (cf. Roters, 1995b Roters , M. ( 1995b ). The Theory of Optimal Sampling in Continuous Time with Applications to Sequentially Planned Decision Procedures, Habilitationsschrift, Universität Trier.  [Google Scholar] 1997 Roters , M. ( 1997 ). The Theory of Optimal Sampling in Continuous Time, Forschungsbericht 97-25, FB IV Mathematik/Informatik, Universit ät Trier. [CSA]  [Google Scholar]). The theory presented here extends the discrete-time results of Berk (1975 Berk , R. H. ( 1975 ). Locally Most Powerful Sequential Tests , Annals of Statistics 3 : 373 – 381 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and Schmitz (1993 Schmitz , N. ( 1993 ). Optimal Sequentially Planned Decision Procedures , Lecture Notes in Statistics 79 , New York : Springer-Verlag .[Crossref] , [Google Scholar]).     

EXPERIMENTAL MEASUREMENTS OF FOAMING TENDENCIES IN AQUEOUS GAS SWEETENING SOLUTIONS CONTAINING MDEA OVER A TEMPERATURE RANGE OF 297–358 K AND A PRESSURE RANGE OF 101–500 KPA

This work presents experimental measurements of foaming tendencies and break times for 50 wt% aqueous solutions of MDEA in contact with nitrogen, methane, and ethane gases from atmospheric pressure to 500 kPa and temperatures from 297 to 358 K. The effect of various contaminants including methanol, HEP (1,4- bis-2-hydroxyethyl piperazine), hexane, and carboxylic acids ranging from formic to dodecanoic acid was investigated. Only those systems containing carboxylic acids heavier than valeric acid exhibited foaming. Foaming tendency in systems containing carboxylic acids was worsened by the addition of methanol and HEP, although the difference was moderate. Foaming was shown to be worse at lower pressure and at higher temperature. Foaming in MDEA systems was shown to be substantially worse than that measured previously by McCarthy and Trebble (1996 McCarthy , J. and Trebble , M. A. ( 1996 ). An experimental investigation into the foaming tendency of diethanolamine gas sweetening solutions , Chemi. Eng. Commun. , 144 , 159 – 171 [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for aqueous systems of DEA.     

Multistage point estimation methodologies for a negative exponential location under a modified linex loss function: Illustrations with infant mortality and bone marrow data

We have designed Stein-type (Stein, 1945 Stein, C. (1945). A Two Sample Test for a Linear Hypothesis Whose Power Is Independent of the Variance, Annals of Mathematical Statistics 16: 243–258.[Crossref] , [Google Scholar], Annals of Mathematical Statistics) two-stage, modified two-stage (Mukhopadhyay and Duggan, 1997 Mukhopadhyay, N. and Duggan, W. T. (1997). Can a Two-Stage Procedure Enjoy Second Order Properties? Sankhya, Series A 59: 435–448. [Google Scholar], Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965 Chow, Y. S. and Robbins, H. (1965). On the Asymptotic Theory of Fixed Width Sequential Confidence Intervals for the Mean, Annals of Mathematical Statistics 36: 457–462.[Crossref] , [Google Scholar], Annals of Mathematical Statistics) to estimate an unknown location parameter of a negative exponential distribution having an unknown scale parameter under a newly defined and modified Linex loss function. We aim at controlling the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, ω, and we emphasize both asymptotic first-order and asymptotic second-order properties for the modified two-stage and purely sequential estimation strategies. In developing asymptotic second-order properties for the modified two-stage methodology, we have heavily relied upon basic ideas rooted in Mukhopadhyay and Duggan (1997 Mukhopadhyay, N. and Duggan, W. T. (1997). Can a Two-Stage Procedure Enjoy Second Order Properties? Sankhya, Series A 59: 435–448. [Google Scholar]). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon nonlinear renewal theory (Lai and Siegmund, 1977 Lai, T. L. and Siegmund, D. (1977). A Nonlinear Renewal Theory with Applications to Sequential Analysis I, Annals of Statistics 5: 946–954.[Crossref], [Web of Science ®] , [Google Scholar], 1979 Lai, T. L. and Siegmund, D. (1979). A Nonlinear Renewal Theory with Applications to Sequential Analysis II, Annals of Statistics 7: 60–76.[Crossref], [Web of Science ®] , [Google Scholar], Annals of Statistics; Woodroofe, 1977 Woodroofe, M. (1977). Second Order Approximation for Sequential Point and Interval Estimation, Annals of Statistics 5: 984–995.[Crossref], [Web of Science ®] , [Google Scholar], Annals of Statistics). Then, we take to extensive data analysis carried out via computer simulations when requisite sample sizes range from small to moderate to large. We find that the Stein-type two-stage estimation methodology oversamples significantly and yet the achieved risk is not close to preset goal ω. On the other hand, both modified two-stage and purely sequential estimation strategies perform remarkably well. We have validated their main theoretical first-order and second-order properties through simulated data. The latter methodologies have been illustrated and implemented using two real data sets from health studies, namely, infant mortality data and bone marrow data.     

EVALUATION OF LAGRANGIAN PARTICLE DISPERSION MODELS IN TURBULENT FLOWS

This work evaluates the performance of Lagrangian turbulent particle dispersion models based on the Langevin equation. A family of Langevin models, extensively reported in the open literature, decompose the fluctuating fluid velocity seen by the particle in two components, one correlated with the previous time step and a second one randomly sampled from a Wiener process, i.e., the closure is at the level of the fluid velocity seen by the particle. We will call those models generically the “standard model.” On the other hand, the model proposed by Minier and Peirano (2001 Minier , J. P. and Peirano , E. ( 2001 ). The PDF approach to turbulent polydisperesed two-phase flows , Phys. Rep. , 352 , 1 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) is considered; this approach is based on the probability density function (PDF) and performs the closure at the level of the acceleration of the fluid seen by the particle. The formulation of a Langevin equation model for the increments of fluid velocity seen by the particle allows capturing some underlying physics of particle dispersion in general turbulent flows while keeping simple the mathematical manipulation of the stochastic model, avoiding some pitfalls, and simplifying the derivation of macroscopic relations. The performance of the previous dispersion models is evaluated in the configurations of grid-generated turbulence (Snyder and Lumley, 1971 Snyder , W. H. and Lumley , J. L. ( 1971 ). Some measurements of particle-velocity autocorrelation functions in a turbulent flow , J. Fluid Mech. , 48 , 41 – 71 .[Crossref], [Web of Science ®] , [Google Scholar]; Wells and Stock, 1983 Wells , M. R. and Stock , D. E. ( 1983 ). The effect of crossing trajectories on the dispersion of particles in a turbulent flow , J. Fluid Mech. , 136 , 31 – 62 .[Crossref], [Web of Science ®] , [Google Scholar]), simple shear flow (Hyland et al., 1999), and confined axisymmetric jet flow laden with solids (Hishida and Maeda, 1987 Hishida , K. and Maeda , M. ( 1987 ). Turbulent characteristics of gas-solids two-phase confined jet: Effect of particle density , Jpn. J. Multiph. Flow , 1 , 56 – 69 . [Google Scholar]).     

BUBBLE COMPUTATION,GRANULAR TEMPERATURES,AND REYNOLDS STRESSES

Bubbles were simulated in a two-dimensional fluidized bed with a constant inlet velocity using two computer codes, the IIT code and the MFIX code. The computational results were compared to the Jung et al. (2005 Jung , J. , Gidaspow , D. , and Gamwo , I. K. (2005). Measurement of two kinds of granular temperatures, stresses and dispersion in bubbling beds, Ind. Eng. Chem. Res. , 44, 1329–1341 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) experiments in a thin bubbling bed of 530 μm glass beads. The use of higher order numerics produces better bubble resolution due to smaller numerical diffusion. The computed bubble sizes and their distributions agreed with the experiments. The simulations show that there is no bubble formation for sufficiently elastic particles.

CFD computations and previous experiments show that in the bubbling fluidized beds there exist two random oscillations. The first kind is due to random oscillations of particles and is measured by the conventional granular temperature. The second one is due to motion of bubbles and gives rise to Reynolds type stresses. It is shown that the particle granular temperature is much smaller than the bubble-like granular temperature computed from the average of the normal Reynolds stresses, measured by Cody using a shot noise technique.     

A Bayesian adaptive dose-finding algorithm for balancing individual- and population-level ethics in Phase I clinical trials

Multiple Bayesian adaptive designs have been proposed for Phase I clinical trials since the continual reassessment method (CRM) was proposed by O’Quigley et al. (1990). Focused on dose-finding in cancer studies, the CRM seeks to allocate new patients to an estimated maximum tolerable dose (MTD). Later, Whitehead and Brunier (1995 Whitehead, J. and Brunier, H. (1995). Continual Reassessment Method: Bayesian Decision Procedures for Dose Determining Experiments, Statistics in Medicine 14: 885–893.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) applied Bayesian decision theory to maximize statistical information for the MTD when allocating new patients. The two allocation rules reflect conflicting perspectives. The CRM emphasizes individual-level ethics, whereas the method of Whitehead and Brunier (1995 Whitehead, J. and Brunier, H. (1995). Continual Reassessment Method: Bayesian Decision Procedures for Dose Determining Experiments, Statistics in Medicine 14: 885–893.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) emphasizes population-level ethics. In the design of a Phase I clinical trial to investigate hyperthermic intraperitoneal chemotherapy (HIPEC) we sought to compromise the two perspectives. To this end, we propose a novel dose allocation design referred to as the balanced information gain method. We first decompose the loss function used by Whitehead and Brunier and then modify it with a tuning parameter that allows a trialist to differentially weigh individual- and population-level ethics based on their particular clinical setting. Simulation studies show that the proposed design provides a reasonable compromise between the distribution of the estimated MTD and the distribution of the number of observed adverse events per trial when compared to the two existing methods.     

Nonparametric partially random sequential test under Phase II sampling: An illustration to monitor water samples for arsenic contamination

We introduce a class of nonparametric two-sample tests based on a new partially sequential sampling scheme. Existing partially sequential procedures based on inverse sampling schemes, pioneered by Wolfe (1977 Wolfe, D. A. (1977). On a Class of Partially Sequential Two Sample Test Procedure, Journal of American Statistical Association 72: 202–205.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Orban and Wolfe (1980 Orban, J. and Wolfe, D. A. (1980). Distribution Free Partially Sequential Placement Procedure, Communications in Statistics - Theory &; Methods 9: 883–902.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), are updated in the light of random sequential sampling techniques, proposed by Mukhopadhyay and de Silva (2008 Mukhopadhyay, N. and de Silva, B. M. (2008). Theory and Applications of a New Methodology for the Random Sequential Probability Ratio Test, Statistical Methodology 5: 424–453.[Crossref] , [Google Scholar]). In a quality control setup, the present procedure can be looked upon as a Phase II on-line monitoring with rational subgroups of variable sizes where standards are unknown. We consider a training sample of prefixed size m as Phase I observations and adopt a random sequential sampling in Phase II. We discuss statistical methodologies in detail and provide some asymptotic results. Numerical results based on Monte Carlo are presented to justify asymptotic theory. We computationally investigate the power performances of the proposed test against some fixed alternative. We illustrate our procedure with real data related to water samples for monitoring arsenic contamination. Some concluding remarks along with possible future research problems are offered     

A simulation model for settling tanks with varying cross-sectional area

A quasi-one-dimensional model of the process of continuous sedimentation in clarifier-thickeners (settlers) with variable cross-sectional area is presented. The partial differential equation (PDE) model extends the settler model advanced by Bürger et al. (2013 Bürger, R., Diehl, S., Farås, S., Nopens, I., and Torfs, E. (2013). A consistent modelling methodology for secondary settling tanks: A reliable numerical method, Water Sci. Technol. 68(1), 192–208.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which assumes a constant cross section. A reliable numerical method that handles the special features of the nonlinear PDE is presented along with an advantageous time-step condition for continuous and batch sedimentation under the condition of a variable cross-sectional area. Simulations of continuous sedimentation show the effect of change of cross-sectional area in the concentration inside the vessel and in the underflow. Simulations of batch settling in cones illustrate the versatility of the numerical scheme to include a vertex, where the area shrinks to zero.     

Adhesive Contact of a Compliant Sphere to an Elastic Coated Substrate: The Thin Film Limit

Experimental results for adhesive contacts on substrates coated with elastomeric thin films have recently been obtained by Tardivat and Léger [1 Tardivat , C. and Léger L. , J. Adhesion Sci. Technol. 15 , 1055 – 1078 ( 2001 ).[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] by the so-called Johnson-Kendall-Roberts (JKR) test, which provides both adhesion energy and elastic modulus. These data show that on substrates coated with thin films the adhesion and effective elastic modulus of the sphere depend upon the film thickness. In keeping with the experimental conditions, we try to interpret these data using a simple model [4 Barthel , E. and Perriot , A. , J. Phys. D: Appl. Phys. 40, 1059–1067 (2007).[Crossref] , [Google Scholar]] in the thin film limit, i.e., when the film thickness is small compared with the contact radius. Although the film does impact the local crack tip stress field, we show that no effect on the macroscopic contact variables is expected for the adhesion to coated substrates in such confined geometries. The deviations from the experimental results are ascribed to the idealized contact boundary conditions assumed in the model.     

Analytical expression for the rotational friction coefficient of DLCA aggregates over the entire Knudsen regime

We apply a self-consistent field method (Corson et al. 2017c Corson, J., Mulholland, G. W., and Zachariah, M. R. (2017c). Calculating the Rotational Friction Coefficient of Fractal Aerosol Particles in the Transition Regime using Extended Kirkwood-Riseman Theory. Phys. Rev. E, 96(1):013110.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to calculate the rotational friction coefficient for fractal aerosol particles in the transition flow regime. Our method considers hydrodynamic interactions between spheres in a rotating aggregate due to the linear velocities of the spheres. Results are consistent with electro-optical measurements of soot alignment. Calculated rotational friction coefficients are also in good agreement with continuum and free molecule results in the limits of small (Kn = 0.01) and large (Kn = 100) primary sphere Knudsen numbers. As we previously demonstrated (Corson et al. 2017b Corson, J., Mulholland, G. W., and Zachariah, M. R. (2017b). Analytical Expression for the Friction Coefficient of DLCA Aggregates based on Extended Kirkwood–Riseman Theory. Aerosol Sci. Technol., 51(6):766–777.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for the translational friction coefficient, the rotational friction coefficient approaches the continuum limit as either the primary sphere size and the number of primary spheres increases. We apply our results to develop an analytical expression Equation (26) for the rotational friction coefficient as a function of the primary sphere size and number of primary spheres. One important finding is that the ratio of the translation to rotational diffusion times is nearly independent of cluster size. We include an extension of previous scaling analysis for aerosol aggregates to include rotational motion.

Copyright © 2018 American Association for Aerosol Research     

MASS TRANSFER WITH HETEROGENEOUS CHEMICAL REACTION OF THE FIRST ORDER IN FALKNER-SKAN FLOW OF A POWER-LAW FLUID

ABSTRACT

Mass transfer with a heterogeneous chemical reaction of the first order in a Falkner-Skan flow of non-Newtonian power-law fluid is investigated. This problem is solved by the method of Laplace transform following the approach suggested by Apelblat (1980 Apelblat , A. ( 1980 ). Chem. Eng. J. , 19 , 19 – 37 . [CSA] [CROSSREF]  [Google Scholar]). The solution is obtained in a closed analytical form.     

ARL-unbiased geometric and CCCG control charts

In order to control increases and decreases in a parameter in a timely fashion, a chart should be set in such way that the average run length (ARL) curve attains a maximum in the in-control situation.

This article proposes not only a geometric (or cumulative count of conforming, CCC) chart but also a CCC chart under group inspection (CCC_G) for which all out-of-control ARL values are smaller than the in-control ARL and thus provides an improvement on the designs described in L. Zhang et al. (2004 Zhang, L., Govindaraju, K., Bebbington, M. and Lai, C. D. (2004). On the Statistical Design of Geometric Control Charts, Quality Technology &; Quantitative Management 2: 233–243.[Taylor &; Francis Online] , [Google Scholar]) and C. W. Zhang et al. (2012 Zhang, C. W., Xie, M., and Jin, T. (2012). An Improved Self-starting Cumulative Count of Conforming Chart for Monitoring High-Quality Processes under Group Inspection, International Journal of Production Research 50: 7026–7043.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). Moreover, by exploring the notions of uniformly most powerful unbiased tests with randomization probabilities, we are able not only to eliminate the bias of the ARL function of the geometric charts but also to bring their in-control ARL exactly to a prespecified value.

Instructive examples are provided to illustrate the e?ciency of the proposed charts.     

PREDICTION OF PRESSURE DROP IN A VENTURI SCRUBBER: EFFECT OF LIQUID FILM AND RE-ENTRAINMENT

A mathematical model has been developed for prediction of pressure drop in a Venturi scrubber. This model includes the effect of the amount of liquid film and re-entrainment of liquid droplets from liquid film. The result of the present model is compared with experimental data of Viswanathan et al. (1985) Viswanathan, S., Gnyp, W. A. and Pierre, St., C. 1985. Annular flow pressure drop model for pease-Anthony-type Venturi scrubbers. AIChE. J., 31(12): 1947–1958.  [Google Scholar] as well as with the other models (Viswanathan et al., 1985 Viswanathan, S., Gnyp, W. A. and Pierre, St., C. 1985. Annular flow pressure drop model for pease-Anthony-type Venturi scrubbers. AIChE. J., 31(12): 1947–1958.  [Google Scholar]; Boll, 1973 Boll, R. H. 1973. Particle collection and pressure drop in Venturi scrubbers. Ind. Eng. Chem. Fundam., 12: 40[Crossref], [Web of Science ®] , [Google Scholar]). Results of this study indicate that at high liquid to gas ratios prediction of pressure drop can be improved by considering re-entrainment and liquid film effects. Also the effects of gas velocity and liquid to gas ratio were investigated on the rate of droplet re-entrainment and pressure drop.     

Asymptotically optimal procedures in multivariate Bayesian sequential estimation

In this article, we consider the problem of multivariate Bayesian sequential estimation of the unknown mean vector. We propose a robust sequential procedure without using the prior information or any auxiliary data, which is similar to multivariate non-Bayesian sequential estimation by M. Ghosh et al. (1976 Ghosh, M., Sinha, B. K., and Mukhopadhyay, N. (1976). Multivariate Sequential Point Estimation, Journal of Multivariate Analysis 6: 281–294.[Crossref], [Web of Science ®] , [Google Scholar]). The proposed procedure, depending only on the present data but not on its distribution, is shown to be asymptotically as well as or better than the optimal fixed-sample-size procedures for the arbitrary distributions and asymptotically pointwise optimal and asymptotically optimal for multivariate exponential family with a large class of prior distributions.     

FLUIDIZED BED PRESSURE DROP IN PROMOTED GAS-SOLID BEDS WITH COAXIAL ROD,DISK, AND BLADE TYPE PROMOTERS

Bed pressure drop equations have been formulated for gas-solid fluidized beds with different types of promoters using Ergun's equation (Ergun, 1952 Ergun, S. 1952. Fluid flow through packed columns. Chem. Eng. Prog., 48(2): 89[Web of Science ®] , [Google Scholar]) and experimental data. Four rod promoters, seven disk promoters, along with one blade promoter were used in beds supported on five different distributors with open areas of 12.9%, 8.96%, 5.74%, 3.23%, and 1.43% of the column section. The predicted values of bed pressure drop using a modified (i.e., modified numerical constant) Burke-Plummer (Burke and Plummer, 1928 Burke, F. E. and Plummer, W. B. 1928. Gas flow through packed columns. Ind. Eng. Chem., 20: 1196[Crossref] , [Google Scholar]) equation were compared with the corresponding experimental as well as the respective values obtained with the help of Kumar et al. (submitted) and traditional gas-solid fluidized bed equations.     

Experimental study and hydrodynamic modeling of countercurrent liquid–solid system with batch liquid

It is known that a transient effluent outlet concentration is obtained with a batch of adsorbent solids in any operation. A preferred steady state outlet concentration can be achieved with a continuous flow of solids. In the present work, information on pressure profiles, the total pressure drop across the column and holdup of solids are experimentally obtained for various solid flow rates, particle sizes and densities in a countercurrent liquid–solid system. These experimental results are compared with the prediction obtained using a phenomenological model containing continuity and momentum balance equations. The dominant drag force term was expressed in terms of various drag equations. The drag expression developed by Foscolo et al. (1983 Foscolo, P. U., Gibilaro, L. G., and Waldram, S. P. (1983). A unified model for particulate expansion of fluidized beds and flow in fixed porous media, Chem. Eng. Sci., 38(8), 1251–1260.[Crossref], [Web of Science ®] , [Google Scholar]) could predict the axial profiles of pressure drop and holdup, and the effect of various parameters on total pressure drop and solid holdup most satisfactorily.     

Two-stage procedures for the bounded risk point estimation of the parameter and hazard rate in two families of distributions

Two families of distributions are considered that cover a large number of probability distributions useful in investigations involving reliability studies and survival analyses. The problem of bounded risk point estimation of the parameter and hazard rate function of the two families of distribution is handled. Motivated by Mukhopadhyay and Pepe (2006 Mukhopadhyay, N. and Pepe, W. (2006). Exact Bounded Risk Estimation When the Terminal Sample Size and Estimator Are Dependent: The Exponential Case, Sequential Analysis 25: 85–101.[Taylor &; Francis Online] , [Google Scholar]), Roughani and Mahmoudi (2015 Roughani, G. and Mahmoudi, E. (2015). Exact Risk Evaluation of the Two-Stage Estimation of the Gamma Scale Parameter under Bounded Risk Constraint, Sequential Analysis 34: 387–405.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Mahmoudi and Lalehzari (2017 Mahmoudi, E. and Lalehzari, R. (2017). Bounded Risk Estimation of the Hazard Rate Function of the Exponential Distribution: Two-Stage Procedure, Sequential Analysis 36: 38–54.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), two-stage procedures are developed based on the maximum likelihood estimator (MLE) as well as uniformly minimum variance unbiased estimator (UMVUE). The estimation problem based on the minimum mean square estimator (MMSE) is also considered. We establish that the MMSE of the parameter and hazard rate provides a smaller risk.     

Asymptotic exponentiality of the first exit time of the Shiryaev–Roberts diffusion with constant positive drift

We consider the first exit time of a Shiryaev–Roberts diffusion with constant positive drift from the interval [0,A] with A>0 fixed. We show that the moment generating function (Laplace transform) of a suitably standardized version of the first exit time converges to that of the unit-mean exponential distribution as A→+∞. The proof is explicit in that the moment generating function of the first exit time is first expressed analytically and in a closed form and then the desired limit as A→+∞ is evaluated directly. The result is of importance in the area of quickest change-point detection, and its discrete-time counterpart has been previously established—although in a different manner—by Pollak and Tartakovsky (2009a Pollak, M. and Tartakovsky, A. G. (2009a). Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection, Theory of Probability and Its Applications 53: 430–442.[Crossref], [Web of Science ®] , [Google Scholar]).