Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

Mixture cumulative count control chart for mixture geometric process characteristics

A statistical process control chart named the mixture cumulative count control chart (MCCC-chart) is suggested in this study, motivated by an existing control chart named cumulative count control chart (CCC-chart). The MCCC-chart is constructed based on the distribution function of a two component mixture of geometric distributions using the number of items inspected until a defective item is observed ‘n’ as plotting statistics. We have observed that the MCCC-chart has the ability to perform equivalent to the CCC-chart when number of defective items follows geometric distribution and better than the CCC-chart when the number of defective items produced by a process follows a mixture geometric model. The MCCC-chart may be considered as a generalized version of CCC-chart.     

Compound gamma bivariate distributions

Summary Bivariate distributions, which may be of special relevance to the lifetimes of two components of a system, are derived using the following approach. As the two components are part of one system and therefore exposed to similar conditions of service, there will be similarity between their lifetimes that is not shared by components belonging to different systems. The lifetime distribution for a given system is assumed to be Gamma in form (this includes the exponential as a special case; extension to the Stacey distribution, which includes the Weibull distribution, is straightforward). The scale parameter of this distribution is itself a random variable, with a Gamma distribution. We thus obtain what might be termed a compound Gamma-Gamma bivariate distribution. The cumulative distribution function of this may be expressed in terms of one of the double hypergeometric functions of Appell.Generalised hypergeometric functions play an important part in this paper, and one of Saran's triple hypergeometric functions is obtained when generalising the above model to permit the scale parameters of the distributions for the two components to be correlated, rather than identical.Work started while the author was with the Transport Studies Group, University College London.     

Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X ₁,...,X _n ) be a sample from an unknown cumulative distribution function F defined on the real line . The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.     

A distribution-free least squares estimator for censored linear regression models

This paper describes a method for estimating simultaneously the parameter vector of the systematic component and the distribution function of the random component of a censored linear regression model. The estimator is obtained by minimizing the sum of the squares of the differences between the observed values of the dependent variable and the corresponding expected values of this variable according to the estimated parameter vector and distribution function. The resulting least squares parameter estimator incorporates information on the distribution of the random component of the regression model that is available from the estimation sample. Hence, it may often be more efficient than are parameter estimators that do not use such information. The results of numerical experiments with the least squares estimator tend to support this hypothesis.     

RANK TESTS IN 2X2 DESIGNS

Abstract. In literature numerous attempts can be found for the evaluation of two factor designs with fixed effects by means of rank tests. The aim of the present article is to show the limits of these methods and to give some new procedures for 2X2 designs. First, functionals of distribution functions shall be defined whose relations to the usual parameters of the linear model are analysed. These functionals are free of nuissance parameters under the respective hypothesis; they are estimated by special ranks of the data. The asymptotic distribution of these statistics is derived by a generalization of the Chemoff–Savage theorem for correlated random variables. The asymptotic variance depends on the parent distribution function but it can be estimated by using special rank methods. Thus, one obtains asymptotically distribution–free tests for two–factor designs with fixed effects. Some counter examples show why it is not possible to construct suitable rank tests for greater designs than the 2X2 design. The paper closes with a discussion of the drawbacks of the well known rank transform.     

A POISSON PROCESS APPROXIMATION FOR GENERALIZED K–S CONFIDENCE REG

One–sided confidence regions for continuous cumulative distribution functions are constructed using empirical cumulative distribution functions and the generalized Kolmogorov–Smimov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. To avoid tedious computation of confidence levels and critical values, an approximation based on the Poisson process is introduced. This approximation provides a conservative confidence region; moreover, the approximation error decreases monotonically to 0 as sample size increases. Critical values necessary for implementation are given. Applications are made to the areas of risk analysis, investment modelling, and analysis of fault–tolerant systems.     

Performance monitoring of credit portfolios using survival analysis

We are interested in detecting changes in the performance of a credit portfolio quickly and robustly. The portfolio is dynamic: customers can either default or pay the full amount, and new customers can be taken on. Robust detection means that changing the number of new customers taken on should not lead to either a false or delayed signal. We investigate the performances of monitoring schemes via a simulation study that uses several scenarios. We consider monitoring based on default rates estimated through a gliding window, cumulative sum (CUSUM) charts based on default rates, CUSUM charts based on defaults within a given follow-up time after arrival, and a survival analysis CUSUM chart. We conclude that using a survival analysis approach is preferable to using the other approaches.     

The long-run performance of initial public offerings: Stochastic dominance criteria

We examine the long-run performance of initial public offerings (IPOs) using the idea of stochastic dominance. The analysis is a first attempt using a non-event study methodology to evaluate long-horizon performance. We find that there is no first-order stochastic dominance relation between the IPO portfolio and the benchmark of a broad index or a portfolio including either small size or low book-to-market stocks. However, those benchmarks second-order stochastically dominate the IPO portfolio. When using a portfolio including both small size and low book-to-market stocks as benchmark, there is a clear dominance of the IPO portfolio over the benchmark for both orders. Our findings generally imply that the question of assessing portfolio performance between IPO firms and benchmark portfolios depends critically on the specific construction or the cumulative distribution function of the benchmark portfolios. The empirical results also potentially explain the extent of sample dependent results in the literature.     

Testing multivariate distributions in GARCH models

In this paper, we consider testing distributional assumptions in multivariate GARCH models based on empirical processes. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions. We then apply the Khmaladze's martingale transformation (K-transformation) to the empirical process in the presence of estimated parameters. The K-transformation eliminates the effect of parameter estimation, allowing a distribution-free test statistic to be constructed. We show that the K-transformation takes a very simple form for testing multivariate normal and multivariate t-distributions. The procedure is applied to a multivariate financial time series data set.     

Constructive identification of the mixed proportional hazards model

We give a new proof of the identifiably of the MPH model. This proof is constructive: it is a recipe for constructing the triple—regression function, base-line hazard, and distribution of the individual effect—from the observed cumulative distribution functions.
We then prove that the triples do not depend continuously on the observed cumulative distribution functions. Uniformly consistent estimators do not exist.
Finally we show that the MPH model is even identifiable from two-sided censored observations. This proof is constructive, too.     

Quasi goodness of fit tests for lifetime distributions

In this paper we propose a general method for the construction of tests that can be used for testing goodness of fit of lifetime distributions. The method is the following: first find an identity which holds for the survival function or the cumulative hazard function of the null distribution. Then replace the function by a consistent estimate. The resulting statistic is asymptotically normal. Estimating its asymptotic variance then gives a test statistic which is underH ₀ asymptotically chi². The method can be used for randomly (right) censored and single type-I (right) censored data. We apply this method to the following distributions: Weibull, Log-logistic, Log-normal, Half-normal, Rayleigh, Gompertz, Pareto.     

Empirically-transformed linear opinion pools

The linear opinion pool (LOP) produces potentially non-Gaussian combination forecast densities. In this paper, we propose a computationally convenient transformation for the LOP to mirror the non-Gaussianity exhibited by the target variable. Our methodology involves a Smirnov transform to reshape the LOP combination forecasts using the empirical cumulative distribution function. We illustrate our empirically transformed opinion pool (EtLOP) approach with an application examining quarterly real-time forecasts for U.S. inflation evaluated on a sample from 1990:1 to 2020:2. EtLOP improves performance by approximately 10% to 30% in terms of the continuous ranked probability score across forecasting horizons.     

Applications of subsampling,hybrid, and size-correction methods

This paper analyzes the properties of subsampling, hybrid subsampling, and size-correction methods in two non-regular models. The latter two procedures are introduced in Andrews and Guggenberger (2009a). The models are non-regular in the sense that the test statistics of interest exhibit a discontinuity in their limit distribution as a function of a parameter in the model. The first model is a linear instrumental variables (IV) model with possibly weak IVs estimated using two-stage least squares (2SLS). In this case, the discontinuity occurs when the concentration parameter is zero. The second model is a linear regression model in which the parameter of interest may be near a boundary. In this case, the discontinuity occurs when the parameter is on the boundary.     

Predicting LGD distributions with mixed continuous and discrete ordinal outcomes

The loss given default (LGD) distribution is known to have a complex structure. Consequently, the parametric approach for its prediction by fitting a density function may suffer a loss of predictive power. To overcome this potential drawback, we use the cumulative probability model (CPM) to predict the LGD distribution. The CPM applies a transformed variable to model the LGD distribution. This transformed variable has a semiparametric structure. It models the predictor effects parametrically. The functional form of the transformation is unspecified. Thus, CPM provides more flexibility and simplicity in modeling the LGD distribution. To implement CPM, we collect a sample of defaulted debts from Moody’s Default and Recovery Database. Given this sample, we use an expanding rolling window approach to investigate the out-of-time performance of CPM and its alternatives. Our results confirm that CPM is better than its alternatives, in the sense of yielding more accurate LGD distribution predictions.     

A Conditional Approach to Measure Mortality Reductions Due to Cancer Screening

The prevailing lack of consensus about the comparative harms and benefits of cancer screening stems, in part, from the inappropriate calculations of the expected mortality impact of a sustained screening programme. There is an inherent, and often substantial, time lag from the time of screening until the resulting mortality reductions begin, reach their maximum and ultimately end. However, the cumulative mortality reduction reported in a randomised screening trial is typically calculated over an arbitrarily defined follow‐up period, including follow‐up time where the mortality impact is yet to realise or where it has already been exhausted. Because of this, the cumulative reduction cannot be used for projecting the mortality impact expected from a sustained screening programme. For this purpose, we propose a new measure, the time‐specific probability of being helped by screening, given that the cancer would have proven fatal otherwise. This can be decomposed into round‐specific impacts, which in turn can be parametrised and estimated from the trial data. This represents a major shift in quantifying the benefits due to a sustained screening programme, based on statistical evidence extracted from existing trial data. We illustrate our approach using data from screening trials in lung and colorectal cancers.     

Planning step-stress life test with progressively type I group-censored exponential data

Accelerated life testing of products is used to get information quickly on their lifetime distributions. This paper discusses a k -stage step-stress accelerated life test under progressive type I censoring with grouped data. An exponential lifetime distribution with mean life that is a log-linear function of stress is considered. A cumulative exposure model is also assumed. We use the maximum likelihood method to obtain the estimators of the model parameters. The methods for obtaining the optimum test plan are investigated using the variance-optimality and D-optimality criteria. Some numerical studies are discussed to illustrate the proposed criteria.     

Asymptotic efficiency in estimation with conditional moment restrictions

In this paper, bounds on asymptotic efficiency are derived for a class of non-parametric models. The data are independent and identically distributed according to some unknown distribution F. There is a given function of the data and a parameter. The restrictions are that a conditional expectation of this function is zero at some point in the parameter space; this point is to be estimated. If F is assumed to be a multinomial distribution with known (finite) support, then the problem becomes parametric and the bound can be obtained from the information matrix. This bound turns out to depend only upon certain conditional moments, and not upon the support of the distribution. Since a general F can be approximated by a multinomial distribution, the multinomial bound applies to the general case.     

Bayesian estimation of inefficiency heterogeneity in stochastic frontier models

Estimation of the one sided error component in stochastic frontier models may erroneously attribute firm characteristics to inefficiency if heterogeneity is unaccounted for. However, unobserved inefficiency heterogeneity has been little explored. In this work, we propose to capture it through a random parameter which may affect the location, scale, or both parameters of a truncated normal inefficiency distribution using a Bayesian approach. Our findings using two real data sets, suggest that the inclusion of a random parameter in the inefficiency distribution is able to capture latent heterogeneity and can be used to validate the suitability of observed covariates to distinguish heterogeneity from inefficiency. Relevant effects are also found on separating and shrinking individual posterior efficiency distributions when heterogeneity affects the location and scale parameters of the one-sided error distribution, and consequently affecting the estimated mean efficiency scores and rankings. In particular, including heterogeneity simultaneously in both parameters of the inefficiency distribution in models that satisfy the scaling property leads to a decrease in the uncertainty around the mean scores and less overlapping of the posterior efficiency distributions, which provides both more reliable efficiency scores and rankings.     

Parametric bootstrap tests for continuous and discrete distributions

Statistical procedures based on the estimated empirical process are well known for testing goodness of fit to parametric distribution families. These methods usually are not distribution free, so that the asymptotic critical values of test statistics depend on unknown parameters. This difficulty may be overcome by the utilization of parametric bootstrap procedures. The aim of this paper is to prove a weak approximation theorem for the bootstrapped estimated empirical process under very general conditions, which allow both the most important continuous and discrete distribution families, along with most parameter estimation methods. The emphasis is on families of discrete distributions, and simulation results for families of negative binomial distributions are also presented.