On estimating the dimensionality in discriminant analysis

In discriminant analysis, the dimension of the hyperplane which population mean vectors span is called the dimensionality. The procedures commonly used to estimate this dimension involve testing a sequence of dimensionality hypotheses as well as model fitting approaches based on (consistent) Akaike's method, (modified) Mallows' method and Schwarz's method. The marginal log-likelihood (MLL) method is developed and the asymptotic distribution of the dimensionality estimated by this method for normal populations is derived. Furthermore a modified marginal log-likelihood (MMLL) method is also considered. The MLL method is not consistent for large samples and two modified criteria are proposed which attain asymptotic consistency. Some comments are made with regard to the robustness of this method to departures from normality. The operating characteristics of the various methods proposed are examined and compared.     

Time series count data regression

The count data model studied in the paper extends the Poisson model by al-lowing for overdispersion and serial correlation. Alternative approaches to esti-mate nuisance parameters, required for the correction of the Poisson maximum likelihood covariance matrix estimator and for a quasi-likelihood estimator, are studied. The estimators are evaluated by finite sample Monte Carlo experi-mentation. It is found that the Poisson maximum likelihood estimator with corrected covariance matrix estimators provide reliable inferences for longer time series. Overdispersion test statistics are wellbehaved, while conventional portmanteau statistics for white noise have too large sizes. Two empirical illustrations are included.     

Estimation of gaussian mixtures with rotationally invariant covariance matrices

Homoscedastic and heteroscedastic Gaussian mixtures differ in the constraints placed on the covariance matrices of the mixture components. A new mixture, called herein a strophoscedastic mixture, is defined by a new constraint, This constraint requires the matrices to be identical under orthogonal trans¬formations, where different transformations are allowed for different matrices. It is shown that the M-step of the EM method for estimating the parameters of strophoscedastic mixtures from sample data is explicitly solvable using singular value decompositions. Consequently, the EM-based maximum likelihood estimation algorithm is as easily implemented for strophoscedastic mixtures as it is for homoscedastic and heteroscedastic mixtures. An example of a “noisy” Archimedian spiral is presented.     

Estimating survival curves with left truncated and interval censored data via the ems algorithm

It is well-known that the nonparametric maximum likelihood estimator (NPMLE) of a survival function may severely underestimate the survival probabilities at very early times for left truncated data. This problem might be overcome by instead computing a smoothed nonparametric estimator (SNE) via the EMS algorithm. The close connection between the SNE and the maximum penalized likelihood estimator is also established. Extensive Monte Carlo simulations demonstrate the superior performance of the SNE over that of the NPMLE, in terms of either bias or variance, even for moderately large Samples. The methodology is illustrated with an application to the Massachusetts Health Care Panel Study dataset to estimate the probability of being functionally independent for non-poor male and female groups rcspectively.     

Reverse Regression and Employment Discrimination

The importance of interval forecasts is reviewed. Several general approaches to calculating such forecasts are described and compared. They include the use of theoretical formulas based on a fitted probability model (with or without a correction for parameter uncertainty), various “approximate” formulas (which should be avoided), and empirically based, simulation, and resampling procedures. The latter are useful when theoretical formulas are not available or there are doubts about some model assumptions. The distinction between a forecasting method and a forecasting model is expounded. For large groups of series, a forecasting method may be chosen in a fairly ad hoc way. With appropriate checks, it may be possible to base interval forecasts on the model for which the method is optimal. It is certainly unsound to use a model for which the method is not optimal, but, strangely, this is sometimes done. Some general comments are made as to why prediction intervals tend to be too narrow in practice to encompass the required proportion of future observations. An example demonstrates the overriding importance of careful model specification. In particular, when data are “nearly nonstationary,” the difference between fitting a stationary and a nonstationary model is critical.     

Unobserved-Components Models for Seasonal Adjustment Filters

Time series models are presented, for which the seasonal-component estimates delivered by linear least squares signal extraction closely approximate those of the standard option of the widely-used Census X-11 program. Earlier work is extended by consideration of a broader class of models and by examination of asymmetric filters, in addition to the symmetric filter implicit in the adjustment of historical data. Various criteria that guide the specification of unobserved- components models are discussed, and a new preferred model is presented. Some nonstandard options in X-11 are considered in the Appendix.     

Tests for Cointegration Breakdown Over a Short Time Period

We derive a speculative trading model with endogenous informed trading that yields a conditionally heteroscedastic time series for trading volume and the squared price changes. We use half-hourly price-change and volume data for IBM during 1988 to test the model and estimate the structural parameters using the simulated method-of-moments estimation procedure. Although the model seems to do a reasonable job fitting the unconditional moments of the volume and the squared price change processes, it fares less well in fitting the relation between current trading volume and lags of trading volume and squared volume's (and its lag's) relation to squared price changes.     

Estimation of gene/haplotype frequencies in genetic marker systems based on phenotype data

Two equivalent methods (gene counting and maximum likelihood) for estimating gene frequencies in a general genetic marker system based on observed phenotype data are derived. Under the maximum likelihood approach, an expression is given for the estimated covariance matrix from which estimated standard errors of the estimators can be found. In addition, consideration is given to the problem of estimating gene frequencies when there are available several independent population data sets.     

Gibbs sampling for marginal posterior expectations

In earlier work (Gelfand and Smith, 1990 and Gelfand et al, 1990) a sampling based approach using the Gibbs sampler was offered as a means for developing marginal posterior densities for a wide range of Bayesian problems several of which were previously inaccessible. Our purpose here is two-fold. First we flesh out the implementation of this approach for calculation of arbitrary expectations of interest. Secondly we offer comparison with perhaps the most prominent approach for calculating posterior expectations, analytic approximation involving application of the LaPlace method. Several illustrative examples are discussed as well. Clear advantages for the sampling based approach emerge.