Mixture experiments with mixing errors

In mixture experiments the properties of mixtures are usually studied by mixing the amounts of the mixture components that are required to obtain the necessary proportions. This paper considers the impact of inaccuracies in discharging the required amounts of the mixture components on the statistical analysis of the data. It shows how the regression calibration approach can be used to minimize the resulting bias in the model and in the estimates of the model parameters, as well as to find correct estimates of the corresponding variances. Its application is made difficult by the complex structure of these errors. We also show how knowledge of the form of the model bias allows for choosing a manufacturing setting for a mixture product that is not biased and has smaller signal to noise ratio.     

Likelihood Estimation and Inference in a Class of Nonregular Econometric Models

We study inference in structural models with a jump in the conditional density, where location and size of the jump are described by regression curves. Two prominent examples are auction models, where the bid density jumps from zero to a positive value at the lowest cost, and equilibrium job‐search models, where the wage density jumps from one positive level to another at the reservation wage. General inference in such models remained a long‐standing, unresolved problem, primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions. This paper develops likelihood‐based estimation and inference methods for these models, focusing on optimal (Bayes) and maximum likelihood procedures. We derive convergence rates and distribution theory, and develop Bayes and Wald inference. We show that Bayes estimators and confidence intervals are attractive both theoretically and computationally, and that Bayes confidence intervals, based on posterior quantiles, provide a valid large sample inference method.     

Product Positioning and Pricing Under Production Cost Considerations*

A framework in a competitive environment is proposed that incorporates production cost and economies of scale in the problem of positioning a product for a market segment. The model facilitates the existence of a Nash equilibrium in prices and product positions. As such, firms can simultaneously choose prices and product positions for the segment. This result improves the traditional theory on equilibria points in prices and product positions where firms choose their product positions first and then set their prices. A sensitivity analysis demonstrates the effects of changes in the unit savings derived from economies of scale or the cost of furnishing a product with its attributes by one firm on the product positions, prices, and profits of all competing firms. More important, the paper examines the effect on prices and profits of competing firms when one of the firms repositions its product closer to the segment's ideal point. It is shown that under certain conditions, the profit of a firm may actually decrease as it redesigns its product closer to the segment's ideal point. These conditions assist management to identify the product design beyond which enhancements of the product would lead to lower profits because of increasing production costs. It is also shown that the price of this firm increases. Past research supports the idea that positioning a brand closer to the ideal point, given fixed product positions of competing firms, would lead to greater buyer preferences and eventually higher profits. The price and profits of the competing firm may increase or decrease. Conditions are derived under which a movement towards the segment's ideal point by one firm would lead to higher profits by the competing firm.     

Principal weighted logistic regression for sufficient dimension reduction in binary classification

Sufficient dimension reduction (SDR) is a popular supervised machine learning technique that reduces the predictor dimension and facilitates subsequent data analysis in practice. In this article, we propose principal weighted logistic regression (PWLR), an efficient SDR method in binary classification where inverse-regression-based SDR methods often suffer. We first develop linear PWLR for linear SDR and study its asymptotic properties. We then extend it to nonlinear SDR and propose the kernel PWLR. Evaluations with both simulated and real data show the promising performance of the PWLR for SDR in binary classification.     

A study of sign adjustment in weighted signed networks

We analyse the adjustment of networks comprising of weighted positive (P) and negative (N) symmetric relations under the impact of various balancing rules. Five kinds of rules are studied: (1) a local minimal edge adjustment which is a special case of, (2) a local pressure based rule, (3) a local sign based rule, (4) a global rule and (5) rules varying on a local to global dimension. The convergence and convergent proportions of different 3-cycles and, thus the impact upon β(3) balance, under the different kinds of adjustment rule are studied both analytically and through simulation. The effects of network size (n), density (d) and the initial proportion of positive edges (α₀) upon the convergence of 3-cycles and, thus, balance and the eventual implications for the process of group formation are explored.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

Inferences on a life distribution by sampling from the ages or the ages at death

Consider a system where units having independent and identically distributed lifetimes enter according to a nonhomogeneous Poisson process. After the unit’s life in the system, the unit departs the system. For a fixed system time, this paper relates the units’ common underlying life distribution with the distribution of the ages of units in the system, the distribution for the system life of units that departed the system and the distribution for the system life of units that have recently departed the system. Results can be used to estimate the underlying life distribution or a truncated version of that distribution based on the ages and/or most recent ages at death in both one sample and two sample situations. Results include a complete characterization of the possible distribution of the ages of those units in the system, how to estimate the underlying life distribution from the most recent ages at death, and how to test for an underlying monotone failure rate function based on independent samples from the ages and most recent ages at death. Two sample inferences that involve a likelihood ratio ordering make use of the results in Dykstra et al. (1995, J Amer Statisc Assoc 90(431):1030–1040), which provides the maximum likelihood estimators and a likelihood ratio test when the two distributions satisfy a likelihood ratio ordering. For the ages of the active units and the ages at death among the departed units, limits for their distributions and strong limiting results for their empirical distributions will be provided.Disclaimer: The views expressed in this paper are those of the author and may not represent the views of the U.S. FDA.     

Reliability computation under dynamic stress–strength modeling with cumulative stress and strength degradation

Reliability function is defined under suitable assumptions for dynamic stress–strength scenarios where strength degrades and stress accumulates over time. Methods for numerical evaluation of reliability are suggested under deterministic strength degradation and cumulative damage due to shocks arriving according to a point process, in particular a Poisson process, using simulation method and inversion theorem. These methods are specifically useful in the scenarios where damage distributions do not possess closure property under convolution. The method is also extended for non-identical, dependent damage distributions as well as for random strength degradation. Results from inversion method is compared with known approximate methods and also verified by simulation. As it turns out, the simulation method seems to have an edge in terms of computational burden and has much wider domain of applicability.     

Acceptance sampling plans based on truncated life tests for weighted exponential distribution

Gupta and Kundu proposed a new class of weighted exponential distributions using the idea of Azzalini. In this article, we develop an acceptance sampling plan for the weighted exponential distribution under a truncated life test. For various acceptance numbers, consumer’s confidence levels and values of the ratio of the experimental time to the specified mean lifetime, the minimum sample size necessary to ensure a certain mean lifetime are obtained. The operating characteristic function values and the associated producer’s risks are also presented. A numerical example is provided to illustrate the acceptance sampling plan.     

A model for estimating point sources in spatial data

We introduce a point source model which may be useful for estimating point sources in spatial data. It may also be useful for modelling general spatial data, and providing a simple explanatory model for some data, whilst in other cases it may give a parsimonious representation. The model assumes that there are point sources (or sinks), usually at unknown positions, and that the mean value at a site depends on the distance from these sources. We discuss the general form of the model, and some methods for estimating the sources and the regression parameters. We demonstrate the methodology using a simulation study, and apply the model to two real data sets. Some possibilities for further research are outlined.