Explicit analytical solutions for ARL of CUSUM chart for a long-memory SARFIMA model

This paper aims to derive explicit analytical solutions for Average Run Length (ARL) of CUSUM chart for the SARFIMA(P,D,Q)_S process with exponential white noise. Measurement of performance was done with the ARL in terms of percentage error and CPU time. The results obtained from the explicit formulas were compared focusing on the performance using the numerical integral equation (NIE) method. Both methods had similarly excellent agreement with the percentage error at less than 0.25%. Meanwhile, the explicit formulas consumed less CPU time than the NIE method. It is clear that the explicit formulas are a good alternative in real applications.     

Ratio test for variance change point in linear process with long memory

In this paper we consider the detection problem of variance change point in linear process with long memory. We propose the ratio test to detect the variance change point. The limiting distribution for test statistics under H ₀ is derived and the consistency of the test is also established. In comparison with the existing CUSUM of squares (SCUSUM) test, the ratio test does not need to estimate the long memory parameter in practical situation and therefore it can be used more conveniently.     

Distribution-Free Exceedance CUSUM Control Charts for Location

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. A Phase II nonparametric cumulative sum (CUSUM) chart based on the exceedance statistics, called the exceedance CUSUM chart, is proposed here for detecting a shift in the unknown 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. Guidelines and recommendations are provided for the chart's design parameters along with an illustrative example. The in- and out-of-control performances of the chart are studied through extensive simulations on the basis of the average run-length (ARL), the standard deviation of run-length (SDRL), the median run-length (MDRL), and some percentiles of run-length. Further, a comparison with a number of existing control charts, including the parametric CUSUM chart and a recent nonparametric CUSUM chart based on the Wilcoxon rank-sum statistic, called the rank-sum CUSUM chart, is made. It is seen that the exceedance CUSUM chart performs well in many cases and thus can be a useful alternative chart in practice. A summary and some concluding remarks are given.     

Comment on “modeling asset returns with alternative stable distributions”

This paper considers a modified CUSUM test, suggested by Dufour (1982) for parameter instability and structural change with an unknown change point in a linear model with serially correlated disturbances, in which a preliminary estimate of the autoregressive coefficient for the error process is obtained, and used to transform the data. Then the standard CUSUM statistic is calculated on the transformed data. This paper derives the asymptotic distribution of the modified CUSUM test. We show that the modified CUSUM test retains its asymptotic significance level, i.e., the modified CUSUM test has the same asymptotic distribution as the CUSUM test with serially uncorrelated errors.     

A smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes

In this paper, we propose a smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes. In contrast to the Q‐learning algorithm in which nonregular inference is involved, we show that, under assumptions adopted in this paper, the proposed smoothed Q‐learning estimator is asymptotically normally distributed even when the Q‐learning estimator is not and its asymptotic variance can be consistently estimated. As a result, inference based on the smoothed Q‐learning estimator is standard. We derive the optimal smoothing parameter and propose a data‐driven method for estimating it. The finite sample properties of the smoothed Q‐learning estimator are studied and compared with several existing estimators including the Q‐learning estimator via an extensive simulation study. We illustrate the new method by analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness–Alzheimer's Disease (CATIE‐AD) study.     

Statistical surveillance of the mean vector and the covariance matrix of nonlinear time series

The purpose of this paper is to jointly monitor the mean vector and the covariance matrix of multivariate nonlinear times series. The underlying target process is assumed to be a constant conditional correlation process Bollerslev (Rev Econ Stat 72:498–505, 1990) or a dynamic conditional correlation model Engle (J Bus Econ Stat 20:339–350, 2002). We introduce several EWMA and CUSUM control charts. These control schemes are based on univariate EWMA statistics, multivariate EWMA recursions, and different types of cumulative sums. The recursions are applied to local measures for means and covariances, e.g. the present observations and the conditional covariances. Further, they are applied to means and covariances of residuals. The control statistics are obtained by computing the Mahalanobis distance between the EWMA or CUSUM statistics and their expectations if no change occurs. Via Monte Carlo simulation the performance of the proposed charts is compared. Our empirical study illustrates an application of these control procedures to bivariate logarithmic returns of the European indices FTSE100 and DAX. In order to assess the performance of the introduced schemes we apply the average run length and the maximum conditional expected delay.     

Best linear unbiased quantile estimators of the logistic distribution using order statistics

This paper deals with the problem of estimating the Quantiles Q(ξ), with ξ=0.01, 0.05(0.05)0.95, 0.99 of the logistic distribution by using order statistics in small samples. Linear unbiased estimators with minimum variance BLUE based on ordered observations are constructed for sample size n = 2(1)10, 15 and 20.     

Asymptotic properties of a goodness-of-fit test based on maximum correlations

We study the efficiency properties of the goodness-of-fit test based on the Q _n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q _n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.     

Bayesian X̄ control limits for exponentially distributed measurements

This article deals with the construction of an X? control chart using the Bayesian perspective. We obtain new control limits for the X? chart for exponentially distributed data-generating processes through the sequential use of Bayes’ theorem and credible intervals. Construction of the control chart is illustrated using a simulated data example. The performance of the proposed, standard, tolerance interval, exponential cumulative sum (CUSUM) and exponential exponentially weighted moving average (EWMA) control limits are examined and compared via a Monte Carlo simulation study. The proposed Bayesian control limits are found to perform better than standard, tolerance interval, exponential EWMA and exponential CUSUM control limits for exponentially distributed processes.     

Cumulative sum (cusum) control schemes

Cumulative Sum (CUSUM) quality control schemes are becoming widely used i n industry because they are powerful, versatile, and easy to use. They cumulate recent process data to quickly detect out-of-control situations. They also serve as a powerful diagnostic tool. There are now more than 10,000 CUSUM control schemes in use daily in Du Pont. This paper describes design and implementation procedures for CUSUM control schemes with emphasison properties that are recorded as counts. The paper will describe recent developments which make CUSUM procedures more useful and more powerful.

The recent developments described are: Fast Initial Response (FIR) CUSUM which gives extra sensitivity to out-of-control situations at start up or after a(possiblyin effective)control action.

A Combined She whart-CUSUM which combines the key features

of CUSUM schemes and She whart Schemes by adding She whart Control Limits to a CUSUM scheme.

ROBUST CUSUM schemes which are no tunduly influenced by a few outliers or fliers occurring in the stream of data.

The philosophy of continual improvement of aprocessisvery compatible with CUSUM procedures.As CUSUM procedures give much more responsive control,a CUSUM signal does notmean that the process is producing bad product.Rather it means that action should be taken sothat the process does not produce bad product.     

A minimum cost exponential cusum test

A minimum cost CUSUM test for an event rate increase when inter-event times are exponentially distributed is presented. Optimal values of the test decision parameters, h and k, are developed from a renewal reward model of the event cycle by combining a non-linear optimization technique with an exact method for determining exponential average run lengths. Test robustness for event cycle parameter estimates and departures from the assumption of exponentially distributed inter-event times are discussed in the context of an injury monitoring scenario. Robustness to positively serially correlated observations emanating from EAR(1) and EMA(1) processes is also examined.     

Variations of Q–Q Plots: The Power of Our Eyes!

In statistical modeling, we strive to specify models that resemble data collected in studies or observed from processes. Consequently, distributional specification and parameter estimation are central to parametric models. Graphical procedures, such as the quantile–quantile (Q–Q) plot, are arguably the most widely used method of distributional assessment, though critics find their interpretation to be overly subjective. Formal goodness of fit tests are available and are quite powerful, but only indicate whether there is a lack of fit, not why there is lack of fit. In this article, we explore the use of the lineup protocol to inject rigor into graphical distributional assessment and compare its power to that of formal distributional tests. We find that lineup tests are considerably more powerful than traditional tests of normality. A further investigation into the design of Q–Q plots shows that de-trended Q–Q plots are more powerful than the standard approach as long as the plot preserves distances in x and y to be the same. While we focus on diagnosing nonnormality, our approach is general and can be directly extended to the assessment of other distributions.     

A binomial CUSUM chart for detecting large shifts in fraction nonconforming

This article studies a unique feature of the binomial CUSUM chart in which the difference (d _t ?d ₀) is replaced by (d _t ?d ₀)² in the formulation of the cumulative sum C _t (where d _t and d ₀ are the actual and in-control numbers of nonconforming units, respectively, in a sample). Performance studies are reported and the results reveal that this new feature is able to increase the detection effectiveness when fraction nonconforming p becomes three to four times as large as the in-control value p ₀. The design of the new binomial CUSUM chart is presented along with the calculation of the in-control and out-of-control Average Run Lengths (ARL₀ and ARL₁).     

Power of the adjusted Q statistic to evaluate heterogeneity in meta-analyses of cluster randomized trials

Because of its simplicity, the Q statistic is frequently used to test the heterogeneity of the estimated intervention effect in meta-analyses of individually randomized trials. However, it is inappropriate to apply it directly to the meta-analyses of cluster randomized trials without taking clustering effects into account. We consider the properties of the adjusted Q statistic for testing heterogeneity in the meta-analyses of cluster randomized trials with binary outcomes. We also derive an analytic expression for the power of this statistic to detect heterogeneity in meta-analyses, which can be useful when planning a meta-analysis. A simulation study is used to assess the performance of the adjusted Q statistic, in terms of its Type I error rate and power. The simulation results are compared to that obtained from the proposed formula. It is found that the adjusted Q statistic has a Type I error rate close to the nominal level of 5%, as compared to the unadjusted Q statistic commonly used to test for heterogeneity in the meta-analyses of individually randomized trials with an inflated Type I error rate. Data from a meta-analysis of four cluster randomized trials are used to illustrate the procedures.     

Estimating the quantile function of a location-scale family of distributions based on few selected order statistics

Some general asymptotic methods of estimating the quantile function, Q(ξ), 0<ξ<1, of location-scale families of distributions based on a few selected order statistics are considered, with applications to some nonregular distributions. Specific results are discussed for the ABLUE of Q(ξ) for the location-scale exponential and double exponential distributions. As a further application of the exponential results, we discuss a nonlinear estimator of Q(ξ) for the scale-shape Pareto distribution.     

Formulas for the Design of CUSUM Quality Control Charts

ABSTRACT

In the design of CUSUM control charts, it is common to use charts, tables, or software to find an appropriate critical threshold (h). This article provides an approximate formula to calculate the threshold directly from prespecified values of the reference value (k) and the in-control average run length (ARL₀). Formulas are also provided for choosing k and h from prespecified values of the in-control and out-of-control average run lengths.     

Existence and Uniqueness of Q-Processes with a Given Finite μ-Invariant Measure

Let Q be a stable and conservative Q‐matrix over a countable state space S consisting of an irreducible class C and a single absorbing state 0 that is accessible from C. Suppose that Q admits a finite μ‐subinvariant measure m on C. We derive necessary and sufficient conditions for there to exist a Q‐process for which m is μ‐invariant on C, as well as a necessary condition for the uniqueness of such a process.     

Test of parameter changes in a class of observation-driven models for count time series

Abstract

This paper investigates the parameter-change tests for a class of observation-driven models for count time series. We propose two cumulative sum (CUSUM) test procedures for detection of changes in model parameters. Under regularity conditions, the asymptotic null distributions of the test statistics are established. In addition, the integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes with conditional negative binomial distributions are investigated. The developed techniques are examined through simulation studies and also are illustrated using an empirical example.     

Pooling operators with the marginalization property

In this paper, we consider the problem of combining a number of opinions which have been expressed as probability measures P₁, …, P_n, over some space. It is shown that a pooling formula which has the marginalization property of McConway (1981) must be of the form T = Σⁿ_i=₁W_i P_i + (1 - Σⁿ_i =₁W_i)Q, where Q is an arbitrary measure and W₁, …, W_n ϵ [—1,1] are weights such that| Σ_J Σ _j w_j | ≤ 1 for every subset J of {1, …, n}. If, in addition, T is required to preserve the independence of arbitrary events A and B whenever these events are independent under each P_i, then either T = P_i for some 1 ≤ i ≤ n or T = Q, in which case Q takes values in {0, l}.