A heuristic approach for near optimal truncated sequential test of exponential distribution

Abstract

Optimal truncated sequential test of the exponential distribution is studied in this paper. Definitions and relative concepts of optimal truncated sequential test are discussed. A heuristic approach, sample space sorting method (SSSM) and the procedures are established to solve a near optimal truncated sequential test. Comparing with the test plans provided by international standard IEC 61124 (2012 IEC 61124 (2012). Reliability Testing - Compliance Tests for Constant Failure Rate and Constant Failure Intensity, International Electrotechnical Commission, Geneva. [Google Scholar]) and Russian national standard GOST R 27.402 (1995 GOST R 27.402 (1995). Dependability Techniques-Compliance Test Plans for Mean Operating Time to Failure or between Failures-Part 1: Exponential Case, Russian National Standard, Moscow (in Russian). [Google Scholar]), the results show that the near optimal tests proposed in this paper can keep the error probabilities almost equivalence to the nominal error levels and save more synthetical expected test times and maximum expected test times simultaneously.     

Necessary and sufficient conditions for bounded stopping time of sequential bayes tests in one parameter exponential families1

The problem of sequently testing one-sided hypotheses about the parameter in a one-parameter exponential family, continuous with respect to Lebesgue measure, is consiered in a Bayesian framework. The paper gives a simple necessary and sufficient condition for the Bayes sampling rule to be bounded. The risk fucntion is taken to be a constant times the number of observations plus a weighted probability of error. The sufficient condition for boundness is generalized to other risk functions as wells.     

Sequential testing of a Wiener process with costly observations

We consider the sequential testing of two simple hypotheses for the drift of a Brownian motion when each observation of the underlying process is associated with a positive cost. In this setting where continuous monitoring of the underlying process is not feasible, the question is not only whether to stop or to continue at a given observation time but also, if continuing, how to distribute the next observation time. Adopting a Bayesian methodology, we show that the value function can be characterized as the unique fixed point of an associated operator and that it can be constructed using an iterative scheme. Moreover, the optimal sequential distribution of observation times can be described in terms of the fixed point.     

Sequential nonparametric procedures for testing the equality of two lifetime distributions based on progressively censored data

Many nonparametric tests have been formulated to test the equality of two lifetime distributions (e.g., Wilcoxon rank-sum test and sign test), and some of these nonparametric tests can be applied to censored samples (e.g., precedence-type tests and log-rank test). In this article, we study several sequential nonparametric test procedures based on the precedence-type statistics when the lifetime data are observed sequentially and one of the two samples is progressively Type-II censored. Exact power function under Lehmann alternatives is derived. Monte Carlo simulation is used to evaluate the performances of these sequential test procedures. Conclusions and recommendations are provided based on these results.     

A Linear Programming Approach to Sequential Hypothesis Testing

Abstract

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. This result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function, with respect to these multipliers, coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and can be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process.     

Sensitivity of the test risk with respect to contamination

In a framework of testing statistical hypotheses the sensitivity of the test error probabilities is studied. The Edgeworth approximations of proposed characteristics of this sensitivity are derived and specialized for the local alternative setting. Results, similar to Stein's lemma and Chernoff's theorem, describing the convergence rate of test risk sensitivity characteristic is given, too. An admissible character of contamination is also discussed.     

Sequential procedures for comparing several medical treatments

The use of sequential methods in clinical trials allows inferior treatments to be eliminated early. From an ethical point of view, the advantages are substantial. However, early stopping induces estimation bias and a deterioration in precision because of reduced sample sizes. This paper considers the problem of determining which of k ≥ 2 treatments with Bernoulli responses has the highest probability of success. Two sequential procedures are investigated and compared with a fixed—sample procedure. Various properties are derived and illustrated for the cases k =2,3 and 5. It is shown that the sequential procedures can achieve a pattern of error probabilities equivalent to the fixed—sample procedure for a much lower level of expected successes lost. Approximations for the bias and standard deviation of estimators of treatment differences are obtained by using results about the distribution of stopping times for a normal process.     

Recursive M-tests for the constancy of multivriate regression relationships over time

For a general rnultivariate linear model, based on a general class of recursive M-estimators of regression parameters and allied M-statistics, some testing procedures for a possible change in the regression relationships occurring at an unknown time point are considered,. The (asymptotic) theory of the proposed tests rests on some invariance principles for some recursive M-estimators and related (residual) M-statistics, and these are studied too.     

On asymptotically Risk-effivient sequential versions of generalized U-statistics

Sequential point estimation of means of generalized U-statistics is considered. Based on a well defined stopping rule, the proposed sequential (estimation) procedure is shown to be asymptotically (first order) risk-efficient. Asymptotic distributions of sequential generalized U-statistics and the allied stopping times are also studied.     

Odds ratio-based group sequential analysis for joint binomial and inverse binomial sampling

We consider two treatment comparison in a clinical trial setup where the responses from one treatment are a priori known for a fixed number of individuals and patients are allocated in a groupsequential way for the other treatment using inverse sampling. We consider the odds ratio as the measure of treatment difference at the end of each group, where for the calculation of odds ratios we consider the full data on one treatment, which are a priori known, and the available data on the other treatment up to that point of time. We have calculated the optimal number of index subjects for inverse sampling and examine the effects of different type I and type II error spending functions on group sequential testing in this context. The methodology is illustrated by using a real data set.     

Explicit solutions of the Bayes sequential estimation problem for a time-transformed exponential distribution

A problem of Bayesian time-sequential estimation of an unknown parameter of a time-transformed exponential distribution is considered. It is supposed that the cost of observing the process is the sum of an increasing function of time and a linear function of the number of observations. Under some general assumptions concerning the loss function associated with the error of estimation, the optimal stopping time is derived using the free-boundary method. As an example, a solution of the problem considered is given in the case of a weighted precautionary loss function.     

A general framework for optimal stopping problems associated with multivariate point processes,and applications

A general framework for optiinal stopping problems associated with multivariate point processes is presented. Such problems arise. for example. in finding optimal selection/detection strategies in discovering new species, error detection and job search problems. The optimal stopping problem is solved in a finite and an infinite horizon setting. and various special models are coilsidered. It turns out that in regenerative type of modlels where the interarrival time distribtutions have properties known from reliability theory. such as NBUE, IFR and DFR, the optimal stopping times have appealirig properties.     

Optimal Sequential Tests for Two Simple Hypotheses

Abstract

A general problem of testing two simple hypotheses about the distribution of a discrete-time stochastic process is considered. The main goal is to minimize an average sample number over all sequential tests whose error probabilities do not exceed some prescribed levels. As a criterion of minimization, the average sample number under a third hypothesis is used (modified Kiefer–Weiss problem). For a class of sequential testing problems, the structure of optimal sequential tests is characterized. An application to the Kiefer–Weiss problem for discrete-time stochastic processes is proposed. As another application, the structure of Bayes sequential tests for two composite hypotheses, with a fixed cost per observation, is given. The results are also applied for finding optimal sequential tests for discrete-time Markov processes. In a particular case of testing two simple hypotheses about a location parameter of an autoregressive process of order 1, it is shown that the sequential probability ratio test has the Wald–Wolfowitz optimality property.     

On sequentially adaptive asymptotically efficient rank statistics

For the simple regression model ( containing the two—sample location model as a special case ), adaptive ( linear ) rank statistics arising in the context of ( asymtotically) efficient testing and estimation procedures are considered. An orthonormal system based on the classical Legendre polynomial system is incorporated in the adaptive determination of the score generating function, and the proposed sequential procedure is based on a suitably posed stopping rule. Various properties of this sequentailly adaptive procedure and the allied stopping rule are studied. Asymptotic linearity results ( in a shift or regression parameter ) of linear rank statistics are studied with special reference to the Legendre polynomial system and some improved rates of convergence are estabilished in this context.     

A laser irradiation disc test for fracture testing of refractory fine ceramics

The critical stress intensity factor, tensile strength and crack stability were analysed for a zirconia refractory by parameter identification based on a laser irradiation test and a finite element simulation. Furthermore, the results for the specific fracture energy were determined for different assumed cohesive behaviours. The tests were carried out on notched discs that were irradiated at their centres. During the tests, temperatures are recorded by a thermo vision camera and the crack propagation by an acoustic emission recorder. Acoustic emission allowed for the determination of the onset of crack initiation (time t₁) and unstable crack propagation (time t₂). A finite element model representing the geometry of the sample was built to determine the fracture mechanical parameters from t₁ and t₂. This method is believed to be favourable for rather brittle refractory ceramics, whereas for less brittle materials, a wedge splitting procedure according to Tschegg is considered more favourable.     

An accelerated bias-corrected sequential estimation for the mean of NEF-PVF distributions

An accelerated sequential procedure for estimating the mean p in the class of the natural exponential family of distributions having power variance function (NEF-PVF), is proposed. The accelerated procedure is conducted under a combined loss of weighted squared estimation error and sarnpling cost. A class of bias-corrected estimators, which are natural for this suggested accelerated sampling scheme is proposed. The asymptotic properties of the suggested estimators are provided. In particular, one realizes the impact of acceleration on the regret and the tradeoff between bias reduction and regret reduction.     

太阳辐射得热量测试台测量不确定度和误差分析

一切测量结果都具有不确定度及误差。本文通过对太阳辐射得热量测试台进行不确定度和误差分析,得到各个参数的不确定度及测量参数对测试结果的影响程度,其中风机盘管换热量的不确定度最大,而风机盘管进出口水温对测试结果误差影响最大。分析结果表明：整个测试台95%的测试误差在7.6%以内,最大测试误差约为110 W,测试台能满足测试精度要求。     

A new test methodology for studying the response of walls to real fire environments

A new test methodology was developed to investigate the response of walls, partitions, and in-wall systems exposed to real fires. The apparatus includes a 3.5 m long, 2.3 m wide, and 2.3 m high fire compartment within a standard sea container. A wall specimen measuring up to 1.8 m wide, 1.8 m tall, and 0.3 m deep is mounted in a steel frame at one end of the fire compartment. Fire exposures to the wall specimen evolve over time depending on the fuel load and ventilation configuration. Gas temperatures and heat flux were characterized for five different fuel and ventilation configurations. Peak exposures ranged from 30 to 75 kW/m² for about 20 minutes. Five additional tests were conducted using a single fuel and ventilation configuration to assess the repeatability of the test methodology. It was found that a 19.3 minute growth period occurred plateauing at a ceiling temperature of 708°C for 8.4 minutes, on average. Compartment gas temperatures were found to be repeatable, having a sample standard deviation less than 32°C for symmetric data. Repeatability improved when account was taken for the rapid fire growth inflection point. The utility of the approach for studying fire performance of building elements was demonstrated.     

A 1/4 scale test facility for PF transport in power station pipelines

Coal fired power station efficiency affects the environment and cost of power. Pulverised fuel (pf) is conveyed from the mill to the furnace through a convoluted network of steel pipes with several junctions and the coal can be badly distributed between the burners. In order to simulate the behaviour of pf in the network and to provide a solution to this problem, a test facility has been constructed at the University of Nottingham. The rig provides air drawn through a 154 mm pipe, which then splits two or three ways. A number of bends prior to the junction point ensure an unbalanced distribution of powder in the downstream pipes. Conventional and novel techniques have been applied to quantify the mal-distribution.