On the Restricted Liu Estimator in the Gauss–Markov Model

In this paper, we compare two estimators, the RLE (restricted Liu estimator) and the RLSE (restricted least squares estimator) of parameters in linear models under Gauss–Markov models. Using generalized inverse of matrices, we found some equivalency conditions for the superiority of the RLE with respect to the MSE criterion.     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

Extended tables of the distribution of friedman’ s s-statistic in the two-way layout

Friedman’s (1937, 1940) S-statistic is designed to test the hypothesis that there is no treatment effect in a randomized-block design with k treatments and n blocks. In this paper we give tables of the null distribution of S for k = 5, n = 6(1)8, and for k = 6, n = 2(1)6. Computational details are discussed.     

A consistent estimator of the smoothing operator in the functional Hodrick–Prescott filter

In this article, we consider a version of the functional Hodrick–Prescott filter for functional time series. We show that the associated optimal smoothing operator preserves the “noise-to-signal ratio” structure. Moreover, as the main result, we propose a consistent estimator of this optimal smoothing operator.     

EEG in the diagnostics of Alzheimer’s disease

Dementia caused by Alzheimer’s disease (AD) is worldwide one of the main medical and social challenges for the next years and decades. An automated analysis of changes in the electroencephalogram (EEG) of patients with AD may contribute to improving the quality of medical diagnoses. In this paper, measures based on uni- and multi-variate spectral densities are studied in order to measure slowing and, in greater detail, reduced synchrony in the EEG signals. Hereby, an EEG segment is interpreted as sample of a (weakly) stationary stochastic process. The spectral density was computed using an indirect estimator. Slowing was considered by calculating the spectral power in predefined frequency bands. As measures for synchrony between single EEG signals, we analyzed coherences, partial coherences, bivariate and conditional Granger causality; for measuring synchrony between groups of EEG signals, we considered coherences, partial coherences, bivariate and conditional Granger causality between the respective first principal components of each group, and dynamic canonic correlations. As measure for local synchrony within a group, the amount of variance explained by the respective first principal component of static and dynamic principal component analysis was investigated. These measures were exemplarily computed for resting state EEG recordings from 83 subjects diagnosed with probable AD. Here, the severity of AD is quantified by the Mini Mental State Examination score.     

Aliasing in 2n–k fractions in the absence of within group interactions

It is often known in advance that certain subsets of factors act independently upon a response. Such information can be used to estimational advantage by aliasing low-order effects with such zero interactions. We find the best 2^n–k fractions for the case when the factors can be partitioned into two classes such that non-zero interactions may exist only between classes but not within a class.     

Estimation of a change point in the variance function based on the χ2-distribution

ABSTRACT

Let us consider that the variance function or its νth derivative in a regression model has a change/discontinuity point at an unknown location. To use the local polynomial fits, the log-variance function which break the positivity is targeted. The location and the jump size of the change point are estimated based on a one-sided kernel-weighted local-likelihood function which is provided by the χ²-distribution. The whole structure of the log-variance function is then estimated using the data sets split by the estimated location. Asymptotic results of the proposed estimators are described. Numerical works demonstrate the performances of the methods with simulated and real examples.     

Testing for random effect in the Fuller–Battese model

We consider the Fuller–Battese model where random effects are allowed to be from non-normal universes. The asymptotic distribution of the F-statistic in this model is derived as the number of groups tends to infinity (is large) and sample size from any group is either fixed or large. The result is used to establish an approximate test for the significance of the random effect variance component. Robustness of the established approximate test is given.     

Selection of the best in 2×2 factorial designs

Selection of the cell in a 2×2 -factorial design with the greatest mean is considered. A general class of ranking and selection procedures (RSP) is constructed to include methods based on the largest marginal cell means (SP1) or on the largest cell mean (SP3). Using the preference zone approach the minimum probability of a correct solution is found, In this paper a RSP which maximizes the minimum probability of a correct solution over the preference zone is found. In this way selection of the cell with the greatest observed mean is proven to be admissible.     

Clear Effects in the 2 n−p Combined Design

The technique of semifolding is used to develop the 2 ^n?p designs. Based on the initial analysis, some factors may be more important than others. In other words, the results from analyzing the original experiment may suggest a specific set of effects to be de-aliased. On the other hand, some previously acquired information might be available for specific factors. In these cases, one may desire to isolate the main effects of these factors and each of their two-factor interactions in the experiments. Four rules that are presented in this article can systematically isolate effects of potential interest, which should serve well for researchers in all disciplines. The combined design, by semifolding, provides estimates of the interactions that involve specific factors so that the alias chains of the two-factor interactions can be broken.     

Interval estimation of the stress–strength reliability in the two-parameter exponential distribution based on records

We consider interval estimation of the stress–strength reliability in the two-parameter exponential distribution based on records. We constructed Bayesian intervals, Bootstrap intervals and intervals using the generalized pivot variable. A simulation study is conducted to investigate and compare the performance of the intervals in terms of their coverage probability and expected length. An example is given.     

A modification of the Box–Meyer method for finding the active factors in screening experiments

Screening experiments are conducted to identify a few active factors among a large number of factors. For the objective of identifying active factors, Box and Meyer provided an innovative approach, the Box–Meyer method (BMM). With the use of means models, we propose a modification of the BMM in this paper. Compared with the original BMM, the modified BMM (MBMM) can circumvent the problem that the original BMM runs into, namely that it may fail to identify some active factors due to the ignorance of higher order interactions. Furthermore, the number of explanatory variables in the MBMM is smaller. Therefore, the computational complexity is reduced. Finally, three examples with different types of designs are used to demonstrate the wide applicability of the MBMM.     

Do we need the constant term in the heterogenous autoregressive model for forecasting realized volatilities?

No-constant strategy is considered for the heterogenous autoregressive (HAR) model of Corsi, which is motivated by smaller biases of its estimated HAR coefficients than those of the constant HAR model. The no-constant model produces better forecasts than the constant model for four real datasets of the realized volatilities (RVs) of some major assets. Robustness of forecast improvement is verified for other functions of realized variance and log RV and for the extended datasets of all 20 RVs of Oxford-Man realized library. A Monte Carlo simulation also reveals improved forecasts for some historic HAR model estimated by Corsi.     

How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?

This paper studies the impact of jumps on volatility estimation and inference based on various realised variation measures such as realised variance, realised multipower variation and truncated realised multipower variation. We review the asymptotic theory of those realised variation measures and present a new estimator for the asymptotic ‘variance’ of the centered realised variance in the presence of jumps. Next, we compare the finite sample performance of the various estimators by means of detailed Monte Carlo studies. Here we study the impact of the jump activity, of the jump size of the jumps in the price and of the presence of additional independent or dependent jumps in the volatility. We find that the finite sample performance of realised variance and, in particular, of log-transformed realised variance is generally good, whereas the jump-robust statistics tend to struggle in the presence of a highly active jump process.     

The Volume of Transactions and the Circulation of Money in the United States, 1950–1979

In the identity of exchange I distinguish between currency and bank payments on one side and several types of transactions and the transfer of idle money on the other. An attempt is made to measure these variables, with varying success. On the payments side I argue that currency velocity is constant (and low) and that the vast rise of bank money velocity is largely due to increased short-term investment of idle funds. The results suggest an upward shift in the level of transactions in 1968–1972, which I attribute to changes in the international role of the dollar.     

Asymptotic efficiency of estimation in the partial Koziol–Green model

In this paper, the convolution theorem and the minimax theorem for estimating the survival function in the partial Koziol–Green model (PKG) are presented. The result indicates that the partial Abdushukurov–Cheng–Lin (ACL) estimator in the PKG model is asymptotically efficient in the sense of being the least dispersed regular estimator. Consequently, the calculation shows that the ACL estimator in the KG model is also asymptotically efficient.