Distribution of ranks in a two-sample problem with right censored data

We consider different censoring models for a two-sample and find the joint distribution of the rank vector and number of uncensored observations under each censoring model when the distributions of life times and/or distributions of censoring of censoring variables satisfy the condition for the Lehmann type of alternatives.     

Belief functions and statistical inference

The Dempster Shafer theory of belief functions is a method of quantifying uncertainty that generalizes probability theory. We review the theory of belief functions in the context of statistical inference. We mainly focus on a particular belief function based on the likelihood function and its application to problems with partial prior information. We also consider connections to upper and lower probabilities and Bayesian robustness.     

Accuracy and fairness of rain rules for interrupted one-day cricket matches

In this paper, we investigate the relative merits of rain rules for one-day cricket matches. We suggest that interrupted one-day matches present a missing data problem: the outcome of the complete match cannot be observed, and instead the outcome of the interrupted match, as determined at least in part by the rain rule in question, is observed. Viewing the outcome of the interrupted match as an imputation of the missing outcome of the complete match, standard characteristics to assess the performance of classification tests can be used to assess the performance of a rain rule. In particular, we consider the overall and conditional accuracy and the predictive value of a rain rule. We propose two requirements for a ‘fair’ rain rule, and show that a fair rain rule must satisfy an identity involving its conditional accuracies. Estimating the performance characteristics of various rain rules from a sample of complete one-day matches our results suggest that the Duckworth–Lewis method, currently adopted by the International Cricket Council, is essentially as accurate as and somewhat more fair than its best competitors. A rain rule based on the iso-probability principle also performs well but might benefit from re-calibration using a more representative data base.     

On constructing T 2 control charts for retrospective examination

In this paper we consider the issue of constructing retrospective T ² control chart limits so as to control the overall probability of a false alarm at a specified value. We describe an exact method for constructing the control limits for retrospective examination. We then consider Bonferroni-adjustments to Alt's control limit and to the standard x ² control limit as alternatives to the exact limit since it is computationally cumbersome to find the exact limit. We present the results of some simulation experiments that are carried out to compare the performance of these control limits. The results indicate that the Bonferroni-adjusted Alt's control limit performs better that the Bonferroni-adjusted x ² control limit. Furthermore, it appears that the Bonferroni-adjusted Alt's control limit is more than adequate for controlling the overall false alarm probability at a specified value.     

High wage workers and low wage firms: negative assortative matching or limited mobility bias?

Summary.  In the empirical literature on assortative matching using linked employer–employee data, unobserved worker quality appears to be negatively correlated with unobserved firm quality. We show that this can be caused by standard estimation error. We develop formulae that show that the estimated correlation is biased downwards if there is true positive assortative matching and when any conditioning covariates are uncorrelated with the firm and worker fixed effects. We show that this bias is bigger the fewer movers there are in the data, which is 'limited mobility bias'. This result applies to any two-way (or higher) error components model that is estimated by fixed effects methods. We apply these bias corrections to a large German linked employer–employee data set. We find that, although the biases can be considerable, they are not sufficiently large to remove the negative correlation entirely.     

Noninformative priors for the common mean in the bivariate normal distribution

In this paper, we consider some noninformative priors for the common mean in a bivariate normal population. We develop the first-order and second-order matching priors and reference priors. We find that the second-order matching prior is also an HPD matching prior, and matches the alternative coverage probabilities up to the second order. It turns out that derived reference priors do not satisfy a second-order matching criterion. Our simulation study indicates that the second-order matching prior performs better than the reference priors in terms of matching the target coverage probabilities in a frequentist sense. We also illustrate our results using real data.     

Bayesian Estimation of Survival Functions under Stochastic Precedence

When estimating the distributions of two random variables, X and Y, investigators often have prior information that Y tends to be bigger than X. To formalize this prior belief, one could potentially assume stochastic ordering between X and Y, which implies Pr(X < or = z) > or = Pr(Y < or = z) for all z in the domain of X and Y. Stochastic ordering is quite restrictive, though, and this article focuses instead on Bayesian estimation of the distribution functions of X and Y under the weaker stochastic precedence constraint, Pr(X < or = Y) > or = 0.5. We consider the case where both X and Y are categorical variables with common support and develop a Gibbs sampling algorithm for posterior computation. The method is then generalized to the case where X and Y are survival times. The proposed approach is illustrated using data on survival after tumor removal for patients with malignant melanoma.     

Expansion estimation by Bayes rules

In the problem of estimating a location parameter in any symmetric unimodal location parameter model, we demonstrate that Bayes rules with respect to squared error loss can be expanders for some priors that belong to the family of all symmetric priors. That generalizes the results obtained by DasGupta and Rubin for the one dimensional case. We also consider symmetric priors which either have an appropriate point mass at 0 or are unimodal, and prove that under the latter condition all Bayes rules are shrinkers. Results of such nature are important, for example, in wavelet based function estimation and data denoising, where shrinkage of wavelet coefficients is associated with smoothing the data. We illustrate the results using FIAT stock market data.     

Unrestricted consumption under a deterministic wealth and an Ornstein–Uhlenbeck process as a discount rate

We consider an individual or household endowed with an initial capital and an income, modeled as a linear function of time. Assuming that the discount rate evolves as an Ornstein–Uhlenbeck process, we target to find an unrestricted consumption strategy such that the value of the expected discounted consumption is maximized. Differently than in the case with restricted consumption rates, we can determine the optimal strategy and the value function.     

Using Liu-Type Estimator to Combat Collinearity

Abstract

Linear regression model and least squares method are widely used in many fields of natural and social sciences. In the presence of collinearity, the least squares estimator is unstable and often gives misleading information. Ridge regression is the most common method to overcome this problem. We find that when there exists severe collinearity, the shrinkage parameter selected by existing methods for ridge regression may not fully address the ill conditioning problem. To solve this problem, we propose a new two-parameter estimator. We show using both theoretic results and simulation that our new estimator has two advantages over ridge regression. First, our estimator has less mean squared error (MSE). Second, our estimator can fully address the ill conditioning problem. A numerical example from literature is used to illustrate the results.     

On count time series prediction

We consider the problem of assessing prediction for count time series based on either the Poisson distribution or the negative binomial distribution. By a suitable parametrization we employ both distributions with the same mean. We regress the mean on its past values and the values of the response and after obtaining consistent estimators of the regression parameters, regardless of the response distribution, we employ different criteria to study the prediction problem. We show by simulation and data examples that scoring rules and diagnostic graphs that have been proposed for independent but not identically distributed data can be adapted in the setting of count dependent data.     

Connectedness of complete block designs under an interference model

We consider an experiment with fixed number of blocks, in which a response to a treatment can be affected by treatments from neighboring units. For such experiment the interference model with neighbor effects is studied. Under this model we study connectedness of binary complete block designs. Assuming the circular interference model with left-neighbor effects we give the condition for minimal number of blocks necessary to obtain connected design. For a specified class of binary, complete block designs, we show that all designs are connected. Further we present the sufficient and necessary conditions of connectedness of designs with arbitrary, fixed number of blocks.     

Summer temperature effects on deaths and hospital admissions among the elderly population in two Italian cities

In developed countries the effects of climate on health status are mainly due to temperature. Our analysis is aimed to deepen statistically the relationship between summer climate conditions and daily frequency of health episodes: deaths or hospital admissions. We expect to find a U-shaped relationship between temperature and frequencies of events occurring in summer regarding the elderly population resident in Milano and Brescia. We use as covariates hourly records of temperature recorded at observation sites located in Milano and Brescia. The analysis is performed using Generalized Additive Models (GAM), where the response variable is the daily number of events, which varies as a possibly non-linear function of meteorological variables measured on the same or previous day. We consider separate models for Milano and Brescia and then we compare temperature effects among the two towns and among different age classes. Moreover we consider separate models for all diagnosed events, for those due to respiratory disease and those due to circulatory pathologies. Model selection is a central problem, the basic methods used are the UBRE and GCV criteria but, instead of conditioning all final conclusions on the best model according to the chosen criterion, we investigated the effect of model selection by implementing a bootstrap procedure.     

Improvements in the Poisson approximation of mixed Poisson distributions

We consider the approximation of mixed Poisson distributions by Poisson laws and also by related finite signed measures of higher order. Upper bounds and asymptotic relations are given for several distances. Even in the case of the Poisson approximation with respect to the total variation distance, our bounds have better order than those given in the literature. In particular, our results hold under weaker moment conditions for the mixing random variable. As an example, we consider the approximation of the negative binomial distribution, which enables us to prove the sharpness of a constant in the upper bound of the total variation distance. The main tool is an integral formula for the difference of the counting densities of a Poisson distribution and a related finite signed measure.     

Simulation-Based Inspection Policies for a One-Shot System in Storage Over a Finite Time Span

In this paper, we consider an inspection policy problem for a one-shot system with two types of units over a finite time span and want to determine inspection intervals optimally with given replacement points of Type 2 units. The interval availability and life cycle cost are used as optimization criteria and estimated by simulation. Two optimization models are proposed to find the optimal inspection intervals for the exponential and general distributions. A heuristic method and a genetic algorithm are proposed to find the near-optimal inspection intervals, to satisfy the target interval availability and minimize the life-cycle cost. We study numerical examples to compare the heuristic method with the genetic algorithm and investigate the effect of model parameters to the optimal solutions.     

Collapsibility of Conditional Graphical Models

Abstract. In this paper, we consider two kinds of collapsibility, that is, the model‐collapsibility and the estimate‐collapsibility, of conditional graphical models for multidimensional contingency tables. We show that these two definitions are equivalent, and propose a sufficient and necessary condition for them in terms of the interaction graph, which allows the collapsibility to be characterized and judged intuitively and conveniently.     

Upper probabilities based only on the likelihood function

In the problem of parametric statistical inference with a finite parameter space, we propose some simple rules for defining posterior upper and lower probabilities directly from the observed likelihood function, without using any prior information. The rules satisfy the likelihood principle and a basic consistency principle ('avoiding sure loss'), they produce vacuous inferences when the likelihood function is constant, and they have other symmetry, monotonicity and continuity properties. One of the rules also satisfies fundamental frequentist principles. The rules can be used to eliminate nuisance parameters, and to interpret the likelihood function and to use it in making decisions. To compare the rules, they are applied to the problem of sampling from a finite population. Our results indicate that there are objective statistical methods which can reconcile three general approaches to statistical inference: likelihood inference, coherent inference and frequentist inference.     

Use of Quasi-Ranges in the Estimation of Scale Parameter of a Symmetric Distribution

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the scale parameter of a symmetric distribution using quasi-ranges is considered. We also prove a sufficient condition for the non negativity of the scale estimator obtained by the above method. Further, we obtain necessary and sufficient conditions for the derived estimators to be constant multiple of the sample range.     

Testing and Confidence estimation of the Mean of a multidimensional Gaussian Process

From the sequential observation of a multidimensional continuous time Gaussian process, whose mean vector depends linearly of a multidimensional parameter, we consider the confidential estimation of the parameter value and the testing problem of a simple hypothesis about the parameter, in presence of a nuisance variance parameter. The method is based on a previously obtained [cf. 4] point estimate for the case of a known covariance structure. We first see that this estimate is, in fact, independent of the variance parameter. For the hypotheses testing problem, the invariance under certain groups of transformations and the partial sufficiency allows to construct optimal terminal tests. Furthermore we determine the observation time necessary to control its power function. These testing results may be translated in terms of most accurate confidence sets. If the observation is stopped according to the diameter of the confidence set, under some condition, the confidence level is preserved.     

Testing for Structural Instability in Moment Restriction Models: An Info-Metric Approach

In this paper, we develop an info-metric framework for testing hypotheses about structural instability in nonlinear, dynamic models estimated from the information in population moment conditions. Our methods are designed to distinguish between three states of the world: (i) the model is structurally stable in the sense that the population moment condition holds at the same parameter value throughout the sample; (ii) the model parameters change at some point in the sample but otherwise the model is correctly specified; and (iii) the model exhibits more general forms of instability than a single shift in the parameters. An advantage of the info-metric approach is that the null hypotheses concerned are formulated in terms of distances between various choices of probability measures constrained to satisfy (i) and (ii), and the empirical measure of the sample. Under the alternative hypotheses considered, the model is assumed to exhibit structural instability at a single point in the sample, referred to as the break point; our analysis allows for the break point to be either fixed a priori or treated as occuring at some unknown point within a certain fraction of the sample. We propose various test statistics that can be thought of as sample analogs of the distances described above, and derive their limiting distributions under the appropriate null hypothesis. The limiting distributions of our statistics are nonstandard but coincide with various distributions that arise in the literature on structural instability testing within the Generalized Method of Moments framework. A small simulation study illustrates the finite sample performance of our test statistics.