Likelihood ratio-based distribution-free sequential change-point detection

Most existing control charts are for monitoring location or scale parameters, rather than any change in process distribution such as shift in shape. Goodness-of-fit (GOF) test can detect any change in distribution. This paper develops a new distribution-free control chart by integrating a powerful two-sample nonparametric likelihood ratio GOF test into the effective change-point model. Our proposed chart is easy in computation, convenient to use, and very efficient in detecting any change in process distribution, including shifts in location, scale, and shape. It is also robust in detecting various magnitudes of shifts and especially powerful in monitoring any distributional change involving a decrease in scale.     

Ancillarity properties of generalized residuals with applications in failure time models

This paper addresses the issue of when residuals from failure time models, which are useful in model validation and diagnostics, possess a conditional ancillarity property. This property states that the distribution of the residuals depends on the model parameters only through a many-to-one function of these parameters, which in certain models turn out to be the censoring proportion. Concrete results are obtained for models which possess an invariance structure, and these results are applied to commonly used failure time models. Aside from furthering our understanding of the distributional structure of residuals, this conditional ancillarity property can be exploited to study in a more efficient manner the distributional properties of residuals either analytically and/or through numerical methods.     

The Govindarajulu Distribution: Some Properties and Applications

In this article, we present various distributional properties and application to reliability analysis of the Govindarajulu distribution. A quantile-based analysis is performed as the distribution function is not analytically tractable. The properties of the distribution like percentiles, L-moments, L-skewness, and kurtosis and order statistics are presented. Various reliability characteristics are derived along with some characterization theorems by relationship between reliability measures. We also make a comparative study with other competing models with reference to real data.     

Multivariate regression models for discrete and continuous repeated measurements

A general class of multivariate regression models is considered for repeated measurements with discrete and continuous outcome variables. The proposed model is based on the seemingly unrelated regression model (Zellner, 1962) and an extension of the model of Park and Woolson(1992). The regression parameters of the model are consistently estimated using the two-stage least squares method. When the out come variables are multivariate normal, the two-stage estimator reduces to Zellner’s two-stage estimator. As a special case, we consider the marginal distribution described by Liang and Zeger (1986). Under this this distributional assumption, we show that the two-stage estimator has similar asymptotic properties and comparable small sample properties to Liang and Zeger's estimator. Since the proposed approach is based on the least squares method, however, any distributional assumption is not required for variables outcome variables. As a result, the proposed estimator is more robust to the marginal distribution of outcomes.     

Discrete Burr and discrete Pareto distributions

In this paper we obtain discrete Burr and Pareto distributions using the general approach of discretizing a continuous distribution and propose them as suitable lifetime models. It may be worth exploring the possibility of developing discrete versions of the Burr and Pareto distributions, so that, the same can be used for modeling discrete data. The equivalence of continuous and discrete Burr distributions has been established. Some important distributional properties and estimation of reliability characteristics are discussed. An application in reliability estimation in series system and a real data example on dentistry using this distribution is also discussed.     

Estimation of Change Point in Generalized Variance Control Chart

In this study, using maximum likelihood estimation, a considerably effective change point model is proposed for the generalized variance control chart in which the required statistics are calculated with its distributional properties. The procedure, when used with generalized variance control charts, would be helpful for practitioners both controlling the multivariate process dispersion and detecting the time of the change in variance-covariance matrix of a process. The procedure starts after the chart issues a signal. Several structural changes for the variance-covariance matrix are considered and the precision and the accuracy of the proposed method is discussed.     

Tests for Structural Changes in Time Series of Counts

We propose methods for detecting structural changes in time series with discrete‐valued observations. The detector statistics come in familiar L2‐type formulations incorporating the empirical probability generating function. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. For both models, we study mainly structural changes due to a change in distribution, but we also comment for the classical problem of parameter change. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is also included along with a real data example.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

Correlated Errors in the Parameters Estimation of the ARFIMA Model: A Simulated Study

Kurtosis, usually as measured by the standardised fourth central moment, has been examined on a number of occasions by observing the effect of contaminating the distribution, that is, mixing in another distribution. However, superficial treatment can lead, and indeed has led, to misunderstandings. This paper considers, firstly for a symmetric distribution contaminated at two points symmetrically placed around its centre and then for a mixture of two continuous symmetric distributions, the behaviour of three measures of kurtosis. This is done in general and not just as the mixing proportion tends to zero as in the influence function approach. It is seen that when both scale and kurtosis change, the latter is not necessarily intuitive. It is also illustrated that parameter interpretation in terms of distributional properties such as shape can be misleading without the use of the appropriate distributional partial ordering     

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.     

Monitoring the parameter vector of regression models with time-to-event response in phase II processes

In most of the existing specialized literature, monitoring regression models are a special case of profile monitoring. However, not every regression model always represents appropriately a profile data structure. This is clearly the case of the Weibull regression model (WRM) with common shape parameter γ. Even though it might be thought that existing methodologies (especially likelihood-ratio (LRT)-based methods) for monitoring generalized linear profiles can also be successfully applied to monitoring regression models with time-to-event response, it will be shown in this paper that those methodologies work fairly acceptable just for data structures with 1000 observations at least approximately. It was found out that some corrections, often referred to as Bartlett's adjustments, are needed to be implemented in order to improve the accuracy of using the asymptotic distributional properties of the LRT statistic for carrying out the monitoring of WRM with relatively small and moderate dimensions of the available datasets. Simulation studies suggest that the use of the aforementioned corrections make the resulting charts work quite acceptable when available data structures contain 30 observations at least. Detection abilities of the proposed schemes improve as dataset dimension increases.     

A new four-parameter discrete distribution with bathtub and unimodal failure rate

In this paper, a discrete counterpart of the general class of continuous beta-G distributions is introduced. A discrete analog of the beta generalized exponential distribution of Barreto-Souza et al. [2], as an important special case of the proposed class, is studied. This new distribution contains some previously known discrete distributions as well as two new models. The hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Some distributional and moment properties of the new distribution as well as its order statistics are discussed. Estimation of the parameters is illustrated using the maximum likelihood method and, finally, the model with a real data set is examined.     

Information,measure shifts and distribution metrics

Reflexive shifting of a given distribution, using its own distribution function, can reveal information. The shifts are changes in measure such that the separation of the resulting left and right unit shifted distributions reveals the binary entropy of position, called locational or partition entropy. This can be used for spread and asymmetry functions. Alternatively, summary metrics for distributional asymmetry and spread can be based on the relative strengths of left- and right-hand shifts. Such metrics are applicable even for long tail densities where distributional moments may not exist.     

Goodness-of-fit-based outlier detection for Phase I monitoring

ABSTRACT

Nonparametric charts are useful in statistical process control when there is a lack of or limited knowledge about the underlying process distribution. Most existing approaches in the literature of Phase I monitoring assume that outliers have the same distributions as the in-control sample but only differ in location or scale parameters, they may not be effective with distributional changes. This article develops a new procedure based on the integration of the classical Anderson–Darling goodness-of-fit test and the stepwise isolation method. Our proposed procedure is efficient in detecting potential shifts in location, scale, or shape, and thus it offers robust protection against variation in various underlying distributions. The finite sample performance of our method is evaluated through simulations and is compared with that of available outlier detection methods for Phase I monitoring.     

Estimation for change point of discretely observed ergodic diffusion processes

We treat the change point problem in ergodic diffusion processes from discrete observations. Tonaki et al. (2021a) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion process models. When any change in the diffusion or drift parameter is detected by this or any other method, the next question to consider is where the change point is located. Therefore, we propose the method to estimate the change point of the parameter for two cases: the case where there is a change in the diffusion parameter, and the case where there is no change in the diffusion parameter but a change in the drift parameter. Furthermore, we present rates of convergence and distributional results of the change point estimators. Some examples and simulation results are also given.     

Goodness of fit tests for the multiple logistic regression model

Several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model. The test statistics are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions. The null distributions of these statistics are examined by applying the theory for chi-square tests of Moore Spruill (1975) and through computer simulations. All statistics are shown to have a chi-square distribution or a distribution which can be well approximated by a chi-square. The degrees of freedom are shown to depend on the particular statistic and the distributional assumptions.

The power of each of the proposed statistics is examined for the normal, linear, and exponential alternative models using computer simulations.     

Quantal models: a review with additional methodological development

Analysis of quantal models is a particular aspect of the general problem of investigating multimodality. The distinction is that the spacings between modes are integral multiples of some unspecified fundamental unit and that the number of modes is not defined. Such semi-structured models arise in a wide variety of contexts such as biology, cosmology, archaeology and molecular physics. This paper presents a brief review of their historical development in such areas as an aid to their recognition in other contexts as well as giving guidance to their analysis from the statistical viewpoint. The available methodology for their analysis is collated into a coherent and self-contained account, establishing various optimality properties under particular parametric distributional assumptions. An illustrative power study shows how dependence on sample size and failure of assumptions such as underlying distribution, origin of measurements and independence affect the power of various analyses. These aspects are illustrated by an example from developmental biology.     

Generalized order statistics from two parameter uniform distribution

In this paper some distributional properties of the generalized order statistics from uniform distribution are given. The minimum variance linear unbiased as well best ( in the sense of minimum mean squared error) invariant estimators of the parameters of the two parameter uniform distribution based on the first m generalized order statistics are presented.     

Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data

We consider the probability-weighted moment and the maximum-likelihood estimators of two parameters in the log-logistic distribution. Quantile estimators are obtained using both methods. The distributional properties of these estimators are studied in large samples, via asymptotic theory, and in small and moderate samples, via Monte Carlo simulation. The distribution is shown to be appropriate for a wide variety of meteorological data.     

EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA

The most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non‐negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non‐negative data. We explore exponential smoothing state space models for non‐negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models that do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non‐Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short‐term forecasting performance of the various models using data on the weekly sales of over 300 items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one‐step‐ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed.