A model with mixed binary responses and censored observations

In this paper, we study the maximum likelihood estimation of a model with mixed binary responses and censored observations. The model is very general and includes the Tobit model and the binary choice model as special cases. We show that, by using additional binary choice observations, our method is more efficient than the traditional Tobit model. Two iterative procedures are proposed to compute the maximum likelihood estimator (MLE) for the model based on the EM algorithm (Dempster et al, 1977) and the Newton-Raphson method. The uniqueness of the MLE is proved. The simulation results show that the inconsistency and inefficiency can be significant when the Tobit method is applied to the present mixed model. The experiment results also suggest that the EM algorithm is much faster than the Newton-Raphson method for the present mixed model. The method also allows one to combine two data sets, the smaller data set with more detailed observations and the larger data set with less detailed binary choice observations in order to improve the efficiency of estimation. This may entail substantial savings when one conducts surveys.     

Outliers in Multi-Response Experiments

Cook-statistic has been developed for detecting outliers in two likely situations of occurrence of outliers in multi-response experiments. In the first situation, more than one outlying observations vector has been considered. Each of these vectors is obtained on the assumption that a particular observation from each of the responses is an outlier. A general expression of Cook-statistic for detecting any such t outlying observations vectors has been obtained. Then some particular cases have been considered. In the second case a situation is considered where observations from all the responses may not be outliers. Here also a general expression of Cook-statistic is obtained for detecting any t observations from each of any k responses as outliers. In both the cases Cook-statistic is applied to real experimental data.     

Robust estimation of location in samples of size three

Youden (1953) discussed the practice of averaging the two most concordant observations in sets of three measurements as a method of estimating location. Distributional results for this estimator can be found in Seth (1950) and Lieblein (1952). It follows from their work that the sample median has smaller variance for normal and uniform populations. In this paper it is shown that themedian stochastically dominates the average of the two closest observations for uniform, normal, double–exponential and Cauchy populations and thus is the superior resistant estimator in these cases for a broad class of loss functions. However, an example is given in which, for a particular contaminaion model and loss function, the mean of the closest two observations has smaller risk than the median.     

Bounds for how much influence an observation can have

That outliers or influential observations can affect the results in a regression is well-known. But it is not clear how much influence a specific observation can have on other statistics. In time series, especially in predictive situations, the effect of additional observations is of singular importance. We here examine bounds for the effect of an additional observation on the mean, variance, Mahalanobis distance, product moment correlation, and coefficients of linearity and monotonicity.     

Updating discriminant functions estimated from inverse gaussian populations

The problem of updating discriminant functions estimated from inverse Gaussian populations is investigated in situations when the additional observations are mixed (unclassified) or classified. In each case two types of discriminant functions, linear and quadratic, are considered. Using simulation experiments the performance of the updating procedures is evaluated by means of relative efficiencies.     

Two zs are Better than One

Given only a random sample of observations, the usual estimator for the population mean is the sample mean. If additional information is provided it might be possible in some situations to obtain a better estimator. The situation considered here is when the variable whose mean is sought is composed of factors that are themselves observable. In the basic case, the variable can be expressed as the product of two, independent, more basic variables, but we also consider the case of more than two, the effect of correlation, and when there are observation costs.     

Asymptotic normality of numbers of observations in random regions determined by order statistics

In this paper, we study the joint limiting behaviour of numbers of observations that fall into regions determined by order statistics and Borel sets. We show that suitably centred and normed versions of these numbers are asymptotically multivariate normal under some conditions. We consider two cases: one where the population distribution function is discontinuous and the other where it is continuous and the order statistics are extreme. Finally, we compare results obtained for the two cases with their analogues for absolutely continuous distribution function and central-order statistics.     

Continuous single attribute control chart for Markov-dependent processes

Most of the times, the observations related to the quality characteristic of a process do not need to be independent. In such cases, control charts based on the assumption of independence of the observations are not appropriate. When the characteristic under study is qualitative, Markov model serves as a simple model to account for the dependency of the observations. For this purpose, we develop an attribute control chart under 100% inspection for a Markov dependent process by controlling the error probabilities. This chart consists of two sub-charts. For a given sample, depending upon the state of the last observation of previous sample (if any), one of these two will be used. Optimal values of the design parameters of the control chart are obtained. Chart’s performance is studied by using its capability (probability) of detecting a shift in process parameters.     

Small-sample performance of a modified least-failures sampling procedure for bernoulli subset selection

We describe two sequential sampling procedures for Bernoulli subset selection which were shown to exhibit desirable behavior for large-sample problems. These procedures have identical performance characteristics in terms of the number of observations taken from any one of the populations under investigation, but one of the procedures employs one-at-a-time sampling while theother allows observations to be taken in blocks during early stages of experimentation. In this paper, a simulation study of their behavior for small-sample cases (n > 25) reveals that they canresult in a savings (sometimes substantial) in the expected total number of observations requiredto terminate the experiment as compared to single-stage procedures. Hence they may be quite usefulto a practitioner for screening purposes when sampling is limited.     

Bayesian look ahead one stage sampling allocations for selecting the largest normal mean

From two independent normal populations with unknown means and a common known variance, samples of unequal sizes are observed at stage 1. The goal is to find that population with the larger mean. Using the Bayes approach, optimum allocations ofm additional observations, at stage 2, are derived under the linear and the 0–1 loss.     

Nonlinear Discriminant Functions for Mixed Random Walk Models

A procedure is presented for finding maximum likelihood estimates of the parameters of a mixture of two random walk distributions in two cases, using classified and unclassified observations. Based on small sample size, estimation of nonlinear discriminant functions is considered. Throughout simulation experiments, the performance of the corresponding estimated nonlinear discriminant functions is investigated. The total probabilities of misclassification and percentage biases are evaluated and discussed.     

Experimental designs and their efficiencies for spatially correlated observations in two dimensions

Optimality of experimental designs for spatially correlated observations is investigated.come two dimensional correlation structures are discussed and an attempt has been made to find optimal or nearly optimal design for each sitution.The solution lend to designs similar to that used for repeated measurements.The relative efficiency of the proposed designs in comparison to randomized latin square designs is tabulated for some cases.     

Updating a nonlinear discriminant function estimated from a mixture of two Burr Type III distributions

The main contribution of this paper is is updating a nonlinear discriminant function on the basis of data of unknown origin. Specifically a procedure is developed for updating the nonlinear discriminant function on the basis of two Burr Type III distributions (TBIIID) when the additional observations are mixed or classified. First the nonlinear discriminant function of the assumed model is obtained. Then the total probabilities of misclassification are calculated. In addition a Monte carlo simulation runs are used to compute the relative efficiencies in order to investigate the performance of the developed updating procedures. Finally the results obtained in this paper are illustrated through a real and simulated data set.     

Extended generalised Pareto models for tail estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.     

Optimal Designs for the Carryover Model with Random Interactions Between Subjects and Treatments

The paper considers a model for crossover designs with carryover effects and a random interaction between treatments and subjects. Under this model, two observations of the same treatment on the same subject are positively correlated and therefore provide less information than two observations of the same treatment on different subjects. The introduction of the interaction makes the determination of optimal designs much harder than is the case for the traditional model. Generalising the results of Bludowsky's thesis, the present paper uses Kushner's method to determine optimal approximate designs. We restrict attention to the case where the number of periods is less than or equal to the number of treatments. We determine the optimal designs in the important special cases that the number of periods is 3, 4 or 5. It turns out that the optimal designs depend on the variance of the random interactions and in most cases are not binary. However, we can show that neighbour balanced binary designs are highly efficient, regardless of the number of periods and of the size of the variance of the interaction effects.     

Statistical Inference in Orthogonal Regression for Three-Part Compositional Data Using a Linear Model with Type-II Constraints

Orthogonal regression is a proper tool to analyze relations between two variables when three-part compositional data, i.e., three-part observations carrying relative information (like proportions or percentages), are under examination. When linear statistical models with type-II constraints (constraints involving other parameters besides the ones of the unknown model) are employed for estimating the parameters of the regression line, approximate variances and covariances of the estimated line coefficients can be determined. Moreover, the additional assumption of normality enables to construct confidence domains and perform hypotheses testing. The theoretical results are applied to a real-world example.     

Further theoretical results and a comparison between two methods for approximating eigenvalues of perturbed covariance matrices

Covariance matrices, or in general matrices of sums of squares and cross-products, are used as input in many multivariate analyses techniques. The eigenvalues of these matrices play an important role in the statistical analysis of data including estimation and hypotheses testing. It has been recognized that one or few observations can exert an undue influence on the eigenvalues of a covariance matrix. The relationship between the eigenvalues of the covariance matrix computed from all data and the eigenvalues of the perturbed covariance matrix (a covariance matrix computed after a small subset of the observations has been deleted) cannot in general be written in closed-form. Two methods for approximating the eigenvalues of a perturbed covariance matrix have been suggested by Hadi (1988) and Wang and Nyquist (1991) for the case of a perturbation by a single observation. In this paper we improve on these two methods and give some additional theoretical results that may give further insight into the problem. We also compare the two improved approximations in terms of their accuracies.     

Estimation of the linear-plateau segmented regression model in the presence of measurement error

It is well known that when the true values of the independent variable are unobservable due to measurement error, the least squares estimator for a regression model is biased and inconsistent. When repeated observations on each x_i are taken, consistent estimators for the linear-plateau model can be formed. The repeated observations are required to classify each observation to the appropriate line segment. Two cases of repeated observations are treated in detail. First, when a single value of y_i is observed with the repeated observations of x_i the least squares estimator using the mean of the repeated x_i observations is consistent and asymptotically normal. Second, when repeated observations on the pair (x_i, y_i ) are taken the least squares estimator is inconsistent, but two consistent estimators are proposed: one that consistently estimates the bias of the least squares estimator and adjusts accordingly; the second is the least squares estimator using the mean of the repeated observations on each pair.     

A test statistic for weighted runs

A new test statistic based on runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic χ² test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute p-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.     

A Parametric Test of Perfect Ranking in Balanced Ranked Set Sampling

Many techniques based on data which are drawn by Ranked Set Sampling (RSS) scheme assume that the ranking of observations is perfect. Therefore it is essential to develop some methods for testing this assumption. In this article, we propose a parametric location-scale free test for assessing the assumption of perfect ranking. The results of a simulation study in two special cases of normal and exponential distributions indicate that the proposed test performs well in comparison with its leading competitors.