Nonparametric Estimation of the Average Availability

The average availability of a repairable system is the expected proportion of time that the system is operating in the interval [0, t]. The present article discusses the nonparametric estimation of the average availability when (i) the data on ‘n’ complete cycles of system operation are available, (ii) the data are subject to right censorship, and (iii) the process is observed upto a specified time ‘T’. In each case, a nonparametric confidence interval for the average availability is also constructed. Simulations are conducted to assess the performance of the estimators.     

Relation of modified power series distributions to lagrangian probability distributions

The class of Lagrangian probability distributions ‘LPD’, given by the expansion of a probability generating function f‘t’ under the transformation u = t/g‘t’ where g ‘t’ is also a p.g.f., has been substantially widened by removing the restriction that the defining functions g ‘t’ and f‘t’ be probability generating functions. The class of modified power series distributions defined by Gupta ‘1974’ has been shown to be a sub-class of the wider class of LPDs     

An improved fuzzy time-series method of forecasting based on L–R fuzzy sets and its application

Classical time-series theory assumes values of the response variable to be ‘crisp’ or ‘precise’, which is quite often violated in reality. However, forecasting of such data can be carried out through fuzzy time-series analysis. This article presents an improved method of forecasting based on L–R fuzzy sets as membership functions. As an illustration, the methodology is employed for forecasting India's total foodgrain production. For the data under consideration, superiority of proposed method over other competing methods is demonstrated in respect of modelling and forecasting on the basis of mean square error and average relative error criteria. Finally, out-of-sample forecasts are also obtained.     

Using randomized response from a complex survey to estimate a sensitive proportion in a dichotomous finite population

In estimating the proportion θ_A of people in a given community bearing a sensitive characteristic A, in order to protect the respondent's privacy, various techniques of generating randomized response (RR) rather than direct response are available in the literature. But the theory concerning them is developed only for samples selected by ‘simple random sampling’ (SRS) ‘with replacement’ or at most by SRS without replacement method. Illustrating two such RR devices we show how an estimator along with an estimated measure of its error may be developed if the sample of persons may be drawn adopting a complex survey design involving unequal selection probabilities with or without replacement.     

FACTORIAL ANALYSIS OF RECIDIVIST DATA

This paper presents a method of fitting factorial models to recidivism data consisting of the (possibly censored) time to ‘fail’ of individuals, in order to test for differences between groups. Here ‘failure’ means rearrest, reconviction or reincarceration, etc. A proportion P of the sample is assumed to be ‘susceptible’ to failure, i.e. to fail eventually, while the remaining 1-P are ‘immune’, and never fail. Thus failure may be described in two ways: by the probability P that an individual ever fails again (‘probability of recidivism’), and by the rate of failure Λ for the susceptibles. Related analyses have been proposed previously: this paper argues that a factorial approach, as opposed to regression approaches advocated previously, offers simplified analysis and interpretation of these kinds of data. The methods proposed, which are also applicable in medical statistics and reliability analyses, are demonstrated on data sets in which the factors are Parole Type (released to freedom or on parole), Age group (≤ 20 years, 20–40 years, > 40 years), and Marital Status. The outcome (failure) is a return to prison following first or second release.     

Reference regions for beat‐to‐beat ECG data

The QT interval is regarded as an important biomarker for the assessment of arrhythmia liability, and evidence of QT prolongation has led to the withdrawal and relabeling of numerous compounds. Traditional methods of assessing QT prolongation correct the QT interval for the length of the RR interval (which varies inversely with heart‐rate) in a variety of ways. These methods often disagree with each other and do not take into account changes in autonomic state. Correcting the QT interval for RR reduces a bivariate observation (RR, QT) to a univariate observation (QTc). The development of automatic electrocardiogram (ECG) signal acquisition systems has made it possible to collect continuous (so called ‘beat‐to‐beat’) ECG data. ECG data collected prior to administration of a compound allow us to define a region for (RR, QT) values that encompasses typical activity. Such reference regions are used in clinical applications to define the ‘normal’ region of clinical or laboratory measurements. This paper motivates the need for reference regions of (RR, QT) values from beat‐to‐beat ECG data, and describes a way of constructing these. We introduce a measure of agreement between two reference regions that points to the reliability of 12‐lead digital Holter data. We discuss the use of reference regions in establishing baselines for ECG parameters to assist in the evaluation of cardiac risk and illustrate using data from two methodological studies. Copyright © 2010 John Wiley & Sons, Ltd.     

Semiparametric Estimators for Limited Dependent Variable (LDV) Models with Endogenous Regressors

This article reviews semiparametric estimators for limited dependent variable (LDV) models with endogenous regressors, where nonlinearity and nonseparability pose difficulties. We first introduce six main approaches in the linear equation system literature to handle endogenous regressors with linear projections: (i) ‘substitution’ replacing the endogenous regressors with their projected versions on the system exogenous regressors x, (ii) instrumental variable estimator (IVE) based on E{(error) × x} = 0, (iii) ‘model-projection’ turning the original model into a model in terms of only x-projected variables, (iv) ‘system reduced form (RF)’ finding RF parameters first and then the structural form (SF) parameters, (v) ‘artificial instrumental regressor’ using instruments as artificial regressors with zero coefficients, and (vi) ‘control function’ adding an extra term as a regressor to control for the endogeneity source. We then check if these approaches are applicable to LDV models using conditional mean/quantiles instead of linear projection. The six approaches provide a convenient forum on which semiparametric estimators in the literature can be categorized, although there are a few exceptions. The pros and cons of the approaches are discussed, and a small-scale simulation study is provided for some reviewed estimators.     

Another look at linear programming for feature selection via methods of regularization

We consider statistical procedures for feature selection defined by a family of regularization problems with convex piecewise linear loss functions and penalties of l ₁ nature. Many known statistical procedures (e.g. quantile regression and support vector machines with l ₁-norm penalty) are subsumed under this category. Computationally, the regularization problems are linear programming (LP) problems indexed by a single parameter, which are known as ‘parametric cost LP’ or ‘parametric right-hand-side LP’ in the optimization theory. Exploiting the connection with the LP theory, we lay out general algorithms, namely, the simplex algorithm and its variant for generating regularized solution paths for the feature selection problems. The significance of such algorithms is that they allow a complete exploration of the model space along the paths and provide a broad view of persistent features in the data. The implications of the general path-finding algorithms are outlined for several statistical procedures, and they are illustrated with numerical examples.     

A new interval mapping approach based on a general sib-pair regression model with a modified Wald test

Various regression models based on sib-pair data have been developed for mapping quantitative trait loci (QTL) in humans since the seminal paper published in 1972 by Haseman and Elston. Fulker and Cardon [D.W. Fulker, L.R. Cardon, A sib-pair approach to interval mapping of quantitative trait loci, Am. J. Hum. Genet. 54 (1994) 1092–1103] adapted the idea of interval mapping [E.S. Lander, D. Botstein, Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps, Genetics 121 (1989) 185–199] to the Haseman–Elston regression model in order to increase the power of QTL mapping. However, in the interval mapping approach of Fulker and Cardon, the statistic for testing QTL effects does not obey the classical statistical theory and hence critical values of the test can not be appropriately determined. In this article, we consider a new interval mapping approach based on a general sib-pair regression model. A modified Wald test is proposed for the testing of QTL effects. The asymptotic distribution of the modified Wald test statistic is provided and hence the critical values or the $p$-values of the test can be well determined. Simulation studies are carried out to verify the validity of the modified Wald test and to demonstrate its desirable power.     

Learning causal structure from mixed data with missing values using Gaussian copula models

We consider the problem of causal structure learning from data with missing values, assumed to be drawn from a Gaussian copula model. First, we extend the ‘Rank PC’ algorithm, designed for Gaussian copula models with purely continuous data (so-called nonparanormal models), to incomplete data by applying rank correlation to pairwise complete observations and replacing the sample size with an effective sample size in the conditional independence tests to account for the information loss from missing values. When the data are missing completely at random (MCAR), we provide an error bound on the accuracy of ‘Rank PC’ and show its high-dimensional consistency. However, when the data are missing at random (MAR), ‘Rank PC’ fails dramatically. Therefore, we propose a Gibbs sampling procedure to draw correlation matrix samples from mixed data that still works correctly under MAR. These samples are translated into an average correlation matrix and an effective sample size, resulting in the ‘Copula PC’ algorithm for incomplete data. Simulation study shows that: (1) ‘Copula PC’ estimates a more accurate correlation matrix and causal structure than ‘Rank PC’ under MCAR and, even more so, under MAR and (2) the usage of the effective sample size significantly improves the performance of ‘Rank PC’ and ‘Copula PC.’ We illustrate our methods on two real-world datasets: riboflavin production data and chronic fatigue syndrome data.

     

Simultaneous Confidence Tubes in Multivariate Linear Regression

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one particular simultaneous confidence band for multivariate linear regression. Because of the shape of the band, the word ‘tube’ is more pertinent and so will be used to replace the word ‘band’. It is shown that the construction of the tube is related to the distribution of the largest eigenvalue. A simulation‐based method is proposed to compute the 1 ? α quantile of this eigenvalue. With the computation power of modern computers, the simultaneous confidence tube can be computed fast and accurately. A real‐data example is used to illustrate the method, and many potential research problems have been pointed out.     

The analysis of ordinal time-series data via a transition (Markov) model

While standard techniques are available for the analysis of time-series (longitudinal) data, and for ordinal (rating) data, not much is available for the combination of the two, at least in a readily-usable form. However, this data type is common place in the natural and health sciences where repeated ratings are recorded on the same subject. To analyse these data, this paper considers a transition (Markov) model where the rating of a subject at one time depends explicitly on the observed rating at the previous point of time by incorporating the previous rating as a predictor variable. Complications arise with adequate handling of data at the first observation (t=1), as there is no prior observation to use as a predictor. To overcome this, it is postulated the existence of a rating at time t=0; however it is treated as ‘missing data’ and the expectation–maximisation algorithm used to accommodate this. The particular benefits of this method are shown for shorter time series.     

Coverage‐adjusted Confidence Intervals for a Binomial Proportion

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap‐type plots for comparing confidence intervals, we show that the coverage‐adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.     

CLONING DATA: GENERATING DATASETS WITH EXACTLY THE SAME MULTIPLE LINEAR REGRESSION FIT

This paper presents a simple computational procedure for generating ‘matching’ or ‘cloning’ datasets so that they have exactly the same fitted multiple linear regression equation. The method is simple to implement and provides an alternative to generating datasets under an assumed model. The advantage is that, unlike the case for the straight model‐based alternative, parameter estimates from the original data and the generated data do not include any model error. This distinction suggests that ‘same fit’ procedures may provide a general and useful alternative to model‐based procedures, and have a wide range of applications. For example, as well as being useful for teaching, cloned datasets can provide a model‐free way of confidentializing data.     

A family of methods for statistical disclosure control

Statistical disclosure control (SDC) is a balancing act between mandatory data protection and the comprehensible demand from researchers for access to original data. In this paper, a family of methods is defined to ‘mask’ sensitive variables before data files can be released. In the first step, the variable to be masked is ‘cloned’ (C). Then, the duplicated variable as a whole or just a part of it is ‘suppressed’ (S). The masking procedure's third step ‘imputes’ (I) data for these artificial missings. Then, the original variable can be deleted and its masked substitute has to serve as the basis for the analysis of data. The idea of this general ‘CSI framework’ is to open the wide field of imputation methods for SDC. The method applied in the I-step can make use of available auxiliary variables including the original variable. Different members of this family of methods delivering variance estimators are discussed in some detail. Furthermore, a simulation study analyzes various methods belonging to the family with respect to both, the quality of parameter estimation and privacy protection. Based on the results obtained, recommendations are formulated for different estimation tasks.     

Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.     

Thermal,viscoelastic and mechanical properties' optimization of polyphenylene sulfide via optimal processing parameters using the Taguchi method

Thermal, viscoelastic and mechanical properties of polyphenylene sulfide (PPS) were optimized as a function of extrusion and injection molding parameters. For this purpose, design of experiments approach utilizing Taguchi's L₂₇ (3⁷) orthogonal arrays was used. Effect of the parameters on desired properties was determined using the analysis of variance. Differential scanning calorimeter (DSC) tests were performed for the analysis of thermal properties such as melting temperature (T_m) and melting enthalpy (ΔH_M). Dynamic mechanical analysis (DMA) tests were performed for the analysis of viscoelastic properties such as damping factor (tan?δ) and glass transition temperature (T_g). Tensile tests were performed for the analysis of mechanical properties such as tensile strength and modulus. With optimized process parameters, verification DSC, DMA and tensile tests were performed for thermal, viscoelastic and mechanical properties, respectively. The Taguchi method showed that ‘barrel temperature’ and its level of ‘340°C’ were seen to be the most effective parameter and its level; respectively. It was suggested that PPS can be reinforced for further improvement after optimized thermal, viscoelastic and mechanical properties.     

A win ratio approach to comparing continuous non‐normal outcomes in clinical trials

Clinical trials are often designed to compare continuous non‐normal outcomes. The conventional statistical method for such a comparison is a non‐parametric Mann–Whitney test, which provides a P‐value for testing the hypothesis that the distributions of both treatment groups are identical, but does not provide a simple and straightforward estimate of treatment effect. For that, Hodges and Lehmann proposed estimating the shift parameter between two populations and its confidence interval (CI). However, such a shift parameter does not have a straightforward interpretation, and its CI contains zero in some cases when Mann–Whitney test produces a significant result. To overcome the aforementioned problems, we introduce the use of the win ratio for analysing such data. Patients in the new and control treatment are formed into all possible pairs. For each pair, the new treatment patient is labelled a ‘winner’ or a ‘loser’ if it is known who had the more favourable outcome. The win ratio is the total number of winners divided by the total numbers of losers. A 95% CI for the win ratio can be obtained using the bootstrap method. Statistical properties of the win ratio statistic are investigated using two real trial data sets and six simulation studies. Results show that the win ratio method has about the same power as the Mann–Whitney method. We recommend the use of the win ratio method for estimating the treatment effect (and CI) and the Mann–Whitney method for calculating the P‐value for comparing continuous non‐Normal outcomes when the amount of tied pairs is small. Copyright © 2016 John Wiley & Sons, Ltd.