Lambert H. Koopmans

Androstenedione, testosterone, and free testosterone concentration in women of various occupations.

Abstract Serum androstenedione (A), testosterone (T), testosterone‐binding globulin (TBG), and free testosterone concentration (FTC) were correlated with occupational status in fifty‐five normal females. With some variation among age groups, A, T, and FTC were positively associated with occupation, mean levels being higher in students and professional, managerial, and technical workers compared to clerical workers and housewives. A and FTC also associated significantly with the degree of job complexity in relation to people, whereas T significantly correlated with the degree of job complexity in relation to things. These data most likely reflect biological and environmental causes in a hormone‐behavior feedback relationship. The genetic potential for increased androgen secretion may affect certain personality characteristics related to career‐orientation in females, i.e., those labeled “masculine” by this culture, such as assertiveness and independence. Moreover, the positive or negative stress associated...

A note on using the moment generating function to teach the laws of large numbers

Abstract The presentation of the weak and strong laws of large numbers in those courses in probability and mathematical statistics that first include proofs provides a golden opportunity to discuss the differences between the two types of convergence involved, and to show why these differences seem unimportant for sequences of sample means. Without a proof of the strong law, this presentation loses much of its intended impact. In this note, an example is provided that isolates the critical condition that distinguishes between the two types of convergence, and an elementary proof of the strong law is given that depends on a simple, but seldom used inequality based on the moment generating function.

CHAPTER 4 – Linear Filters—General Properties with Applications to Continuous-Time Processes

This chapter discusses the basic properties of linear filters. Linear filters provide an important class of models for physical transformations. For example, to a good degree of approximation, the earth behaves like a linear filter to seismic waves and the ocean to ocean waves. To a more restricted extent, phenomena, such as economic systems and biological systems, behave like linear filters for restricted lengths of time. Linear filters transform time series into new time series where the term time series can be interpreted in the broadest sense as meaning any numerical function of time whether continuous or discrete, random or non-random. Because of this feature, an important topic is the construction of special purpose linear filters to modify data to meet particular objectives or to display specific features of the data. The chapter discusses the methods for combining linear filters, by which a variety of special filters can be obtained. Moreover, it is presented in the chapter how high-pass, band-pass, and notch filters can be constructed from low-pass filters.

CHAPTER 3 – Sampling, Aliasing, and Discrete-Time Models

This chapter discusses the basic operation of sampling a time series that converts a time series in continuous time to one in discrete time. This is a necessary step in the preparation of the series for manipulation by a digital computer. When the original series is modeled by a weakly stationary stochastic process, the sampled series will be a weakly stationary process in discrete time, complete with its own spectral representation and power spectrum. The relationship between the spectrum of the continuous-time series and that of the sampled series is quite important, as estimates of the spectrum for the sampled series must be used to estimate the spectrum of the original series. Proper selection of the sampling rate guarantees good agreement between the two spectra, thus, the possibility of forming good estimates of the continuous-time spectrum. Improper selection of the sampling rate introduces the problem of aliasing where the agreement between the two spectra is, to varying degrees, destroyed. The chapter presents the aliasing problem and the means by which it can be avoided. Moreover, the chapter presents a sampling theorem that relates the required rate of sampling to recover all of the information in the time series to bounds on the width of the spectrum.

CHAPTER 1 – Preliminaries

Publisher Summary This chapter presents a preliminary idea of the scope of applicability of time series analysis. It illustrates the physical processes that are designed for models of time series and presents some of the basic features of the models. The central feature of all models is the existence of a spectrum by which the time series is decomposed into a linear combination of sines and cosines. Generally, several kinds of spectral or Fourier decompositions are used in time series analysis and it is somewhat of a problem to remember them clearly. They all have properties in common that are essentially geometric in character. Moreover, the same geometry, which is basically the geometry of vector spaces, plays a central role in the construction and interpretation of the important stochastic time series model.

CHAPTER 7 – Finite Parameter Models, Linear Prediction, and Real-Time Filtering

This chapter discusses the moving average processes that are important collections of finite parameter models. The equivalence between mixed autoregressive-moving average processes and processes with rational spectral densities are established in the chapter; these processes have predictors that can be put in recursive form. Finite-order autoregressions are quite useful as models for observed time series and, as in the case of moving averages, a great deal of effort has gone into devising schemes for fitting them to time series data. The most popular method is based on the Yule–Walker equations, which comprise a set of linear equations relating the coefficients of the autoregression to its autocovariances. By estimating the autocovariances, using observations from the given time series, the coefficients of the autoregression that best fits this series can be obtained by solving the equations with the estimated autocovariances in place of the real ones.

CHAPTER 8 – The Distribution Theory of Spectral Estimates with Applications to Statistical Inference

This chapter discusses the distribution theory of spectral estimates with applications to statistical inference. The distribution theory of the periodogram based on a trigonometric polynomial regression function with a white noise residual was derived by Fisher and was used as the basis for testing for the existence of periodicities. The chapter describes the distribution theory for standard estimators of the spectral density of a one-dimensional time series and the theory for multivariate time series. The theory is applied to the calculation of confidence intervals and the testing of hypotheses for these parameters. The chapter also presents the time series that are discrete, zero-mean, and stationary Gaussian processes with continuous spectra. The restriction to Gaussian processes simplifies the theory substantially and makes it possible to present a rather intuitive yet precise account of results. These results are actually valid for a much broader class of time series.

CHAPTER 6 – Digital Filters

Publisher Summary This chapter describes the basic features all digital filters—linear filter in discrete time—share and presents the rationale for a variety of standard filter construction techniques. Digital filters are used for the purposeful modification of discrete-time data. Consequently, it is not only important to understand the operation of these filters but also to know the way to select the parameters of the filters to achieve specific objectives of data modification. This selection of parameters is called filter design. Because of the many kinds of filtering operations investigators have found necessary or useful, the literature on digital filtering is extensive and diverse. In economics, for example, much attention is given to isolating or removing seasonal trends to detect weaker features of the spectrum. Geophysicists are concerned with removing tidal effects and other low-frequency power to improve the characteristics of spectral estimates. In many fields, the suppression of extraneous noise to better define a weak signal is a problem of importance.

A Note on the Estimation of Amplitude Spectra for Stochastic Processes with Quasi-Linear Residuals

Abstract It is shown that when a deterministic signal is observed in quasi-linear noise (herein defined), the usual estimate of the signal amplitude spectrum for a finite time interval converges uniformly and exponentially fast in probability to this spectrum as the length of the time interval becomes infinite. When the limiting amplitude spectrum exists, the estimate tends to this function exponentially fast but no longer uniformly, in general.