A conditionally-unbiased estimator of population size based on plant-capture in continuous time

This paper proposes an estimator of the unknown size of a target population to which has been added a planted population of known size. The augmented population is observed for a fixed time and individuals are sighted according to independent Poisson processes. These processes may be time-inhomogeneous, but, within each population, the intensity function is the same for all individuals. When the two populations have the same intensity function, known results on factorial series distributions suggest that the proposed estimator is approximately unbiased and provide a useful estimator of standard deviation. Except for short sampling times, computational results confirm that the proposed population-size estimator is nearly unbiased, and indicate that it gives a better overall performance than existing estimators in the literature. The robustness of this performance is investigated in situations in which it cannot be assumed that the behaviour of the plants matches that of individuals from the target population.     

Sequential estimation for a family of counting processes in the nuisance parameter case

The paper deals with the problem of sequential estimation for stochastic processes in the presence of a nuisance parameter. Using the approach to estimation through estimating equations, optimum estimating functions based on a random observation time are investigated in some models for processes appearing in reliability systems theory.     

Incorporating scientific,ethical and economic considerations into the design of clinical trials in the pharmaceutical industry: a sequential approach

A flexible sequential approach to the design of clinical trials is discussed herein. This approach is based on a “confidence sequence” viewpoint instead of the rigid stopping and terminal decision rules in conventional sequential testing theory. By using an appropriate confidence sequence, one can always ensure a prescribed degree of scientific rigor (confidence) in establishing the drug to be effective. Moreover, one also has the option of terminating the trial early when there is already enough statistical evidence for concluding that the drug is effective, or when the drug shows uniorseen harmful effects, or when the data predict that there is little chance of arriving at a definitive conclusion in favor of the drug by the scheduled end of the trial. We discuss how these and other ethical and economic considerations can be readily incorporated into the stopping criteria of the trial.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

Fixed-width sequential confidence interval for the correlation coefficient of a bivariate normal distribution

A sequential confidence interval of fixed width 2d d > 0, is constructed for the correlation coefficient of a bivariate normal distribution. It is shown that the coverage probability is approximately equal to a preassigned number γ, 0 < γ < as d → 0.     

A New Goodness-of-Fit Test Based on the Laplace Statistic for a Large Class of NHPP Models

ABSTRACT

Nonhomogeneous Poisson processes (NHPP) provide many models for hardware and software reliability analysis. In order to get an appropriate NHPP model, goodness-of-Fit (GOF for short) tests have to be carried out. For the power-law processes, lots of GOF tests have been developed. For other NHPP models, only the Conditional Probability Integral Transformation (CPIT) test has been proposed. However, the CPIT test is less powerful and cannot be applied to some NHPP models. This article proposes a general GOF test based on the Laplace statistic for a large class of NHPP models with intensity functions of the form αλ(t, β). The simulation results show that this test is more powerful than CPIT test.     

On the Superiority of a Variable Sampling Interval Control Chart

The paper establishes the analytical grounds of the uniform superiority of a variable sampling interval (VSI) Shewhart control chart over the conventional fixed sampling interval (FSI) control chart, with respect to the zero-time performance, for a wide class of process distributions. We provide a sufficient condition on the distribution of a control chart statistic, and propose a criterion to determine the control limits and the regions in the in-control area of the VSI chart, corresponding to the different sampling intervals used by it. The condition and the criterion together ensure the uniform zero-time superiority of the VSI chart over the matched FSI chart, in detecting a process shift of any magnitude. It is shown that normal, Student's t and Laplace distributions satisfy the sufficient condition. In addition, chi-square, F and beta distributions satisfy it, provided that these are not extremely skewed. Further, it is illustrated that the superiority of the VSI feature is not trivial and cannot be assured if the sufficient condition is not satisfied or the control limits and the regions are not determined according to the proposed criterion. An application of the result to confirm the superiority of the VSI feature is demonstrated for the control chart for individual observations used to monitor a milk-pouch filling process.     

On the improvement of one-sided S control charts

The most common charting procedure used for monitoring the variance of the distribution of a quality characteristic is the S control chart. As a Shewhart-type control chart, it is relatively insensitive in the quick detection of small and moderate shifts in process variance. The performance of the S chart can be improved by supplementing it with runs rules or by varying the sample size and the sampling interval. In this work, we introduce and study one-sided adaptive S control charts, supplemented or not with one powerful runs rule, for detecting increases or decreases in process variation. The properties of the proposed control schemes are obtained by using a Markov chain approach. Furthermore, a practical guidance for the choice of the most suitable control scheme is also provided.     

A semi-Markov decision model for the optimal control of a simple immigration-birth-death process through the introduction of a predator

ABSTRACT

This article is concerned with the problem of controlling a simple immigration-birth-death process, which represents a pest population, by the introduction of a predator in the habitat of the pests. The optimization criterion is the minimization of the expected long-run average cost per unit time. It is possible to construct an appropriate semi-Markov decision model with a finite set of states if and only if the difference between the per capita birth rate and the per capita death rate of the pests is smaller than half of the rate at which the predator is introduced in the habitat.     

A Bayesian approach for locating change points in a compound Poisson process with application to detecting DNA copy number variations

This work examines the problem of locating changes in the distribution of a Compound Poisson Process where the variables being summed are iid normal and the number of variable follows the Poisson distribution. A Bayesian approach is developed to identify the location of significant changes in any of the parameters of the distribution, and a sliding window algorithm is used to identify multiple change points. These results can be applied in any field of study where an interest in locating changes not only in the parameter of a normally distributed data set but also in the rate of their occurrence. It has direct application to the study of DNA copy number variations in cancer research, where it is known that the distances between the genes can affect their intensity level.     

A comparison of maximum likelihood and least squares for the estimation of a cumulative distribution

Monte Carlo methods are used to compare the methods of maximum likelihood and least squares to estimate a cumulative distribution function. When the probabilistic model used is correct or nearly correct, the two methods produce similar results with the MLE usually slightly superior When an incorrect model is used, or when the data is contaminated, the least squares technique often gives substantially superior results.     

A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process

In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these results.     

A regression approach for estimating the parameters of the covariance function of a stationary spatial random process

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions, and various methods for estimating the parameters have been proposed in the literature. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h>0$h>0$ which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. Simulation studies show that our method outperforms all well-known least squares-based approaches to the variogram estimation and is comparable to the maximum likelihood estimation of the parameters of the covariance function. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions.     

A Combination of CUSUM Charts for Monitoring a Zero-Inflated Poisson Process

The Zero-inflated Poisson distribution (ZIP) is used to model the defects in processes with a large number of zeros. We propose a control charting procedure using a combination of two cumulative sum (CUSUM) charts to detect increases in the parameters of ZIP process, one is a conforming run length (CRL) CUSUM chart and another is a zero truncated Poisson (ZTP) CUSUM chart. The control limits of the control charts are obtained using both Markov chain-based methods and simulations. Simulation experiments show that the proposed method outperforms an existing method. Finally, a real example is presented.     

A confidence interval for the median of a finite population under unequal probability sampling: A model-assisted approach

This paper presents a method for constructing confidence intervals for the median of a finite population under unequal probability sampling. The model-assisted approach makes use of the _L1$L_1$-norm to motivate the estimating function which is then used to develop a unified approach to inference which includes not only confidence intervals but hypothesis tests and point estimates. The approach relies on large sample theory to construct the confidence intervals. In cases when second-order inclusion probabilities are not available or easy to compute, the Hartley–Rao variance approximation is employed. Simulations show that the confidence intervals achieve the appropriate confidence level, whether or not the Hartley–Rao variance is employed.     

A new approach for joint modelling of longitudinal measurements and survival times with a cure fraction

The joint analysis of longitudinal measurements and survival data is useful in clinical trials and other medical studies. In this paper, we consider a joint model which assumes a linear mixed $tt$ model for longitudinal measurements and a promotion time cure model for survival data and links these two models through a latent variable. A semiparametric inference procedure with an EM algorithm implementation is developed for the parameters in the joint model. The proposed procedure is evaluated in a simulation study and applied to analyze the quality of life and time to recurrence data from a clinical trial on women with early breast cancer. The Canadian Journal of Statistics 40: 207–224; 2012 © 2012 Statistical Society of Canada