All Forecasters Are Equal

From an analysis of the track records of U.S. economic forecasters, Stekler (1987) concluded that “all forecasters are not equal” (p. 158). This article shows that his result is based on an incorrectly defined test statistic. When a more appropriate test is conducted, the figures suggest that accuracy rankings are not significantly different from those that might be expected as a result of sampling error in a population of equally accurate forecasters.     

The conditional level of confidence intervals for the normal variance and applications

Suppose we observe two independent random vectors each having a multivariate normal distribution with covariance matrix known up to an unknown scale factor σ . The first random vector has a known mean vector while the second has an unknown mean vector. Interest centers around finding confidence intervals for σ² with confidence coefficient 1 ? α. Standard results show that, when we only observe the first random vector, an optimal (i.e., smallest length) confidence interval C, based on the well-known chi- squared statistic, can be constructed for σ² . When we additionally observe the second random vector, the confidence interval C is no longer optimal for estimating σ². One criterion useful for detecting the non-optimality of a confidence interval C concerns whether C admits positively or negatively biased relevant subsets. This criterion has recently received a good deal of attention. It is shown here that under some conditions the confidence interval C admits positively biased relevant subsets.

Applications of this result to the construction of ‘better‘ unconditional confidence intervals for σ² are presented. Some simulation results are given to indicate the typical extent of improvement attained.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

Complexity of penalized likelihood estimation

We show that for a class of penalty functions, finding the global optimizer in the penalized least-squares estimation is equivalent to the ‘exact cover by 3-sets’ problem, which belongs to a class of NP-hard problems. The NP-hardness result is then extended to the cases of penalized least absolute deviations regression and a special class of penalized support vector machines. We discuss its implication in statistics. To the best of our knowledge, this is the first formal documentation on the complexity of this type of problem.     

Optimum design-based ratio estimators of the distribution function

The ratio method is commonly used to the estimation of means and totals. This method was extended to the problem of estimating the distribution function. An alternative ratio estimator of the distribution function is defined. A result that compares the variances of the aforementioned ratio estimators is used to define optimum design-based ratio estimators of the distribution function. Different empirical results indicate that the optimum ratio estimators can be more efficient than alternative ratio estimators. In addition, we show by simulations that alternative ratio estimators can have large biases, whereas biases of the optimum ratio estimators are negligible in this situation.     

The Effects of Misconceptions on the Properties of Friedman's Test

Friedman's test is a widely used rank-based alternative to the analysis of variance (ANOVA) F-test for identifying treatment differences in a randomized complete block design. Many texts provide incomplete or misleading information about when Friedman's test may be appropriately applied. We discuss the assumptions needed for the test and common misconceptions. We show via simulation that when the variance or skew of the treatment distributions differ, application of Friedman's test to detect differences in treatment location can result in Type I error probabilities larger than the nominal α, and even when α is unaffected, the power of the test can be less than expected.     

Interpretation of the coefficient of determination of an ANOVA model as a measure of polarization

In this paper, it is demonstrated that coefficient of determination of an ANOVA linear model provides a measure of polarization. Taking as the starting point the link between polarization and dispersion, we reformulate the measure of polarization of Zhang and Kanbur using the decomposition of the variance instead of the decomposition of the Theil index. We show that the proposed measure is equivalent to the coefficient of determination of an ANOVA linear model that explains, for example, the income of the households as a function of any population characteristic such as education, gender, occupation, etc. This result provides an alternative way to analyse polarization by sub-populations characteristics and at the same time allows us to compare sub-populations via the estimated coefficients of the ANOVA model.     

Design effect of randomized systematic sampling

In statistical practice, systematic sampling (SYS) is used in many modifications due to its simple handling. In addition, SYS may provide efficiency gains if it is well adjusted to the structure of the population under study. However, if SYS is based on an inappropriate picture of the population a high decrease of efficiency, i.e. a high increase in variance may result by changing from simple random sampling to SYS. In the context of two-stage designs SYS so far seems often in use for subsampling within the primary units. As an alternative to this practice, we propose to randomize the order of the primary units, then to select systematically a number of primary units and, thereafter, to draw secondary units by simple random sampling without replacement within the primary units selected. This procedure is more efficient than simple random sampling with replacement from the whole population of all secondary units, i.e. the variance of an adequate estimator for a total is never increased by changing from simple random sampling to randomized SYS whatever be the values associated by a characteristic with the secondary units, while there are values for which the variance decreases for the change mentioned. This result should hold generally, even if our proof, so far, is not complete for general sample sizes.     

Model selection for infinite variance time series

In this we consider the problem of model selection for infinite variance time series. We introduce a group of model selection critera based on a general loss function Ψ. This family includes various generalizations of predictive least square and AIC Parameter estimation is carried out using Ψ. We use two loss functions commonly used in robust estimation and show that certain criteria out perform the conventional approach based on least squares or Yule-Walker estima­tion for heavy tailed innovations. Our conclusions are based on a comprehensive study of the performance of competing criteria for a wide selection of AR(2) models. We also consider the performance of these techniques when the ‘true’ model is not contained in the family of candidate models.     

Generalized L-statistics: Correction to the Form of the Asymptotic Variance Presented by Helmers et al. (1990)

We show that the asymptotic variance of a "generalized L -statistic" is a function of the difference between the conditional and unconditional cumulative distribution functions of the kernel used to form the statistic.     

A note on bounding monte carlo variances michael lavine

A density bounded class P of probability distributions on a space χ is the set of all probability distributions corresponding to probability densities bounded below by a given subprob-ability density and bounded above by a given superprobability density. Density bounded classes arise in robust Bayesian analysis (Lavine 1991) and also in Monte Carlo integration (Fishman Granovsky and Rubin 1989). Finding upper and lower bounds on the variance over all p? P allows one to bound the Monte Carlo variance. Fishman Granovsky and Rubin (1989) find bounds on the variance over all p ? P and also find the densities in P achieving those bounds in the case where χ is discrete; that is, where P is actually a set of probability mass functions. This article generalizes their result by showing how to bound the variance and find the densities achieving the bounds when χ is continuous.     

MSE-matrix superiority of the mixed over the least squares estimator in the presence of outliers

In his recent paper, Ali (1991) has shown that the mixed regression estimator, when data contain mean-shift or variance inflation outliers, is uniformly superior to the ordinary least squares estimator in terms of scalar-valued mean square error. However, when using the matrix-valued mean square error criterion, this dominance fails to hold in general. The subsequent investigation gives a complete characterization of the situation where the mixed estimator is superior to the LS-estimator when the comparison is made with respect to this stronger MSE-property. Vice versa, the LS-estimator never dominates the mixed estimator relative to this criterion.     

On the umvu estimators after using a normalizing transformation

The conditional distribution given complete sufficient statistics is used along with the Rao-Blackwell theorem to obtain uniformly minimum variance unbiased (UMVU) estimators after a transformation to normality has been applied to data. The estimators considered are for the mean, the variance and the cumulative distribution of the original non-normal data. Previous procedures to obtain UMVU estimators have used Laplace transforms, Taylor expansions and the jackknife. An integration method developed in this paper requires only integrability of the normalizing transformation function. This method is easy to employ and it is always possible to obtain a numerical result.     

An alternative approximation to the variance of transformation score

An alternative approximation to the variance of transformation score is given, based on an asymptotic expansion of the transformation estimator. It is then compared with the variance approximation given by Lawrance (1987) in terms of standardized scores. Simulations show that the two standardized scores behave very similarly when model error standard deviation is small. However,when error standard deviation is not small, the new standardized score outperforms that of Lawrance,especially in the structured models.     

Sampling Returns for Realized Variance Calculations: Tick Time or Transaction Time?

This article introduces a new model for transaction prices in the presence of market microstructure noise in order to study the properties of the price process on two different time scales, namely, transaction time where prices are sampled with every transaction and tick time where prices are sampled with every price change. Both sampling schemes have been used in the literature on realized variance, but a formal investigation into their properties has been lacking. Our empirical and theoretical results indicate that the return dynamics in transaction time are very different from those in tick time and the choice of sampling scheme can therefore have an important impact on the properties of realized variance. For RV we find that tick time sampling is superior to transaction time sampling in terms of mean-squared-error, especially when the level of noise, number of ticks, or the arrival frequency of efficient price moves is low. Importantly, we show that while the microstructure noise may appear close to IID in transaction time, in tick time it is highly dependent. As a result, bias correction procedures that rely on the noise being independent, can fail in tick time and are better implemented in transaction time.     

Sampling Returns for Realized Variance Calculations: Tick Time or Transaction Time?

This article introduces a new model for transaction prices in the presence of market microstructure noise in order to study the properties of the price process on two different time scales, namely, transaction time where prices are sampled with every transaction and tick time where prices are sampled with every price change. Both sampling schemes have been used in the literature on realized variance, but a formal investigation into their properties has been lacking. Our empirical and theoretical results indicate that the return dynamics in transaction time are very different from those in tick time and the choice of sampling scheme can therefore have an important impact on the properties of realized variance. For RV we find that tick time sampling is superior to transaction time sampling in terms of mean-squared-error, especially when the level of noise, number of ticks, or the arrival frequency of efficient price moves is low. Importantly, we show that while the microstructure noise may appear close to IID in transaction time, in tick time it is highly dependent. As a result, bias correction procedures that rely on the noise being independent, can fail in tick time and are better implemented in transaction time.     

Strong orthogonal decompositions and non‐linear impulse response functions for infinite‐variance processes

The author proves that Wold‐type decompositions with strong orthogonal prediction innovations exist in smooth, reflexive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. His theory includes as special cases all previous Wold‐type decompositions of discrete time processes, completely characterizes when non‐linear heavy‐tailed processes obtain a strong‐orthogonal moving average representation, and easily promotes a theory of non‐linear impulse response functions for infinite‐variance processes. The author exemplifies his theory by developing a non‐linear impulse response function for smooth transition threshold processes, and discusses how to test decomposition innovations for strong orthogonality and whether the proposed model represents the best predictor. A data set on currency exchange rates allows him to illustrate his methodology.     

Testing for shifts in mean with monotonic power against multiple structural changes

It is known that several widely used structural change tests have non-monotonic power because the long-run variance is poorly estimated under the alternative hypothesis. In this paper, we propose a modified long-run variance estimator to alleviate this problem. We theoretically show that the tests with our long-run variance estimator are consistent against large multiple structural changes. Simulation results show that the proposed test performs well in finite samples.     

Moving Average-Based Estimators of Integrated Variance

We examine moving average (MA) filters for estimating the integrated variance (IV) of a financial asset price in a framework where high-frequency price data are contaminated with market microstructure noise. We show that the sum of squared MA residuals must be scaled to enable a suitable estimator of IV. The scaled estimator is shown to be consistent, first-order efficient, and asymptotically Gaussian distributed about the integrated variance under restrictive assumptions. Under more plausible assumptions, such as time-varying volatility, the MA model is misspecified. This motivates an extensive simulation study of the merits of the MA-based estimator under misspecification. Specifically, we consider nonconstant volatility combined with rounding errors and various forms of dependence between the noise and efficient returns. We benchmark the scaled MA-based estimator to subsample and realized kernel estimators and find that the MA-based estimator performs well despite the misspecification.