Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

Cyclostationarity: half a century of research

Ecological data are often lognormally distributed. Nutrient concentrations, population densities and biomasses, rates of production and other flows are always positive, and generally have standard deviations that increase as the mean increases. Lognormally distributed variables have these characteristics, whereas normally distributed variables can be negative and have a standard deviation that does not change as the mean changes. Lognormal errors arise when sources of variation accumulate multiplicatively, whereas normal errors arise when sources of variation are additive. Given a normally distributed random variable Y, one can calculate a lognormally distributed random variable Z = exp(Y) where exp stands for the exponential function (exp(x) = e x). The mean Z and the standard deviation s Z of the lognormal variable are related to the mean Y and standard deviation s Y of the normal variable by) 2 / exp() exp(2 Y s Y Z = [1] 5. 0 2 ] 1) [exp(− = Y Z s Z s [2] Equation 1 can be used to correct for transformation bias in logarithmic regression. Suppose that lognormally-distributed observations Z have been log transformed as Y = log(Z) to fit a regression model such as ε + =) , (ˆ b X f Y [3] where Y is the log-transformed response variable which is predicted to be Y ˆ computed from the function f, X is a matrix of predictors, b is a vector of parameters, and the errors ε are normally distributed with mean zero and standard deviation s ε. Predictions Z ˆ in the original units are calculated using equation 1 as ] 2) ˆ exp[(ˆ 2 ε s Y Z + = [4] Note that estimates the median prediction of Z, which will be smaller than the mean for a lognormally distributed variate. Thus it makes sense to adjust the median upward, as in equation 4.) ˆ exp(Y Equation 1 is also used in drawing random numbers from a lognormal distribution. Generators for normally-distributed random variables Y are common. Suppose we draw many values of Y with mean zero and standard deviation s Y. Then from equation 1, the mean of exp(Y) will not be 1 = e 0 ; instead the mean of exp(Y) will be. Generally, however, one would prefer to have the mean of a set of lognormally distributed random numbers be 1. This can be accomplished by shifting the random numbers to Y) 2 / exp(2 Y …

対数正規分布(Lognormal Distribution)のあてはめについて

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Operations research: applications and algorithms / Wayne L. Winston

Standard signal extraction results for both stationary and nonstationary time series are expressed as linear filters applied to the observed series. Computation of the filter weights, and of the corresponding frequency response function, is relevant for studying properties of the filter and of the resulting signal extraction estimates. Methods for doing such computations for symmetric, doubly infinite filters are well established. This study develops an algorithm for computing filter weights for asymmetric, semi-infinite signal extraction filters, including the important case of the concurrent filter (for signal extraction at the current time point). The setting is where the time series components being estimated follow autoregressive integrated moving-average (ARIMA) models. The algorithm provides expressions for the asymmetric signal extraction filters as rational polynomial functions of the backshift operator. The filter weights are then readily generated by simple expansion of these expressions, and the corresponding frequency response function is directly evaluated. Recursive expressions are also developed that relate the weights for filters that use successively increasing amounts of data. The results for the filter weights are then used to develop methods for computing mean squared error results for the asymmetric signal extraction estimates.

Computation of Asymmetric Signal Extraction Filters and Mean Squared Error for ARIMA Component Models

Markovian start-up demonstration tests with rejection of units upon observing d failures

. Standard signal extraction results for both stationary and nonstationary time series are expressed as linear filters applied to the observed series. Computation of the filter weights, and of the corresponding frequency response function, is relevant for studying properties of the filter and of the resulting signal extraction estimates. Methods for doing such computations for symmetric, doubly infinite filters are well established. This study develops an algorithm for computing filter weights for asymmetric, semi-infinite signal extraction filters, including the important case of the concurrent filter (for signal extraction at the current time point). The setting is where the time series components being estimated follow autoregressive integrated moving-average (ARIMA) models. The algorithm provides expressions for the asymmetric signal extraction filters as rational polynomial functions of the backshift operator. The filter weights are then readily generated by simple expansion of these expressions, and the corresponding frequency response function is directly evaluated. Recursive expressions are also developed that relate the weights for filters that use successively increasing amounts of data. The results for the filter weights are then used to develop methods for computing mean squared error results for the asymmetric signal extraction estimates.

Donald E. K. Martin

Papers

Computation of Asymmetric Signal Extraction Filters and Mean Squared Error for ARIMA Component Models

Markovian start-up demonstration tests with rejection of units upon observing d failures

Computation of asymmetric signal extraction filters and mean squared error for ARIMA component models

Application of auxiliary Markov chains to start-up demonstration tests

On the distribution of the number of successes in fourth- or lower-order Markovian trials