NON-PARAMETRIC APPROACH IN TIME SERIES ANALYSIS

Abstract. Suppose that { X _t } is a Gaussian stationary process with spectral density f ( Λ ). In this paper we consider the testing problem , where K (Λ) is an appropriate function and c is a given constant. This test setting is unexpectedly wide and can be applied to many problems in time series. For this problem we propose a test based on K { f _n ( Λ )} dΛ where f _n ( Λ ) is a non-parametric spectral estimator of f ( Λ ), and we evaluate the asymptotic power under a sequence of non-parametric contiguous alternatives. We compare the asymptotic power of our test with the other and show some good properties of our test. It is also shown that our testing problem can be applied to testing for independence. Finally some numerical studies are given for a sequence of exponential spectral alternatives. They confirm the theoretical results and the goodness of our test.     

Some Results Concerning the Asymptotic Distribution of Sample Fourier Transforms and Periodograms for a Discrete-time Stationary Process with a Continuous Spectrum

We consider a stationary process ( X_t , t = 0, ±1, ...) with a continuous spectrum. Denote by D_n (λ) a tapered Fourier transform of ( X ₀, X ₁, ..., X _{n −1}) at (angular) frequency λ. We obtain the asymptotic distribution of D_n (λ) and the joint asymptotic distribution of { D_n (λ _j ), 1 ≤ j ≤ k } with continuity of the spectral density f (.) at the relevant frequencies as the only assumption concerning the second-order structure of ( X_t ); all other assumptions required are easily stated. The results are extended to processes for which f (.) is continuous except at λ = 0, with lim_λ←0 f (λ)λ^{2 d} = K , a constant, where 0 < d < ½, as is typical of certain types of processes with long-range dependence. Results for the sample periodogram, proportional to | D_n (λ)|², follow immediately.     

AUTOREGRESSIVE PROCESSES WITH NORMAL STATIONARY DISTRIBUTIONS

Abstract. For the strictly stationary AR( k ) process Z _t = Λ ( Z _{t -1}) + α _t , with Λ : R ^k → R , Z _{t -1}= [ Z _{t -1}, Z _{t -2},…, Z _t-k ] and { α _t } an independent identically distributed white noise process, we partially characterize the Λ for which the stationary distribution of Z _t is normal.     

Regression Models with Time Series Errors

In models of the form Y_t = r ( X_t ) + Z_t , where r is an unknown function and { X_t } is a covariate process independent of the stationary error { Z_t }, we give conditions under which estimators based on residuals Z ₁, ..., Z _n obtained from linear smoothers are asymptotically equivalent to those based on the actual errors Z ₁, ..., Z_n .     

ON STATIONARITY OF THE SOLUTION OF A DOUBLY STOCHASTIC MODEL

Abstract. Consider the discrete parameter process {X_I} satisfying the doubly stochastic model X_t =ø_tX_t-1+ε_t where {ø} and {ε_t} are also stochastic processes. Necessary and sufficient conditions on {ø} are given for { X₁ } to be a second order process. When {ø_t} is a strictly stationary process, some sufficient conditions in terms of {ø} are given which guarantee the wide sense stationarity of {X_t} . It turns out that for these problems the distribution and dependence structure of the process {log |ø|} play an important role.     

A note on L1 density estimation for linear processes

In this paper, we consider the L ₁ performance of a kernel estimator, f^_n of the density of a linear process X_t ∑^∞ _{k =0} a _k Z _t−k , a ₀ = 1, where { Z _t } is a sequence of independent and identically distributed (i.i.d.) random variables with E | Z ₁|^ε< ∞, for some ε > 1, and { a_k } is a sequence of reals converging to zero at a certain rate. Asymptotic minimizations of the integrated L ₁ risk of f_n and its upper bounds are considered. This paper extends the earlier results for the i.i.d. case by Devroye and Gyorfi ( Nonparametric Density Estimation: The L₁ View. New York: Wiley, 1985) and by Hall and Wand (Minimizing the L ₁ distance in nonparametric density estimation, J. Multivariate Anal. 26 (1988), 59–88) to the linear process case. Numerical examples to illustrate the performance of f_n are also presented.     

PARTIAL AUTOCORRELATION PROPERTIES FOR NON-STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. We are primarily interested in relating the partial autocorrelation behaviour of an autoregressive integrated moving-average process of order ( p, d, q ), { Z _i } say, with those of its D -differenced processes {(1 - B ) ^D Z _i } ( D = 1, …, d ). To this end, we evaluate the early partial correlations corresponding to serial correlations which initially follow a slow linear decline from unity. These partials, to a first approximation, take a small constant negative value from lag 2 onwards. We also demonstrate a relationship between the theoretical partials π _k and π _k (Δ) for a once-integrated process { Z _i } and its first-differenced process {(1 - B ) Z _i } respectively. These results carry over to cases where the non-stationary zeros are at -1, and differencing is replaced by the corresponding 'simplifying' transformation implicit in the operator 1 + B.     

ESTIMATION FOR THE FIRST-ORDER DIAGONAL BILINEAR TIME SERIES MODEL

Abstract. The problem of estimation of the parameter b in the simple diagonal bilinear model { X _t }, X_t = e_t + be _{t -1} X_{t -1}, is considered, where { e_t } is Gaussian white noise with zero mean and possibly unknown variance ². The asymptotic normality of the moment estimator of b is established for the two cases when ² is known and ² is unknown. It is noted that the limit distribution of the least-squares cannot easily be derived analytically. A bootstrap comparison of the sampling distributions of the least-squares and moment estimates shows that both are asymptotically normal with the least-squares estimate being the more efficient.     

STATIONARITY OF THE SOLUTION OF Xt= AtXt-1+εt AND ANALYSIS OF NON-GAUSSIAN DEPENDENT RANDOM VARIABLES

Abstract. We give general and concrete conditions in terms of the coefficient (stochastic) process {A_t} so that the (doubly) stochastic difference equation X_t= A_tX_t-1+ε_t has a second-order strictly stationary solution. It turns out that by choosing {A_t} and the "innovation" process {ε_t} properly, a host of stationary processes with non-Gaussian marginals and long-range dependence can be generated using this difference equation. Examples of such nowGaussian marginals include exponential, mixed exponential, gamma, geometric, etc. When {A_t} is a binary time series, the conditional least-squares estimator of the parameters of this model is the same as those of the parameters of a Galton-Watson branching process with immigration.     

THE DISTRIBUTION OF PERIODOGRAM ORDINATES

Abstract. The empirical distribution function of y _i = I (ω _j )/{2π f (ω _j )}, ω j = 2π j / T , where I (ω) is the periodogram for a set of observations from a stationary time series with spectral density f (ω), is shown to converge, almost surely, to the distribution with density exp(- x), under appropriate conditions. The same methods are used to prove the convergence, almost surely, of an estimate of the prediction error variance constructed from the I (ω_j) and of the complex empirical distribution function based on the Fourier coefficients.