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## Multivariate Time Series

### Introduction to Time Series and Forecasting (2016-01-01): 227-257 , January 01, 2016

Many time series arising in practice are best considered as components of some vector- valued (multivariate) time series {*X*_{t}} having not only serial dependence within each component series {*X*_{ti}} but also interdependence between the different component series {*X*_{ti}} and {*X*_{tj}}, *i* ≠ *j*. Much of the theory of univariate time series extends in a natural way to the multivariate case; however, new problems arise.

## An Empirical Bayes Approach to Statistics

### Breakthroughs in Statistics (1992-01-01): 388-394 , January 01, 1992

Let *X* be a random variable which for simplicity we shall assume to have discrete values *x* and which has a probability distribution depending in a known way on an unknown real parameter A,
(1)
$$
p\left( {x|\lambda } \right) = Pr[X = x|\Lambda = \lambda ],
$$
A *itself being a random variable with a priori* distribution function
(2)
$$
G\left( \lambda \right) = \operatorname{P} r[\Lambda {\text{ }}\underline \leqslant {\text{ }}\lambda ].
$$

## Remarks on the Burgers Equation

### Selected Works of Murray Rosenblatt (2011-01-01): 231-238 , January 01, 2011

Periodic and aperiodic solutions of the Burgers equation
$$u_t + uu_x=\mu u_{xx}, \ \mu > 0,$$
are studied in this paper. A harmonic analysis of the solutions is carried out and the form of the spectrum is estimated for large time. Corresponding estimates of energy decay are also made, In Burgers’ work on this equation. the case in which
$$\mu \downarrow 0$$
with *t* fixed, and one then lets *t*→∞, is studied. In our investigation, a fixed value of *μ* > 0 is taken and then one lets *t*→∞. A similar analysts is also carried out for an irrotaticnal solulion of a similar 3-dimensionat system of equations. For large time and moderate wavenumbers there is. to the first order. a drift of spectral mass from low wavenumbers to higher wavenumbers. Comments are also made on the asymptotic distribution of a class of random solutions.

## Front Matter - SAS for Epidemiologists

### SAS for Epidemiologists (2013-01-01) , January 01, 2013

## Strong consistency of least squares estimates in multiple regression

### Herbert Robbins Selected Papers (1985-01-01): 510-512 , January 01, 1985

The strong consistency of least squares estimates in multiple regression models with independent errors is obtained under minimal assumptions on the design and weak moment conditions on the errors.

## Optimal Selection Based on Relative Rank

### Herbert Robbins Selected Papers (1985-01-01): 443-452 , January 01, 1985

*n* rankable persons appear sequentially in random order. At the ith stage we observe the relative ranks of the first i persons to appear, and must either select the ith person, in which case the process stops, or pass on to the next stage. For that stopping rule which minimizes the expectation of the absolute rank of the person selected, it is shown that as *n*→ ∞ this tends to the value
$$\mathop \Pi \limits_{j = 1}^\infty {\left( {\frac{{j + 2}}{j}} \right)^{1/j + 1}} \cong 3.8695.$$

## Back Matter - Statistics for Lawyers

### Statistics for Lawyers (2001-01-01) , January 01, 2001

## Regression Diagnostics

### SAS for Epidemiologists (2013-01-01): 213-229 , January 01, 2013

In this chapter, we consider how we may determine if there are problems with our underlying assumptions for the use of linear regression. We learn that residuals are the key to regression diagnostics, that SAS provides many tools, from plots to statistics, that help us examine whether our data meet assumptions such as normal distribution, linear relationships, and homoscedasticity, and that if there are outliers influencing summary statistics.

## Asymptotic Expansions

### Mathematics for Econometrics (2013-01-01): 393-409 , January 01, 2013

This chapter deals with situations in which we wish to approximate the limiting distribution of an estimator. As such it is different from other chapters in that it does not discuss topics in core econometrics and the ancillary mathematics needed to develop and fully understand them. Moreover, its purpose is different from that of the earlier (theoretical) chapters. Its aim is not only to introduce certain (additional) mathematical concepts but also to derive certain results that may prove useful for econometric applications involving hypothesis testing.

## Some Regression Problems in Time Series Analysis

### Selected Works of Murray Rosenblatt (2011-01-01): 101-123 , January 01, 2011

Estimates of the regression coefficients which are unbiased and linear in the observations are discussed in this paper. The residual is assumed to be a stationary process. Two specific estimates are discussed, the least-squares estimate and the Markov estimate. I call the estimate which is computed under the assumption that the residual is an orthogonal process the least-squares estimate. The Markov estimate is the linear unbiased estimate with minimal covariance matrix. The basic assumptions made in the paper are discussed in section 2 and are held to throughout the paper. In section 3 some remarks about the approximation of a continuous positive definite matrix-valued function by finite trigonometric forms are made. These remarks are used in section 4 to obtain the main results about the asymptotic behavior of the covariance matrices of the least-squares and Markov estimates. The next section discusses the many interesting cases in which the least-squares estimate is asymptotically as good as the Markov estimate. The first really systematic discussion of some of these problems was given by U. Grenander [1]. Further work was carried out by U. Grenander and M. Rosenblatt in [2], [3], and [4]. The author considers some of these problems in the case of a vector-valued time series in [5]. Some of the results of this paper are a generalization of some of those obtained in [5].