Distribution of linear combination of exponential variates

The exact distribution of a linear combination of n independent negative exponential random variables, when the coefficients of the linear combination are distinct and positive quantities, is well-known. This paper extends the above result to the general case, namely when the coefficients are arbitrary real numbers, positive or negative, distinct or coincident.     

On the distribution of quadratic forms in normal random variables

This paper provides necessary and sufficient conditions for a quadratic form in singular normal random variables to be distributed as a given linear combination of independent noncentral chi-square variables. Using this result, an extension of Cochran's theorem to quadratic forms of noncentral chi-square variables is derived.     

Bayes Estimates of Markov Trends in Possibly Cointegrated Series

Stylized facts show that average growth rates of U.S. per capita consumption and income differ in recession and expansion periods. Because a linear combination of such series does not have to be a constant mean process, standard cointegration analysis between the variables to examine the permanent income hypothesis may not be valid. To model the changing growth rates in both series, we introduce a multivariate Markov trend model that accounts for different growth rates in consumption and income during expansions and recessions and across variables within both regimes. The deviations from the multivariate Markov trend are modeled by a vector autoregression (VAR) model. Bayes estimates of this model are obtained using Markov chain Monte Carlo methods. The empirical results suggest the existence of a cointegration relation between U.S. per capita disposable income and consumption, after correction for a multivariate Markov trend. This result is also obtained when per capita investment is added to the VAR.     

The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables

This article gives the exact distribution of a statistic whose numerator and denominator are independent, the former being a linear combination of independent chi-square variables and the latter being the kth root of a product of k independent chi-square variables. This structure appears in the study of multivariate linear functional relationship models. The technique of the inverse Mellin transform is used in order to obtain the density of this test statistic in a computable form.     

On the Linear Combination of Laplace and Logistic Random Variables

The distribution of linear combinations of random variables arises explicitly in many areas of engineering. This has increased the need to have available the widest possible range of statistical results on linear combinations of random variables. In this note, the exact distribution of the linear combination α X+β Y is derived when X and Y are Laplace and logistic random variables distributed independently of each other. Extensive tabulations of the associated percentage points obtained by inverting the derived distribution are also given.     

VARIABLE SELECTION AND INTERPRETATION OF COVARIANCE PRINCIPAL COMPONENTS

In practice, when a principal component analysis is applied on a large number of variables the resultant principal components may not be easy to interpret, as each principal component is a linear combination of all the original variables. Selection of a subset of variables that contains, in some sense, as much information as possible and enhances the interpretations of the first few covariance principal components is one possible approach to tackle this problem. This paper describes several variable selection criteria and investigates which criteria are best for this purpose. Although some criteria are shown to be better than others, the main message of this study is that it is unwise to rely on only one or two criteria. It is also clear that the interdependence between variables and the choice of how to measure closeness between the original components and those using subsets of variables are both important in determining the best criteria to use.     

Functional linear discriminant analysis for irregularly sampled curves

We introduce a technique for extending the classical method of linear discriminant analysis (LDA) to data sets where the predictor variables are curves or functions. This procedure, which we call functional linear discriminant analysis ( FLDA ), is particularly useful when only fragments of the curves are observed. All the techniques associated with LDA can be extended for use with FLDA. In particular FLDA can be used to produce classifications on new (test) curves, give an estimate of the discriminant function between classes and provide a one- or two-dimensional pictorial representation of a set of curves. We also extend this procedure to provide generalizations of quadratic and regularized discriminant analysis.     

Variance estimation using refitted cross-validation in ultrahigh dimensional regression

Variance estimation is a fundamental problem in statistical modelling. In ultrahigh dimensional linear regression where the dimensionality is much larger than the sample size, traditional variance estimation techniques are not applicable. Recent advances in variable selection in ultrahigh dimensional linear regression make this problem accessible. One of the major problems in ultrahigh dimensional regression is the high spurious correlation between the unobserved realized noise and some of the predictors. As a result, the realized noises are actually predicted when extra irrelevant variables are selected, leading to serious underestimate of the level of noise. We propose a two-stage refitted procedure via a data splitting technique, called refitted cross-validation, to attenuate the influence of irrelevant variables with high spurious correlations. Our asymptotic results show that the resulting procedure performs as well as the oracle estimator, which knows in advance the mean regression function. The simulation studies lend further support to our theoretical claims. The naive two-stage estimator and the plug-in one-stage estimators using the lasso and smoothly clipped absolute deviation are also studied and compared. Their performances can be improved by the reffitted cross-validation method proposed.     

Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.     

The jack-knife with a stepwise discriminant algorithm-a warning to BMDP usesr

Program 7M in the BMDP statistical package carries out Fisher's linear discriminant analysis and offers a number of options including stepwise variable selection and jack-knife cross-validation. If these two options are used in combination, then a difficulty arises. A Monte Carlo demonstration was performed to demonstrate the problem. The results showed that the classification performance of the stepwise technique under jack-knife appeared to improve when pure noise variables were added to the data set. After studying the manual, it was hypothesized that this phenomenon was due to the stepwise variable selection phase of the procedure being conducted before the jack-knife, rather than being subject to it. This was verified by a second Monte Carlo experiment which compared the jack-knife in BMDP with a complete jack-knife, in which the variable selection phase of the procedure was also subject to the jack-knife.     

On the distribution of some test statistics connected with the multivariate linear functional relationship model

An alternate representation of the densities of some test statistics for the structural coefficients of the multivariate linear functional relationship model is proposed in this article. These statistics are distributed as the ratio of a linear combination of chi-square variÂtes over the root of a product of chi-square variÂtes. A computable representation of their densities has already been derived by Provost (1984) with the help of the technique of the inverse Mellin transform. The connection of the alternate representation to the densities of products of independent beta type-2 and of independent F-random variables is also discussed.     

A note on Stein's lemma for multivariate elliptical distributions

When two random variables are bivariate normally distributed Stein's original lemma allows to conveniently express the covariance of the first variable with a function of the second. Landsman and Neslehova (2008) extend this seminal result to the family of multivariate elliptical distributions. In this paper we use the technique of conditioning to provide a more elegant proof for their result. In doing so, we also present a new proof for the classical linear regression result that holds for the elliptical family.     

Generalized linear models with functional predictors

Summary. We present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function. The technique is particularly useful when only fragments of each curve have been observed. We demonstrate, on both simulated and real data sets, how this approach can be used to perform linear, logistic and censored regression with functional predictors. In addition, we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors. Finally, we extend the methodology to apply generalized linear models and principal components to standard missing data problems.     

Influence measures in affine combination type regression

The detection of outliers and influential observations has received a great deal of attention in the statistical literature in the context of least-squares (LS) regression. However, the explanatory variables can be correlated with each other and alternatives to LS come out to address outliers/influential observations and multicollinearity, simultaneously. This paper proposes new influence measures based on the affine combination type regression for the detection of influential observations in the linear regression model when multicollinearity exists. Approximate influence measures are also proposed for the affine combination type regression. Since the affine combination type regression includes the ridge, the Liu and the shrunken regressions as special cases, influence measures under the ridge, the Liu and the shrunken regressions are also examined to see the possible effect that multicollinearity can have on the influence of an observation. The Longley data set is given illustrating the influence measures in affine combination type regression and also in ridge, Liu and shrunken regressions so that the performance of different biased regressions on detecting and assessing the influential observations is examined.     

Some New Results on the Multinomial Randomized Response Model

ABSTRACT

The randomized response technique is an effective survey method designed to elicit sensitive information while ensuring the privacy of the respondents. In this article, we present some new results on the randomization response model in situations wherein one or two response variables are assumed to follow a multinomial distribution. For a single sensitive question, we use the well-known Hopkins randomization device to derive estimates, both under the assumption of truthful and untruthful responses, and present a technique for making pairwise comparisons. When there are two sensitive questions of interest, we derive a Pearson product moment correlation estimator based on the multinomial model assumption. This estimator may be used to quantify the linear relationship between two variables when multinomial response data are observed according to a randomized-response protocol.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

Interpretable dimension reduction

The analysis of high-dimensional data often begins with the identification of lower dimensional subspaces. Principal component analysis is a dimension reduction technique that identifies linear combinations of variables along which most variation occurs or which best “reconstruct” the original variables. For example, many temperature readings may be taken in a production process when in fact there are just a few underlying variables driving the process. A problem with principal components is that the linear combinations can seem quite arbitrary. To make them more interpretable, we introduce two classes of constraints. In the first, coefficients are constrained to equal a small number of values (homogeneity constraint). The second constraint attempts to set as many coefficients to zero as possible (sparsity constraint). The resultant interpretable directions are either calculated to be close to the original principal component directions, or calculated in a stepwise manner that may make the components more orthogonal. A small dataset on characteristics of cars is used to introduce the techniques. A more substantial data mining application is also given, illustrating the ability of the procedure to scale to a very large number of variables.     

NON-SYMMETRICAL CORRESPONDENCE ANALYSIS WITH CONCATENATION AND LINEAR CONSTRAINTS

Correspondence analysis is a popular statistical technique used to identify graphically the presence, and structure, of association between two or more cross-classified categorical variables. Such a procedure is very useful when it is known that there is a symmetric (two-way) relationship between the variables. When such a relationship is known not to exist, non-symmetrical correspondence analysis is more appropriate as a method of establishing the source of association. This paper highlights some tools that can be used to explore the behaviour of asymmetric categorical variables. These tools consist of confidence regions, the link between non-symmetrical correspondence analysis and the analysis of variance of categorical variables, and the effect of imposing linear constraints. We also explore the application of non-symmetrical correspondence analysis to three-way contingency tables.     

Asymptotics of M-estimator in multivariate linear regression models for a class of random errors

It is known that linear regression models have immense applications in various areas such as engineering technology, economics and social sciences. In this paper, we investigate the asymptotic properties of M-estimator in multivariate linear regression model based on a class of random errors satisfying a generalised Bernstein-type inequality. By using the generalised Bernstein-type inequality, we obtain a general result on almost sure convergence for a class of random variables and then obtain the strong consistency for the M-estimator in multivariate linear regression models under some mild conditions. The result extends or improves some existing ones in the literature. Moreover, we also consider the case when the dimension $p$ tends to infinity by establishing the rate of almost sure convergence for a class of random variables satisfying generalised Bernstein-type inequality. Some numerical simulations are also provided to verify the validity of the theoretical results.     

Quasi-likelihood ratio tests for cointegration,cobreaking, and cotrending

We consider a set of variables with two types of nonstationary features, stochastic trends and broken linear trends. We develop tests that can determine whether there is a linear combination of these variables under which the nonstationary features can be canceled out. The first test can determine whether stochastic trends can be eliminated and thus whether cointegration holds, regardless of whether structural breaks in linear trends are eliminated. The second test can determine whether both stochastic trends and breaks in linear trends are simultaneously removed and thus whether cointegration and cobreaking simultaneously hold. The third test can determine whether not only breaks in linear trends but also linear trends themselves are eliminated along with stochastic trends and thus whether both cointegration and cotrending hold.