Two-stage point estimation with a shrinkage stopping rule

Consider the problem of estimating a mean vector in ap-variate normal distribution under two-stage sequential sampling schemes. The paper proposes a stopping rule motivated by the James-Stein shrinkage estimator, and shows that the stopping rule and the corresponding shrinkage estimator asymptotically dominate the usual two-stage procedure under a sequence of local alternatives forp3. Also the results of Monte Carlo simulation for average sample sizes and risks of estimators are stated.     

Sequential estimation of a linear function of normal means under asymmetric loss function

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions. Received: August 1999     

Sequential point estimation of normal mean under LINEX loss function

A sequential point estimation of the mean of a normal distribution is considered under LINEX loss function. The regret of sequential procedures are obtained. Furthermore, it is shown that a sequential procedure with the sample mean as an estimate is asymptotically inadmissible. An accerelated stopping time is also considered. Received: December 1999     

The sequential probability ratio test under random censorship

We investigate properties of the sequential probability ratio test when the data are at risk of being censored. It turns out that the stopping boundaries are the same as for completely observable data, but that the average sample size increases as censoring becomes more substantial.     

On purely sequential estimation of an inverse Gaussian mean

The first part of this paper deals with developing a purely sequential methodology for the point estimation of the mean \(\mu \) of an inverse Gaussian distribution having an unknown scale parameter \(\lambda \). We assume a weighted squared error loss function and aim at controlling the associated risk function per unit cost by bounding it from above by a known constant \(\omega \). We also establish first-order and second-order asymptotic properties of our stopping rule. The second part of this paper deals with obtaining a purely sequential fixed accuracy confidence interval for the unknown mean \(\mu \), assuming that the scale parameter \(\lambda \) is known. First-order asymptotic efficiency and asymptotic consistency properties are also built of our proposed procedures. We then provide extensive sets of simulation studies and real data analysis using data from fatigue life analysis to show encouraging performances of our proposed stopping strategies.     

How to Identify and Forecast Bull and Bear Markets?

Because the state of the equity market is latent, several methods have been proposed to identify past and current states of the market and forecast future ones. These methods encompass semi‐parametric rule‐based methods and parametric Markov switching models. We compare the mean‐variance utilities that result when a risk‐averse agent uses the predictions of the different methods in an investment decision. Our application of this framework to the S&P 500 shows that rule‐based methods are preferable for (in‐sample) identification of the state of the market, but Markov switching models for (out‐of‐sample) forecasting. In‐sample, only the mean return of the market index matters, which rule‐based methods exactly capture. Because Markov switching models use both the mean and the variance to infer the state, they produce superior forecasts and lead to significantly better out‐of‐sample performance than rule‐based methods. We conclude that the variance is a crucial ingredient for forecasting the market state. Copyright © 2016 John Wiley & Sons, Ltd.     

On estimating the reliability after sequentially estimating the mean: The exponential case

Sequential estimation problems for the mean parameter of an exponential distribution has received much attention over the years. Purely sequential and accelerated sequential estimators and their asymptotic second-order characteristics have been laid out in the existing literature, both for minimum risk point as well as bounded length confidence interval estimation of the mean parameter. Having obtained a data set from such sequentially designed experiments, the paper investigates estimation problems for the associatedreliability function. Second-order approximations are provided for the bias and mean squared error of the proposed estimator of the reliability function, first under a general setup. An ad hoc bias-corrected version is also introduced. Then, the proposed estimator is investigated further under some specific sequential sampling strategies, already available in the literature. In the end, simulation results are presented for comparing the proposed estimators of the reliability function for moderate sample sizes and various sequential sampling strategies.     

Distributions of stopping times in some sequential estimation procedures

A class of sequential estimation procedures is considered in the case when relevant data may become available only at random times. The exact distributions of the optimal stopping time and the number of observations at the moment of stopping are derived in some sequential procedures. The results obtained in an explicit form are applied to derive the expected time of observing the process, the average number of observations and the expected loss of sequential estimation procedures based on delayed observations. The use of the results is illustrated in a special model of normally distributed observations and the Weibull distributed lifetimes. The probabilistic characteristics are also derived for an adaptive sequential procedures and the behavior of the adaptive procedure is compared with the corresponding optimal sequential procedure.     

On the stopping boundary for sequential selection of the normal population with the largest mean

This paper provides an improved stopping boundary for open sequential selection of the normal population with the largest mean when all populations have common known variance. The proof relies on the theory of Brownian motion processes with drift.     

A robust asymptotically optimal sequential estimation procedure for the Poisson process

The problem of estimating sequentially the intensity parameter of a homogeneous Poisson process with quadratic loss and fixed cost per unit time is considered within the Bayesian framework. Without using both the prior information and any auxiliary data, this paper proposes a sequential procedure as that suggested by Vardi (Ann Statist 7:1040?C1051, 1979) in classical non-Bayesian sequential estimation. The proposed sequential procedure is robust in the sense that it does not depend on the prior. The second order approximations to the expected sample size and the Bayes risk of the proposed sequential procedure are established for a large class of prior distributions.     

Sequential estimation of a location parameter and powers of a scale parameter from delayed observations

The problem of sequentially estimating a location parameter and powers of a scale parameter is considered in the case when the observations become available at random times. Certain classes of sequential estimation procedures are derived under an invariant balanced loss function and with the observation cost determined by a convex function of the stopping time and the number of observations up to that time.     

The size distribution of cities: An examination of the Pareto law and primacy

This paper examines the Pareto and primacy measures of the size distribution of cities. The mean Pareto exponent for a sample of 44 countries is 1.136, somewhat greater than the exponent of one implied by the rank-size rule. We find that value of the Pareto exponent is quite sensitive to the definition of the city and the choice of city sample size. The significance of non-linear terms in variants of the Pareto distribution also indicate that the rank-size rule is only a first approximation to a complete characterization of the size distribution of cities within a country. The relatively low correlation between primacy and Pareto measures confirms the need for a variety of measures of city size distributions. This paper also suggests that large cities are growing faster than small cities in most of the countries in our sample. This is indicated by the positive coefficient on the first non-linear term introduced into the Pareto equation. Finally, variations in the Pareto exponent and measures of primacy are partly explained by economic, demographic, and geographic factors.     

Asymptotic non-deficiency of the Bayes sequential estimation in a family of transformed Chi-square distributions

Bayes sequential estimation in a family of transformed Chi-square distributions using a linex loss function and a cost c > 0 for each observation is considered in this paper. It is shown that an asymptotic pointwise optimal rule (A.P.O.) is asymptotically non-deficient, i.e., the difference between the Bayes risk of the A.P.O. rule and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of single observation, as c → 0.     

Sequential control of time series by functionals of kernal-weighted empirical processes under local alternatives

Motivated in part by applications in model selection in statistical genetics and sequential monitoring of financial data, we study an empirical process framework for a class of stopping rules which rely on kernel-weighted averages of past data. We are interested in the asymptotic distribution for time series data and an analysis of the joint influence of the smoothing policy and the alternative defining the deviation from the null model (in-control state). We employ a certain type of local alternative which provides meaningful insights. Our results hold true for short memory processes which satisfy a weak mixing condition. By relying on an empirical process framework we obtain both asymptotic laws for the classical fixed sample design and the sequential monitoring design. As a by-product we establish the asymptotic distribution of the Nadaraya-Watson kernel smoother when the regressors do not get dense as the sample size increases.Acknowledgements The author is grateful to two anonymous referees for their constructive comments, which improved the paper. One referee draws my attention to Lifshits paper. The financial support of the Collaborative Research Centre Reduction of Complexity in Multivariate Data Structures (SFB 475) of the German Research Foundation (DFG) is greatly acknowledged.     

Jumps in cross‐sectional rank and expected returns: a mixture model

We propose a new nonlinear time series model of expected returns based on the dynamics of the cross‐sectional rank of realized returns. We model the joint dynamics of a sharp jump in the cross‐sectional rank and the asset return by analyzing (1) the marginal probability distribution of a jump in the cross‐sectional rank within the context of a duration model, and (2) the probability distribution of the asset return conditional on a jump, for which we specify different dynamics depending upon whether or not a jump has taken place. As a result, the expected returns are generated by a mixture of normal distributions weighted by the probability of jumping. The model is estimated for the weekly returns of the constituents of the SP500 index from 1990 to 2000, and its performance is assessed in an out‐of‐sample exercise from 2001 to 2005. Based on the one‐step‐ahead forecast of the mixture model we propose a trading rule, which is evaluated according to several forecast evaluation criteria and compared to 18 alternative trading rules. We find that the proposed trading strategy is the dominant rule by providing superior risk‐adjusted mean trading returns and accurate value‐at‐risk forecasts. Copyright © 2008 John Wiley & Sons, Ltd.     

Sequential estimation of means of linear processes

First the minimum risk point estimation as well as the fixed-width confidence interval problems for the mean parameter of a linear process are addressed under the framework of Fakhre-Zakeri and Lee (1992). The accelerated versions of their full sequential methodologies are introduced in order to achieve operational savings. Next, multi-sample analogs are discussed along the lines of Mukhopadhyay and Sriram (1992) both under full sequential as well as accelerated sequential sampling. In either setup, the first-order asymptotic characteristics are highlighted.     

Risk, uncertainty, and option exercise

Many economic decisions can be described as an option exercise or optimal stopping problem under uncertainty. Motivated by experimental evidence such as the Ellsberg Paradox, we follow Knight (1921) and distinguish risk from uncertainty. To capture this distinction, we adopt the multiple-priors utility model. We show that the impact of ambiguity on the option exercise decision depends on the relative degrees of ambiguity about continuation payoffs and termination payoffs. Consequently, ambiguity may accelerate or delay option exercise. We apply our results to investment and exit problems, and show that the myopic NPV rule can be optimal for an agent having an extremely high degree of ambiguity aversion.     

Why do some combinations perform better than others?

The evidence from the literature on forecast combination shows that combinations generally perform well. We discuss here how the accuracy and diversity of the methods being combined and the robustness of the combination rule can influence performance, and illustrate this by showing that a simple, robust combination of a subset of the nine methods used in the M4 competition’s best combination performs almost as well as that forecast, and is easier to implement. We screened out methods with low accuracy or highly correlated errors and combined the remaining methods using a trimmed mean. We also investigated the accuracy risk (the risk of a bad forecast), proposing two new accuracy measures for this purpose. Our trimmed mean and the trimmed mean of all nine methods both had lower accuracy risk than either the best combination in the M4 competition or the simple mean of the nine methods.     

Optimality conditions¶and bubbles in sequential economies¶and bounded relative risk-aversion

We provide specific qualifications in order that Kuhn–Tucker type Euler equations and transversality conditions at infinity hold in stochastic equilibrium models with heterogeneous agents and where assets are traded in sequential markets. It is not assumed that uncertainty is modeled as an event-tree structure or that preferences are necessarily bounded. We also describe an important class of preferences based on bounded relative risk aversion which yields relevant simplifications. Our results are used to establish conditions that rule out asset pricing bubbles. Specific examples of economies with bubbles are also discussed. Received: 28 January 2002 / Accepted: 19 July 2002 We are grateful to the editor and an anonymous referee for their valuable comments. This research was partially supported by MURST (Italy), National Group on “Nonlinear Dynamics and Stochastic Models in Economics and Finance”.     

Remarks on sequential estimation of a linear function of two means: The normal case

Asymptotic normality of the stopping time ofMukhopadhyay [1976] relating to the point estimation problem is proved. Also moderate sample size behaviour of this stopping time has been studied by Monte-Carlo methods.