Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains xviii, 262 p.; also includes graphics (some col.). Includes bibliographical references (p. 246-262).

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. The term nonparametric is not intended to imply that there are no parameters involved, in fact, quite the contrary. While the term distribution-free seems to be a better description of what we expect from these charts, that is, they remain valid for a large class of distributions, nonparametric is perhaps the term more often used. In the statistics literature there is now a rather vast collection of nonparametric tests and confidence intervals and these methods have been shown to perform well compared to their normal theory counterparts. Remarkably, even when the underlying distribution is normal, the efficiency of some nonparametric tests relative to the corresponding (optimal) normal theory methods can be as high as 0.955 (see e.g. Gibbons and Chakraborti (2010) page 218). For some other heavy-tailed and skewed distributions, the efficiency can be 1.0 or even higher. It may be argued that nonparametric methods will be ‘less efficient’ than their parametric counterparts when one has a complete knowledge of the process distribution for which that parametric method was specifically designed. However, the reality is that such information is seldom, if ever, available in practice. Thus it seems natural to develop and use nonparametric methods in statistical process control (SPC) and the quality practitioners will be well advised to have these techniques in their toolkits. In this thesis we only propose univariate nonparametric control charts designed to track the location of a continuous process since very few charts are available for monitoring the scale and simultaneously monitoring the location and scale of a process. Chapter 1 gives a brief introduction to SPC and provides background information regarding the research conducted in this thesis. This will aid in familiarizing the reader with concepts and terminology that are helpful to the following chapters. Details are given regarding the three main classes of control charts, namely the Shewhart chart, the cumulative sum (CUSUM) chart and the exponentially weighted moving average (EWMA) chart. We begin Chapter 2 with a literature overview of Shewhart-type Phase I control charts followed by the design and implementation of these charts. A nonparametric Shewhart-type Phase I control chart for monitoring the location of a continuous variable is proposed. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. The derivations recognize that in Phase I the signalling events are dependent and that more than one comparison is © University of Pretoria v made against the same estimated limits simultaneously; this leads to working with the joint distribution of a set of dependant random variables. An exact expression for the false alarm probability is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits. Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. In Chapter 3 Phase II control charts are introduced and considered for the case when the underlying parameters of the process distribution are known or specified. This is referred to as the ‘standard(s) known’ case and is denoted Case K. Two nonparametric Phase II control charts are considered in this chapter, with the first one being a nonparametric exponentially weighted moving average (NPEWMA)-type control chart based on the sign (SN) statistic. A Markov chain approach (see e.g. Fu and Lou (2003)) is used to determine the run-length distribution of the chart and some associated performance characteristics (such as the average, standard deviation, median and other percentiles). In order to aid practical implementation, tables are provided for the chart’s design parameters. An extensive simulation study shows that on the basis of minimal required assumptions, robustness of the in-control run-length distribution and out-of-control performance, the proposed NPEWMA-SN chart can be a strong contender in many applications where traditional parametric charts are currently used. Secondly, we consider the NPEWMA chart that was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see e.g. Gibbons and Chakraborti (2010) page 195). This is called the nonparametric exponentially weighted moving average signed-rank (NPEWMA-SR) chart. In their article important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this thesis we address these issues with a more in-depth study of the NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the chart’s design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. In Chapter 4 Phase II control charts are introduced and considered for the case when the underlying parameters of the process distribution are unknown and need to be estimated. This is referred to as the ‘standard(s) unknown’ case and is denoted Case U. Two nonparametric Phase II control charts are proposed in this chapter. They are a Phase II NPEWMA-type control chart and a nonparametric cumulative sum (NPCUSUM)-type control chart, based on the exceedance statistics, © University of Pretoria vi respectively, for detecting a shift in the location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and / or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. We also investigate the choice of the order statistics (percentile), from the reference (Phase I) sample that defines the exceedance statistic. It is observed that other choices, such as the third quartile, can play an important role in improving the performance of these exceedance charts. It is seen that these exceedance charts perform as well as and, in many cases, better than its competitors and thus can be a useful alternative chart in practice. Chapter 5 wraps up this thesis with a summary of the research carried out and offers concluding remarks concerning unanswered questions and / or future research opportunities. © University / Thesis (PhD)--University of Pretoria, 2013. / gm2013 / Statistics / restricted

The small-sample bias and root mean squared error of several distribution-free estimators of the variance of the sample median are examined. A new estimator is proposed that is easy to compute and tends to have the smallest bias and root mean squared error.

The main objective of this dissertation is the development of CUSUM procedures based on signed and unsigned sequential ranks. These CUSUMs can be applied to detect changes in the location or dispersion of a process. The signed and unsigned sequential rank CUSUMs are distribution-free and robust against the effect of outliers in the data. The only assumption that these CUSUMs require is that the in-control distribution is symmetric around a known location parameter. These procedures specifically do not require the existence of any higher order moments. Another advantage of these CUSUMs is that Monte Carlo simulation can readily be applied to deliver valid estimates of control limits, irrespective of what the underlying distribution may be. Other objectives of this dissertation include a brief discussion of the results and refinements of the CUSUM in the literature. We justify the use of a signed sequential rank statistic. Also, we evaluate the relative efficiency of the suggested procedure numerically and provide three real-world applications from the engineering and financial industries. / MSc (Risk Analysis), North-West University, Potchefstroom Campus, 2015

The main objective of this dissertation is the development of CUSUM procedures based on signed and unsigned sequential ranks. These CUSUMs can be applied to detect changes in the location or dispersion of a process. The signed and unsigned sequential rank CUSUMs are distribution-free and robust against the effect of outliers in the data. The only assumption that these CUSUMs require is that the in-control distribution is symmetric around a known location parameter. These procedures specifically do not require the existence of any higher order moments. Another advantage of these CUSUMs is that Monte Carlo simulation can readily be applied to deliver valid estimates of control limits, irrespective of what the underlying distribution may be. Other objectives of this dissertation include a brief discussion of the results and refinements of the CUSUM in the literature. We justify the use of a signed sequential rank statistic. Also, we evaluate the relative efficiency of the suggested procedure numerically and provide three real-world applications from the engineering and financial industries. / MSc (Risk Analysis), North-West University, Potchefstroom Campus, 2015

The thesis examines technical efficiency using frontier efficiency estimation techniques from parametric and non-parametric approaches. Five different frontier efficiency estimation techniques are considered which are SFA, DFA, DEA-CCR, DEA-BCC and DEA-RAM. These techniques are then used on an artificially generated panel dataset using a two-input two-output production function framework based on characteristics of German life-insurers. The key contribution of the thesis is firstly, a study that uses simulated panel dataset to estimate frontier efficiency techniques and secondly, a research framework that compares multiple frontier efficiency techniques across parametric and non-parametric approaches in the context of simulated panel data. The findings suggest that, as opposed to previous studies, parametric and non-parametric approaches can both generate comparable technical efficiency scores with simulated data. Moreover, techniques from parametric approaches, i.e. SFA and DFA are consistent with each other whereas the same applies to non-parametric approaches, i.e. DEA models. The research study also discusses some important theoretical and methodological implication of the findings and suggests some ways whereby future research can enable to overcome some of the restrictions associated with current approaches.

Nous considérons deux problèmes. Le premier est la construction des tests d’ajustement (goodness-of-fit) pour les modèles de processus de diffusion ergodique. Nous considérons d’abord le cas où le processus sous l’hypothèse nulle appartient à une famille paramétrique. Nous étudions les tests de type Cramer-von Mises et Kolmogorov- Smirnov. Le paramètre inconnu est estimé par l’estimateur de maximum de vraisemblance ou l’estimateur de distance minimale. Nous construisons alors les tests basés sur l’estimateur du temps local de la densité invariante, et sur la fonction de répartition empirique. Nous montrons alors que les statistiques de ces deux types de test convergent tous vers des limites qui ne dépendent pas du paramètre inconnu. Par conséquent, ces tests sont appelés asymptotically parameter free. Ensuite, nous considérons l’hypothèse simple. Nous étudions donc le test du khi-deux. Nous montrons que la limite de la statistique ne dépend pas de la dérive, ainsi on dit que le test est asymptotically distribution free. Par ailleurs, nous étudions également la puissance du test du khi-deux. En outre, ces tests sont consistants. Nous traitons ensuite le deuxième problème : l’approximation des équations différentielles stochastiques rétrogrades. Supposons que l’on observe un processus de diffusion satisfaisant à une équation différentielle stochastique, où la dérive dépend du paramètre inconnu. Nous estimons premièrement le paramètre inconnu et après nous construisons un couple de processus tel que la valeur finale de l’un est une fonction de la valeur finale du processus de diffusion donné. Par la suite, nous montrons que, lorsque le coefficient de diffusion est petit, le couple de processus se rapproche de la solution d’une équations différentielles stochastiques rétrograde. A la fin, nous prouvons que cette approximation est asymptotiquement efficace. / We consider two problems in this work. The first one is the goodness of fit test for the model of ergodic diffusion process. We consider firstly the case where the process under the null hypothesis belongs to a given parametric family. We study the Cramer-von Mises type and the Kolmogorov-Smirnov type tests in different cases. The unknown parameter is estimated via the maximum likelihood estimator or the minimum distance estimator, then we construct the tests in using the local time estimator for the invariant density function, or the empirical distribution function. We show that both the Cramer-von Mises type and the Kolmogorov-Smirnov type statistics converge to some limits which do not depend on the unknown parameter, thus the tests are asymptotically parameter free. The alternatives as usual are nonparametric and we show the consistency of all these tests. Then we study the chi-square test. The basic hypothesis is now simple The chi-square test is asymptotically distribution free. Moreover, we study also power function of the chi-square test to compare with the others. The other problem is the approximation of the forward-backward stochastic differential equations. Suppose that we observe a diffusion process satisfying some stochastic differential equation, where the trend coefficient depends on some unknown parameter. We try to construct a couple of processes such that the final value of one is a function of the final value of the given diffusion process. We show that when the diffusion coefficient is small, the couple of processes approximates well the solution of a backward stochastic differential equation. Moreover, we present that this approximation is asymptotically efficient.

This thesis is about high–dimensional problems considered under the so{called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p &gt; n, assuming that the ratio <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%5Cfrac%7Bp%7D%7Bn%7D" /> converges when the number of parameters and the sample size increase. We focus on the eigenvalue distribution of the considered matrices, since it is a well–known information–carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non–commutative space of random matrices of size p x p equipped with the functional <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cfrac%7B1%7D%7Bp%7DE%5BTr%5C%7B%5Ccdot%5C%7D%5D" />. Here, the connections with free probability theory arise. In the relation to that eld we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant–moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant{moment relation given through non{crossing partitions ofthe set. Furthermore, we investigate the normalized <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" /> and derive, using the dierentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re–establish moments of the Marcenko–Pastur distribution, and hence the recursive relation for the Catalan numbers. In this thesis we also prove that the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D" />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20W%5Csim%5Cmathcal%7BW%7D_p(I_p,n)" />, is a consistent estimator of the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" />. We consider <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20Y_t=%5Csqrt%7Bnp%7D%5Cbig(%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D-m%5E%7B(t)%7D_1%20(n,p)%5Cbig)," />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20m%5E%7B(t)%7D_1%20(n,p)=E%5Cbig%5B%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D%5Cbig%5D" />, which is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness{of{t approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high–dimensional data (p &gt; n) and the multivariate data (p ≤ n).

The currently available distribution-free confidence interval for a difference of medians in a within-subjects design requires an unrealistic assumption of identical distribution shapes. A confidence interval for a general linear function of medians is proposed for within-subjects designs that do not assume identical distribution shapes. The proposed method can be combined with a method for linear functions of independent medians to provide a confidence interval for a linear function of medians in mixed designs. Simulation results show that the proposed methods have good small-sample properties under a wide range of conditions. The proposed methods are illustrated with examples, and R functions that implement the new methods are provided.

In studies where the response variable is measured on a ratio scale, a ratio of means or medians provides a standardized measure of effect size that is an alternative to the popular standardized mean difference. Confidence intervals for ratios of population means and medians in independent-samples designs and paired-samples designs are proposed as supplements to the independent-samples t test and paired-samples t test. The performance of the proposed confidence intervals are evaluated in a simulation study. The proposed confidence interval methods are extended to the case of a 2 × m factorial design that includes propensity score stratification and meta-analysis as special cases. R functions that implement the recommended confidence intervals are provided in the Supplemental Material file, available in the online version of this article, and are illustrated with several examples.