A Saddlepoint Approximation to the Distribution of the Half-Life Estimator in a Stationary Autoregressive Model

We derive saddlepoint approximations for the distribution and density functions of the half-life estimated by OLS from autoregressive time-series models. Our results are used to prove that none of the integer-order moments of these half-life estimators exist. This provides an explanation for the very large estimates of persistency, and the extremely wide confidence intervals, that have been reported by various authors, i.e., in the empirical economics literature relating to purchasing power parity.     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

A statistically adaptive sampling policy to the Hotelling's T2 control chart: Markov chain approach

ABSTRACT

We present an alternative sampling scheme for the Hotelling's T² control chart with variable parameters (VP T²) which allows the sampling interval h, the sample size n, and control limit k to vary between minimum and maximum values while keeping the warning line fixed over time. Our method uses only one measurement scale to overcome the difficulties of using two scales in practice. Later, we demonstrate the merits of the method in terms of its performance in detecting small-to-moderate shifts and its ease of application.     

A class of robust stepwise alternatives to Hotelling's T 2 tests

Hotelling's T 2 test is known to be optimal under multivariate normality and is reasonably validity-robust when the assumption fails. However, some recently introduced robust test procedures have superior power properties and reasonable type I error control with non-normal populations. These, including the tests due to Tiku & Singh (1982), Tiku & Balakrishnan (1988) and Mudholkar & Srivastava (1999b, c), are asymptotically valid but are useful with moderate size samples only if the population dimension is small. A class of B-optimal modifications of the stepwise alternatives to Hotellings T 2 introduced by Mudholkar & Subbaiah (1980) are simple to implement and essentially equivalent to the T 2 test even with small samples. In this paper we construct and study the robust versions of these modified stepwise tests using trimmed means instead of sample means. We use the robust one- and two-sample trimmed- t procedures as in Mudholkar et al. (1991) and propose statistics based on combining them. The results of an extensive Monte Carlo experiment show that the robust alternatives provide excellent type I error control and a substantial gain in power.     

A SADDLEPOINT APPROXIMATION TO THE DISTRIBUTION FUNCTION OF THE ANDERSON-DARLING TEST STATISTIC

The Anderson-Darling goodness-of-fit test has a highly skewed and non-standard limit distribution. Various attempts have been made to tabulate the associated critical points, using both theoretical approximations and simulation methods. We show that a standard saddlepoint approximation performs well in both tails of the distribution. It is markedly superior to other theoretical approximations in the lower tail of the distribution.     

SMALL SAMPLE INFERENCE IN LOCATION SCALE MODELS WITH STUDENT(λ) ERRORS

ABSTRACT

A third order accurate approximation to the p value in testing either the location or scale parameter in a location scale model with Student(λ) errors is introduced. The third order approximation is developed via an asymptotic method, based on exponential models and the saddlepoint approximation. Techniques are presented for the numerical computation of all quantities required for the third order approximation. To compare the accuracy of various asymptotic methods a numerical example and simulation study are included. The numerical example and simulation study illustrate that the third order method presented leads to a more accurate p value approximation compared to first order methods in Student(λ) models with small samples.     

Bivariate copulas on the Hotelling's T2 control chart

In this paper, we propose five types of copulas on the Hotelling's T² control chart when observations are from exponential distribution and use the Monte Carlo simulation to compare the performance of the control chart, which is based on the Average Run Length (ARL) for each copula. Five types of copulas function for specifying dependence between random variables are used and measured by Kendall's tau. The results show that the copula approach can be fitted the observation and we can use copula as an option for application on Hotelling's T² control chart.     

Adaptive Hotelling's T 2 Control Charts with Run Rules

This study investigates the statistical properties of the adaptive Hotelling's T ² charts with run rules in which the sample size and sampling interval are allowed to vary according on the current and past sampling points. The adaptive charts include variable sample size (VSS), variable sampling interval (VSI), and variable sample size and sampling interval (VSSI) charts. The adaptive Hotelling's T ² charts with run rules are compared with the fixed sampling rate Hotelling's T ² chart with run rules. The numerical results show that the VSS, VSI, and VSSI features improve the performance of the Hotelling's T ² chart with run rules.     

Saddlepoint density and distribution functions for the ratio of two linear functions and the product of generalized gamma variates

The generalized gamma distribution is a flexible and attractive distribution because it incorporates several well-known distributions, i.e., gamma, Weibull, Rayleigh, and Maxwell. This article derives saddlepoint density and distribution functions for the ratio of two linear functions of generalized gamma variables and the product of n independent generalized gamma variables. Simulation studies are used to evaluate the accuracy of the saddlepoint approximations. The saddlepoint approximations are fast, easy, and very accurate.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

A Comparison of Permutation Hotelling's T 2 Test and Log-Ratio Test for Analyzing Compositional Data

An adding-back model is constructed for studying multiple outliers and influential observations. A logarithmic functional form for some influence measures having a better justification for plotting purposes is suggested. Two graphical methods with contours of constant measure values are proposed. They provide valuable information about the interrelationship of multiple influential observations and influence measures.     

Misspecified T2 tests. i. location and scale

Properties of Hotelling's (1931) T ² are studied under model misspecification in the model for a multivariate experiment. Stochastic bounds on T ² and further properties of the T ² test are studied under misspecified location and scale. The bounds are evaluated numerically in selected cases.     

Efficiency of t-Test and Hotelling's T 2-Test After Box-Cox Transformation

Early investigations of the effects of non-normality indicated that skewness has a greater effect on the distribution of t-statistic than does kurtosis. When the distribution is skewed, the actual p-values can be larger than the values calculated from the t-tables. Transformation of data to normality has shown good results in the case of univariate t-test. In order to reduce the effect of skewness of the distribution on normal-based t-test, one can transform the data and perform the t-test on the transformed scale. This method is not only a remedy for satisfying the distributional assumption, but it also turns out that one can achieve greater efficiency of the test. We investigate the efficiency of tests after a Box-Cox transformation. In particular, we consider the one sample test of location and study the gains in efficiency for one-sample t-test following a Box-Cox transformation. Under some conditions, we prove that the asymptotic relative efficiency of transformed t-test and Hotelling's T ²-test of multivariate location with respect to the same statistic based on untransformed data is at least one.     

Asymptotic null and nonnull distribution of Hotelling's T2-statistic under the elliptical distribution

This paper is concerned with asymptotic distribution of Hotelling's T²-statistic under the elliptical distribution for the null hypothesis and the local alternative under the elliptical distribution. Asymptotic expansions for the distribution of T² for the null case and the local alternative are given up to the order N⁻¹, where N is the sample size. The percentiles of T² and the approximate powers are calculated to evaluate the effect of the elliptical distribution for some numerical examples.Also to evaluate the effect of an adjustment of Bartlett type to Hotelling's T² for the local alternative, the approximate power of adjusted T² is calculated in comparison with one of nonadjusted T².     

A note on connectedness in fixed effects manova and gmanova models withmissing cells

Murray and Smith (1985) and Hocking (1985) give a generalized definition and test of connectedness in the case of missing cells using the univariate cell-means model with linear restrictions on the cell-means. The test of connectedness is here extended to multivariate fixed effects models, including the usual MANOVA model with linear restrictions, the MANOVA model with double linear restrictions, and the GMANOVA model.     

A Proof of the Asymptotic Equivalence of Two-Tail Probability Approximations

A modified efficient jump algorithm is proposed for the Markov Chain Monte Carlo draws of the exponential power distribution. Bayesian inference based on the exponential power error term and that on the normal error term are compared. Unbiasedness of the LAD estimator is proven.     

Robust Hotelling's T2 control charts under non-normality: the case of t-Student distribution

Traditional multivariate control charts are based upon the assumption that the observations follow a multivariate normal distribution. In many practical applications, however, this supposition may be difficult to verify. In this paper, we use control charts based on robust estimators of location and scale to improve the capability of detection observations out of control under non-normality in the presence of multiple outliers. Concretely, we use a simulation process to analyse the behaviour of the robust alternatives to Hotelling's T ², which use minimum volume ellipsoidal (MVE) and minimum covariance determinant (MCD) in the presence of observations with a Student's t-distribution. The results show that these robust control charts are good alternatives for small deviations from normality due to the fact that the percentage of out-of-control observations detected for these charts in the Phase II are higher.     

Improved Measures of the Spread of Data for Some Unknown Complex Distributions Using Saddlepoint Approximations

Measures of the spread of data for random sums arise frequently in many problems and have a wide range of applications in real life, such as in the insurance field (e.g., the total claim size in a portfolio). The exact distribution of random sums is extremely difficult to determine, and normal approximation usually performs very badly for this complex distributions. A better method of approximating a random-sum distribution involves the use of saddlepoint approximations.

Saddlepoint approximations are powerful tools for providing accurate expressions for distribution functions that are not known in closed form. This method not only yields an accurate approximation near the center of the distribution but also controls the relative error in the far tail of the distribution.

In this article, we discuss approximations to the unknown complex random-sum Poisson–Erlang random variable, which has a continuous distribution, and the random-sum Poisson-negative binomial random variable, which has a discrete distribution. We show that the saddlepoint approximation method is not only quick, dependable, stable, and accurate enough for general statistical inference but is also applicable without deep knowledge of probability theory. Numerical examples of application of the saddlepoint approximation method to continuous and discrete random-sum Poisson distributions are presented.     

On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis

In this article, we propose a new generalized multivariate log-gamma distribution. We consider the usage of the proposed multivariate distribution as the prior distribution in the Bayesian analysis. The generalized multivariate log-gamma distribution allows for the inclusion of prior knowledge on correlations between model parameters when likelihood is not in the form of a normal distribution. Use of the proposed distribution in the Bayesian analysis of log-linear models is also discussed.     

Saddlepoint p-values for group of linear rank tests

Many nonparametric tests in one sample problem, matched pairs, and competingrisks under censoring have the same underlying permutation distribution. This article proposes a saddlepoint approximation to the exact p-values of these tests instead of the asymptotic approximations. The performance of the saddlepoint approximation is assessed by using simulation studies that show the superiority of the saddlepoint methods over the asymptotic approximations in several settings. The use of the saddlepoint to approximate the p-values of class of two sample tests under complete randomized design is also discussed.