covariance function, 174 probability structure of; Yule-Walker equations, 174 Moving average process, fluctuations, 5-6, 175 probability structure of

Transcription

1 Index* The Statistical Analysis of Time Series by T. W. Anderson Copyright 1971 John Wiley & Sons, Inc. Aliasing, Autoregressive {continued) Amplitude, 4, 94 case of first-order, 174 Associated polynomial equation, case of second-order, 174 see Autoregressive process, covariance function, 174 probability structure of; Yule-Walker equations, 174 Moving average process, fluctuations, 5-6, 175 probability structure of infinite order, Autocorrelation, definition of, 255 in prediction, 423 See also Covariance, sample; representation, 423 Covariance function; prediction with minimum mean Serial correlation, square error, general representation, , I, Autocovariance, see Covariance, sample; 177, , , Covariance function Autoregressive process, probability as doubly-infinite moving average, structure of, 5-6, , I , 256, , as first-order vector process, 177, , , , as infinite moving average, See also Autoregressive process, , , , statistical inference in; Auto regressive process with moving in terms of a lag operator, 168 average residuals, probability spectral density, , 436 structure of; Stochastic process, case of first-order, st at ionary case of second-order, 408 associated polynomial equation, 169, stochastic difference equation, 172, 180, , 378, , 248, 256, , 436, as characteristic equation of vector process, first-order, vector process, 180 covariance matrix of, with a unit root, 172 moving average representation covariance structure, , 378 of, 178 *References to books and papers are listed by author for each section at the end of the respective chapter. Therefore, the index does not give author references. 689

2 690 INDEX Autoregressive (continued) stochastic difference equation for, 177 with independent variables, 183 with independent variables, with superimposed error, , 379,436 Autoregressive process, statistical inference in, , , , , 295, , 358,369 See also Autoregressive process, probability structure of; Serial correlation, general confidence regions, large-sample, 214 maximum likelihood estimate in scalar case, , , 200, 203, , , , , 369 asymptotic' normality of, 200, 203, consistency of, estimate of asymptotic covariance matrix of, in first-order stationary process, , 369 in terms of partial correlations, 188 modified equations for, normal equations for, 184 numerical examples, , maximum likelihood estimate in vector case, first-order, asymptotic equivalence of secondorder sample moments from two processes, asymptotic normality of, consistency of, sufficient statistics, 185, , 358 test for several coefficients,, simultaneous, 214 test for several coefficients (test of order), , in terms of partial correlation, 222 Autoregressive (continued) large-sample chi-square tests, likelihood ratio, test for single coefficient, consistency of, 215 Autoregressive process generated by random numbers, numerical examples of, , , estimated spectral densities of, , fitted autoregressive processes, 243 observations, sample spectral densities of, 495, , spectral densities of, 495, spectral densities of fitted processes, 495, tests of order, 243 variances and correlations of, 495, Autoregressive process with moving average residuals, estimation of' parameters of, using moving average representation, using sample correlations, asymptotic normal distribution, 241 Autoregressive process with moving average residuals, probability structure of, , , parametrization in terms of correlations, 237 representation, 236 in terms of lag operator, 236 moving average, 236 spectral density, 409 Bandwidth, 533 Barycentric coordinates, 127, 161, 363, Bessel function, 117 Best linear unbiased estimates, see Linear regression

3 Beta-type density function, Beveridge Wheat Price index, , , 495, 547, 551, correlations of, 628 data, estimated spectral density of, 547, fitted autoregressive process, spectrogram of, 495, Bias, see Spectral density estimate (general theory); Spectral density estimates, examples of Buijs-Ballot table, , 11 2 Cauchy-Schwarz inequality, Central limit theorems, for finitely dependent stationary process, , for finitely dependent stationary vector process, 429 for independently identically distributed random variables, 427 for stationary linear process, Liapounov's, Lindeberg, 426 Cedro summation, 460,499 Characteristic function, See also Quadratic forms Choice of order of dependence in serial correlation models, See also Tests of order of dependence in serial correlation models case of known mean, based on one-sided tests, based on two-sided unbiased tests, Neyman structure and similarity of regions, case of unknown mean, 299 Circular serial correlation, see Serial correlation, circular INDEX 691 Convergence in mean (quadratic mean), 377, Correlation, partial, 188, 222, 251, 267, ,361,369 of stationary process, 361 sample serial, 188, 222, 251, 267, , 369 Correlation, sample, , See also Serial correlation, general asymptotic normality of, , Correlation coefficient, product moment (Pearson), 109,255, 340 distribution of, 340 Correlation function, see Covariance function; Stochastic process, stationary Cosine, see Trigonometric functions Cospectral density, 391 Covariance, sample, 440, , , I:, 498, case of known mean, 440,45042, , as unbiased estimate, 440 asymptotic covariance of, asymptotic distribution of, asymptotic variance of, 468 covariance of, case of unknown mean, 440, , , ,481,498 asymptotic covariance of, asymptotic distribution of, 481 asymptotic mean of, covariance of, definitions, 440 expected values of, ,498 using residuals from trend, asymptotic distribution of, asymptotic mean of, definition of, 589 Covariance function, 48, , , ,

4 692 INDEX Covariance (con tin ued) 373, 380, , 436 See also Autoregressive process, probability structure of; Moving average process, probability structure of; Spectral density (of stationary process) approximation of, by covariance functions of moving average and autoregressive processes, 436 as Fourier transform of spectral distribution function, 380 generating function of, 224 in circular model, of autoregressive process, of complex-valued process, 389 of differenced time series, 64 of moving average process, of stationary process, , of stochastic process, 373 of vector process, 390 of weighted average, 48 Covariance matrix, estimation by least squares residuals, Cumulants of stationary stochastic process, fourth-order, , 467,497 of moving average process, 467 Cyclic regression, see Trigonometric regression when periods are integral divisors of series length; Trigonometric regression when periods are not integral divisors of series length Deflated Aggregate Disposable Income, , Dependence, see Tests of order of dependence in serial correlation models Determining degree of polynomial regression, see Polynomial regression Determining order of dependence, Determining (continued) see Choice of order of dependence in serial correlation models Difference operator, 62, 88, 165 Dow Jones Industrial Stock Price Averages, 85 Durbin-Watson statistic, see Serial correlation based on successive differences Efficiency of linear estimates, see Linear regression Estimates of coefficients of a utoregressive process, see Autoregressive process, statistical inference in coefficients of autoregressive process with moving average residuals, see Autoregressive process with moving average residuals, estimation of parameters of coefficients of moving average process, see Moving average process, estimation of parameters of covariance function, see Covariance, sample covariance matrix, see Covariance matrix, estimation by least squares residuals mean, see Linear regression; Mean, sample regression coefficients, see Linear regression; Nonlinear regression; Polynomial regression; Trigonometric regression when periods are integral divisors of series length; Trigonometric regression when periods are not integral divisors of series length spectral density, see Spectral density, sample; Spectral density estimate (general theory); Spectral density estimate based on fitted autoregressive process; Spectral density estimates, examples of

5 [NDEX 693 Estimates (continued) spectral distribution function, see Spectral distribution function, sample variance, see Covariance, sample; Variate difference method Fast Fourier Transform, , 550 Fej6r s kernel, 454, 461, 498, , 552 Filter, linear, 399 See also Linear transform of stationary process Fisher s test, see Trigonometric regression when periods are integral divisors of series length Folding frequency, Forecasting, see Prediction Fourier coefficients, sample, see Trigonometric coefficients, sample; Trigonometric regression when periods are not integral divisors of series length Fourier series, , 388, 397 See also Trigonometric functions convergence of, 388 for step functions, 397 Fourier transform, See also Fast Fourier transform; Trigonometric functions Frequency, 4,93 Frequency response function, 400 Gauss-Markov Theorem, see Linear regression Gaussian, 257, 371 See also Stochastic process, stationary Gibbs phenomenon, 514 Graduation, 56 See also Smoothing, linear Hardy class H2, 424 Hermitian matrix, 391, Hilbert space, See also Linear vector space Cauchy criterion for convergence in, 414 Hilbert (continued) completeness of, 4 14 distance in, 414 generated by a stationary process, 414 inner product, 4 14 linear manifold, 4 17 norm in, 414 orthogonality in, projection on subspace of, 418 projection theorem, 418 residual from subspace, 4 18 Hilbert space generated by a stationary process, see Hilbert space Independence, tests of, see Tests of order of dependence in serial correlation models Innovation, 420 Intensity, 108 See also Spectral density, sample; Spectrogram; Trigonometric regression when periods are not integral divisors of series length Kantorovich inequality, 569 Kronecker matrix product, 200 Lag operator, 168 Least squares estimates, see Linear regression; Nonlinear regression; Polynomial regression; Trigonometric regression when periods are integral divisors of series length Limit theorems, general, See also Autoregressive process, statistical inference in; Autoregressive process with moving average residuals, estimation of parameters of; Central limit theorems; Covariance, sample; Linear regression; Mean, sample; Moving average process, estimation of parameters of; Quadratic forms, ratios of; Spectral density, sample; Spectral density estimate (general theory); Trigonometric coefficients, sample;

6 694 INDEX Limit theorems (continued) Trigonometric regression when periods are integral divisors of series length; Variate difference method approximating sequence of random variables, matrix-valued functions of matrices, 429 Linear operator, 61,88,666 Linear process, see Moving average process, probability structure of Linear regression, 8-29,478, , 618 See also Nonlinear regression; Polynomial regression; Trigonometric regression when periods are integral divisors of series length; Trigonometric regression when periods are not integral divisors of series length confidence regions for coefficients in case of normality, least squares estimates in case of uncorrelated variables with common variance, 8-18, asymptotic normal distribution of, consistency of estimate of variance of, covariances of, 9 expected value of, 9 Gauss-Markov Theorem, 9,27 geometric representation of, linear transformation of independent variables, minimum variance unbiased linear estimates, 9 normality of, in normal case, 10 orthogonality of residuals, 11 sufficiency of, in normal case, 10, 27 linear estimates in case of general covariance matrix, 18-20, ,618 best linear unbiased estimates, 18-20, Linear (continued) concentration ellipsoids of, 566 efficiency of least squares estimates, , 618 efficiency of least squares estimates, lower bound on, equivalence of Markov and least squares estimates for all matrices of independent variables, conditions for, 570 equivalence of Markov and least squares estimates of parameter subset, conditions for, 19, Markov estimates] 18-20, maximum likelihood estimates in case of normality, 19 linear estimates, in case of stationary random terms, 478, asymptotic efficiency of least squares estimates, conditions for given spectral density, 581, asymptotic efficiency of least squares estimates, conditions for independence of spectral density, 58 1, asymptotic efficiency of least squares estimates for trigonometric and polynomial sequences of independent variables, asymptotic normality, limiting covariance matrices of least squares and Markov estimates, 478 limiting covariance matrices of least squares estimate, consistent estimate of, 587 prediction, in case of, 19-23, 29 intervals for, 29 minimum variance linear predictor, tests for coefficients, Linear transform of stationary process,

7 INDEX 695 Linear f co n timed) covariance function of, 399 effects on spectral density, spectral distribution function of, stationarity of, 399 Linear vector space, See also Hilbert space Cauchy sequence in, 416 Cauchy-Schwarz inequality, completeness of, 41 6 distance in, 4 I6 inner product, 415 metric, 416 norm, 416 triangle inequality, 416 Logistic curve, 79 Markov estimates, see Linear regression Matrix, , 250, , 363, 391, See also Quadratic forms characteristic roots and vectors of polynomial of, 278 characteristic roots of, inequalities for, diagonalizable, , 363 diagonalization of symmetric, 278, 363 Hermitian, 391, Jordan canonical form of, , 250,671 Maximum likelihood estimates, see Autoregressive process, statistical inference in; Linear regression; Serial correlation, circular; Trigonometric regression when periods are not integral divisors of series length Mean, sample, , 444, , , 498 asymptotic distribution of, asymptotic mean and variance of, covariance of sample mean and sample variance for linear process, 498 Mean (continued) variance of, 444 Meat, annual consumption of, 43-46, 78 Moving average of sample series, see Smoothing, linear Moving average process, estimation of parameters of, See also Moving average process, probability structure of asymptotic normal distribution of sample correlations, using approximate normal distributions of sample correlations, using approximating au toregressive process, asymptotic normality, Moving average process, probability structure of, , , , , 422, 436, See also Moving average process, estimation of parameters of as representative of general process, , , 422 defined by spectral density, for regular process, 422 with a finite number of non-zero given covariances, associated polynomial equation, 224 covariance structure, covariance function, 223 covariance generating function, 224 covariance generating function, zeroes of, 224 cumulant, fourth-order, prediction with minimum mean square error, 406 representation, , 406 in terms of linear operators, 406 infinite autoregressive, spectral density, , 436 alternative forms in general, 405 first-order, second-order,

8 6% mmx Multipledecision procedures, see Autoregressive process, statistical inference in; Choice of order of dependence in serial correlation models; Polynomial regression; Quadratic forms; Serial correlation, general; Spectral density estimate (general theory); Tests; Trigonometric regression when periods are integral divisors of series length; Variate difference method Multiplicative model, New York Commercial Paper Rate Series, spectral density estimate, Neyman structure, 39.84, 26 1, 272,360 Neyman-Pearson Fundamental Lemma, Noise, see Stochastic process, stationary Noncentral chi-square, 117, 16 1 bounded completeness of, 161 Nonlinear regression, least squares estimates, steepest descent method for, Numerical examples, see Autoregressive process generated by random numbers, numerical examples of; Beveridge Wheat Price Index; Deflated Aggregate Disposable Income; Dow Jones Industrial Stock Price Averages; Meat, annual consumption of; New York Commercial Paper Rate Series; Population of the United States; Receipts of butter at five markets; Sunspot series; Votes, ratio of total Republican to Democratic Nyquist frequency, Orthogonal polynomials, 16-17, 32-33, 82.87, See also Polynomial regression Partial correlation, see Correlation, partial Period, 4,93 Periodic functions, See also Trigonometric functions Periodogram, 107 See also Spectral density, sample; Spectrogram Phase, 4,93 Polynomial regression, 16-17, 31-46,82, 8447, , determining the degree of, forward sequential test in, 43 similar regions in, sufficient statistics in, 39, least squares estimates in, 32-33, asymptotic efficiency of (for stationary random terms), orthogonal polynomials, 14-17, 31-34,82,86-87, prediction, in case of, tests for a single coefficient in, uniformly most powerful invariant test, uniformly most powerful unbiased test, Population of the United States, 91 Power spectrum, see Spectral density (of stationary process) Power transfer function, 400 Prediction, see Autoregressive process, probability structure of; Linear regression; Moving average process, probability structure of; Polynomial regression; Prediction in stationary processes, linear Prediction in stationary processes, linear, 417,419420,422 See also Autoregressive process, probability structure of as projection, minimum mean square error, 4 17,419 variance of, 422 Prewhitening, 546, 550

9 Quadratic forms, 67, , , , , , , 365, ,497, ,605, 610 See also Quadratic forms, ratios of canonical forms for, characteristic roots and vectors of, in independent random variables, 67,73,90-91 covariance of, 73 variances of, 67, in normal random variables, , 308, ,365,497,605, 610 characteristic functions of, , 308 cumulantsof, 332, 605,610 examples of, 497 moment generating function of, 332,365 moments of, in probability models of dependence, , , , systems of, , systems with double roots, in random variables forming a stationary process, , covariances of, , expected value of, 445, variance of, in residuals, independence of sample mean and residuals, 296 Quadratic forms, ratios of, 91, 306, 500 See also Serial correlation, general in independent random variables, 91,306 asymptotic normal distribution of, 91 distribution of, 306 in random variables forming a stationary process, asymptotic INDEX 697 normal distribution of, 500 Quadrature spectral density, 39 1 Randomly generated series, see Autoregressive process generated by random numbers, numerical examples Ratio to moving average, Receipts of butter at five markets, , 302 Regression, see Linear regression; Nonlinear regression; Polynomial regression; Trigonometric regression when periods are integral divisors of series length; Trigonometric regression when periods are not integral divisors of series length Regular process, 420 Sample covariance, see Covariance, sample Sample mean, see Mean, sample Sample spectral density, see Spectral density, sample Sample spectral distribution function, see Spectral distribution function, sample Sample trigonometric coefficients, see Trigonometric coefficients, sample Schuster test, see Trigonometric regression when periods are integral divisors of series length Seasonal variation, Serial correlation, circular, , 295, , , , , , 348, 353, , 362, ,607608,620 See also Serial correlation, general: Serial correlation, based on successive differences approximate distributions of, , 607 numerical comparison of, 342,344

10 698 INDEX Serial (continued) using Beta-type density, 343, 607 using Fisher s z-transformation, case of known mean, , , 322, , , 341, approximate moment generating function of, 341 definitions and canonical forms of, distribution of, , 364 moment generating function of, , moments of first-order, 322, case of unknown mean, 295, , 322, 328, , 337, , 362, conditional distributions of, definition of, 295 distribution of, , 362 distribution of, under dependence, joint distribution of, moment generating function of, 328, 330, 366 moments of first-order, , 337, significance points for, table of, 319 model for, covariance function in, 357 likelihood ratio test of independence in, 356 maximum likelihood estimate, asymptotic equivalence to, maximum likelihood estimate in, partial, , using residuals from trigonometric trend, , , 620 definition of, 301 distribution of, 620 example of, 302 Serial (con tin ued) moments of, 607 significance points for, table of, 608 Serial correlation, general, ,267, , 299, , 364, See also Serial correlation, circular; Serial correlation based on successive differences; Tests of order of dependence in serial correlation models approximate distributions of, case of known mean, , 267 case of unknown mean, , 299 independence of sample mean and, 296 distribution of, for one non-paired root, 314 distribution of, for paired roots, , 364 using characteristic functions, 3 17, 364 moment generating functions of first-order, , 365 moments of, 322, using residuals from trend, distribution of, 61 1 inequalities for distribution of, Serial correlation based on successive differences, , 295, , 328, , , , , 620 See also Serial correlation, circular; Serial correlation, general approximate distributions of, numerical comparisons of, 346 related to Pearson correlation, 346 using Beta-type density, 346 case of known mean, definition and canonical forms of,

11 INDEX 699 Serial (con tin ued) case of unknown mean, 295, , 328, , definition of, 295 distribution of, , 365 moment generating function of, 328 moments of, , 366 using residuals from trend (Durbin- Watson statistic), , 620 Shift operator, 61 Sine, see Trigonometric functions Smoothing, linear, 46-60, 64-66, 86-87, 89 See also Linear transform of stationary process coefficients for, 5 1, 64-66, 86-87, 89 elimination of seasonal variation by, in terms of difference operators, in terms of polynomial regression, Spencer s 15-point and 2 1 -point moving average, 56 variances and covariances of smoothed values, 48, Spectral density (of stationary process), 380, 386, 388, 391, , 422, 436, 500, See also Linear transform of stationary process approximation by autoregressive and moving average spectral densities, , as Fourier transform of covariance function, 380 as proportional to multiple correlation coefficient between time series and pair of trigonometric functions, 386 cospectral, 39 I Fourier representation of, normalized, 388 Spectral (con tin ued) of autoregressive process, ,436 examples of, 436 first-order, second-order, 408 with superimposed error, 436 of autoregressive process with moving average residuals, of differenced series, 412 of differenced smoothed series, of moving average, finite, , 500 continuity of, 500 first-order, second-order, of moving average process, convergence in mean, 400 of regular process, 422 of simple moving average, of smoothed series, 4 12 of vector process, 391 quadrature, 391 Spectral density, sample, 382, 386, 422, , , , 499, See also Spectral density estimate (general theory) case of known mean, 382, 386, 442, , , , 499 as proportional to multiple correlation with pair of trigonometric functions, 386 asymptotic covariance of, asymptotic distribution of, 477, asymptotic mean of, 471 asymptotic representation of, covariance of, ,499 definition of, 442 mean of, 382,

12 700 Spectral (continued) case of unknown mean, , ,485 asymptotic covariance of, 476 asymptotic distribution of, 485 asymptotic mean of, mean of, inconsistency of, 474, 477, 485 using residuals from trend, asymptotic mean of, definition of, 594 Spectral density estimate (general theory), , , , , ,661 See also Spectral density, sample; Spectral density estimate based on fitted autoregressive process; Spectral density estimates, examples of as density of a finite moving average process, 602 case of known mean, , approximate chi-square d istribution for, as linear combination of sample covariances, , asymptotic bias of, asymptotic mean square error of, asymptotic normality of, asymptotic normality of logarithm of, 541 asymptotic variance and covariances of, confidence intervals in large samples, 541 consistency of, in terms of quadratic forms, mean of, as expected value of a quadratic form, windows for, 508 case of unknown mean, INDEX Spectral (continued) asymptotic bias of, asymptotic normality of, 545 asymptotic variance and covariance of, consistency of, 545 choice of number of lags, 547, normalized, numerical examples of, , , , ,661 of a given average, asymptotic covariance of, asymptotic mean of, 5 19 asymptotic normality of, 521 scale for plotting, 547,550 using residuals from trend, asymptotic bias of, asymptotic normality of, 602 consistency of, Spectral density estimate based on fitted autoregressive process, 547, numerical examples of, 547 Spectral density estimates, examples of, ,527, 532, 534 averaging for discrete frequencies, 5 I8 Bartlett, ,527, 532, 534 asymptotic bias factor of, 527,534 asymptotic variance factor of, related to segmented series, 512 windows for, Bartlett, modified, , 527, asymptotic bias factor of, 527,534 asymptotic variance factor of, windows for, Blackman-Tukey, , 527.

13 INDEX 70 1 Spectral (continued) asymptotic bias factor of, 527,534 asymptotic variance factor of, windows for, Daniell, , 527, 532, 534 asymptotic bias factor of, 527,534 asymptotic variance factor of, windows for, hamming, 516,527,532, 534 asymptotic bias factor of, 527,534 asymptotic variance factor of, hanning, , 527, 532, 534 asymptotic bias factor of, 527,534 asymptotic variance factor of, windows for, 5 16 Parzen, , 527, 532, 534 asymptotic bias factors of, 527,534 asymptotic variance factors of, windows for, rectangular, see Spectral density estimates, examples of, Daniell sample spectral density, windows for, truncated sample spectral density, , 527, 532, 534 asymptotic bias factor of, 527, 534 asymptotic variance factor of, windows for, Spectral distribution function, 38 1, aliasing, decomposition of, 386 definition of, 38 1, 383 discrete, Spectral (continuedl examples of, folding of spectrum, folding frequency, Nyquist frequency, Spectral distribution function, sample, 496,546,556 as estimate of spectral distribution function, 546 asymptotic covariance of, 556 consistency of, 496, 556 Spectral representation of a stationary process, convergence in mean square, 397 covariance of component process, Fourier series of component process, spectral distribution function of component process, Spectrogram, 108 See also Periodogram; Spectral density, sample Spectrum of increase of matrix function, 580 elements of, 580 Spencer s moving averages, 5 6 Stirling numbers of the second kind, 88 Stochastic difference equation, see Autoregressive process, probability structure of Stochastic integrals, , Stochastic process, 164, , 380, See also Stochastic process, stationary consistent family of cdf s for, 373 convariance function of, 373 mean function of, 373 of independent increments, of uncorrelated increments, realization of, 372 with continuous time parameter, 380 with discrete time parameter, 371 Stochastic process, stationary, 164, , , , 4 2Q-424,

14 702 INDEX Stochastic (con tin ued) correlation function of, 173 covariance function of, cumulants of, fourth-order, 444,497 definition of (in strict sense), 164,373 definition of (in wide sense), 166 deterministic, 420 examples of, , Gaussian, 371, 373, purely indeterministic, 421 regular, 420,422 condition for, 422 spectral representation of, with cyclic components, Wold decomposition of, Stochastic process, vector, , See also Autoregressive process, probability structure of autoregress've, first-order, cospectral density of, 391 covariance function of, 390 quadrature spectral density of, 391 spectral distribution function of, 391 Sufficient statistics, see Autoregressive process, statistical inference in; Linear regression; Polynomial regression; Tests of order of dependence in serial correlation models Sums of powers, 83 Sunspot series, ,547, correlations of, data, 245, 660 estimated spectral density of, 547,661 spectrogram of, test of order of dependence, Systematic part, 2 Tables of significance points and distributions, 318, 342, , 608,614 serial correlation, circular, 318, 342,608 serial correlation based on successive differences, , 614 Tests, see Autoregressive process, statistical inference in; Choice Tests (continued) of order of dependence in serial correlation models; Linear regression; Polynomial regression; Tests of order of dependence in serial correlation models; Trigonometric regression when periods are integral divisors of series length; Trigonometric regression when periods are not integral divisors of series length; Variate difference method Tests of order of dependence, see Serial correlation, general; Tests of order of dependence in serial correlation models Tests of order of dependence in serial correlation models, , , case of known mean, complete family of distributions, 261 Neyman structure of test with similar region, 261 serial correlation coefficient, use of, 263,267 sufficient statistics for, 260 uniformly most powerful onesided, , uniformly most powerful unbiased, case of unknown mean, serial correlation coefficient, use of 299 sufficient statistics for, 295 uniformly most powerful onesided, uniformly most powerful unbiased, using residuals from trend, uniformly most powerful, 604 Tetrahedra, regular, , 363 uniform distributions on, volumes of, 363 Transfer function, 400 Trend, 4,30 See also Regression

15 INDEX 703 Trigonometric coefficients, sample, 137, , , ,441, ,477, See also Trigonometric regression when periods are integral divisors of series length; Trigonometric regression when periods are not integral divisors of series length case of known mean, 137, , ,441,457459,477, asymptotic covariances of, 477 asymptotic normality of, covariances of, 139, definition of, 137,441 mean of, case of unknown mean, , 441,484 asymptotic normality of, 484 covariance of, 153 definition of, 152,441 mean of, Trigonometric functions, 4, 16-17, amplitude, 4, 94 frequency, 4,93 in Fourier representation of a finite sequence, in Fourier representation of a periodic function, orthogonality of, 94-95, 100 period, 4,93 phase, 4, 93 Trigonometric polynomial, 410 Trigonometric regression when periods are integral divisors of series length, , , See also Trigonometric regression when periods are not integral divisors of series length confidence regions on trigonometric parameters, deciding inclusion of trigonometric terms, Trigonometric (continued) Fisher s test, invariance of procedure for, 116 procedure for choosing at most two positive amplitudes (Bayes), procedure for choosing one positive amplitude, known variance, procedure for choosing one positive amplitude, unknown variance, procedure for choosing positive amplitudes, known variance (Bayes, uniformly most powerful symmetric invariant), procedure for choosing positive amplitudes among a subset, unknown variance, 124 Schuster test, 118 Walker test, 120 least squares estimates in, , 110, asymptotic efficiency of (for stationary random terms), unbiasedness of, 110 variances of, 110 tests of amplitude of specified frequency, tests of hypotheses of single coefficients, 110 uniformly most powerful invariant test, unknown variance, 111 uniformly most powerful invariant test for a single frequency, known mean, 118 Trigonometric regression when periods are not integral divisors of series length, , , 160 See also Trigonometric coefficients, sample; Trigonometric regression when periods are integral divisors of series length

16 704 INDEX Trigonometric (continued) intensity function, , population, when trend has single frequency, , sample, 141 sample, chi-square distribution of, sample, maximum for uncorrelated variables, convergence of, maximum likelihood (least squares) estimates of parameters with arbitrary frequency, consistency and asymptotic normality of, 156 sample Fourier coefficients, known mean, independent errors, , , 160 consistency and asymptotic normality of, 156, 160 expression of, in terms of Fourier coefficients for periods that are integral divisors of series length, means, variances, and covariances of, when trend has single frequency, 139 orthogonal transformation of sample Fourier coefficients, quadratic form in, sample Fourier coefficients, unknown mean, independent errors, means, variances, and covariances of, Variance, see Covariance,sample; Covariance function Variate difference method, 60-79, 91 covariance of a differenced time series, determining degree of trend, estimate of variance, consistency and asymptotic normality of, linear combination of, 73 variance of, test of degree of trend, 74-79, 91 asymptotic normality of statistic for, distribution of statistic for, 77 Vector process, see Stochastic process, vector Votes, ratio of total Republican to total Democratic, 347 Walker test, see Trigonometric regression when periods are integral divisors of series length Weierstrass Trigonometric Approximation Theorem, 410 Weighted average of sample series, see Smoothing, linear Window, see Spectral density estimate (general theory); Spectral density estimates, examples of Wold decomposition theorem, Wolfer sunspot series, see Sunspot series Yule-Walker equations, see Autoregressive process, probability structure of