A numerical nonmetric approach for analyzing time series data

A numerical nonmetric approach to data analysis of periodic series with polytone trend is suggested. Estimation is made of thesmallest number of tone (monotone segments) possible for the trend. The seasonal component is estimated without need for first removing the (estimated) polytone trend. A computer program has been developed which enables analysis of arbitrary series, either by a prespecified length of period or by estimating the period length if not known in advance. Robustness of the proposed approach enables analysis of very short series, series with missing values, and other series with limitations that cannot be easily handled otherwise. In a separate appendix some empirical results obtained by this approach are compared with those from the X-ll program; this appendix will be sent upon request.     

Observable trend-projecting state-space models

Much attention has focused in recent years on the use of state-space models for describing and forecasting industrial time series. However, several state-space models that are proposed for such data series are not observable and do not have a unique representation, particularly in situations where the data history suggests marked seasonal trends. This raises major practical difficulties since it becomes necessary to impose one or more constraints and this implies a complicated error structure on the model. The purpose of this paper is to demonstrate that state-space models are useful for describing time series data for forecasting purposes and that there are trend-projecting state-space components that can be combined to provide observable state-space representations for specified data series. This result is particularly useful for seasonal or pseudo-seasonal time series. A well-known data series is examined in some detail and several observable state-space models are suggested and compared favourably with the constrained observable model.     

Observable trend-projecting state-space models

Much attention has focused in recent years on the use of state-space models for describing and forecasting industrial time series. However, several state-space models that are proposed for such data series are not observable and do not have a unique representation, particularly in situations where the data history suggests marked seasonal trends. This raises major practical difficulties since it becomes necessary to impose one or more constraints and this implies a complicated error structure on the model. The purpose of this paper is to demonstrate that state-space models are useful for describing time series data for forecasting purposes and that there are trend-projecting state-space components that can be combined to provide observable state-space representations for specified data series. This result is particularly useful for seasonal or pseudo-seasonal time series. A well-known data series is examined in some detail and several observable state-space models are suggested and compared favourably with the constrained observable model.     

Lifetime distributions motivated by series and parallel structures

According to Ross, any system can be represented either as a series arrangement of parallel structures or as a parallel arrangement of series structures. Motivated by this, we propose new three-parameter lifetime distributions by compounding geometric, power series, and exponential distributions. The distributions can allow for decreasing, increasing, bathtub-shaped, and upside down bathtub-shaped hazard rates. A mathematical treatment of the new distributions is provided including expressions for their density functions, Shannon and Rényi entropies, mean residual life functions, hazard rate functions, quantiles, and moments. The method of maximum likelihood is used for estimating parameters. Five of the new distributions are studied in detail. Finally, two illustrative data examples and a sensitivity analysis are presented.     

Parameter inference for time series with regular and seasonal unit roots

It is common to have both regular and seasonal roots present in many time series data. It may occur that one or both of the roots are just close but not equal to unity. Parameter inference for this situation is considered both when the time series has a finite or an infinite variance. Asymptotic char-acterizations of the test statistics were obtained via functionals of Ornstein-Uhlenbeck processes and Lévy processes. Tabulations for the large sample distributions are obtained. The results will be useful in applications deciding whether both regular and seasonal differencing are needed in fitting a time series model.     

L'analyse de la variation saisonniére quand l'amplitude et la taille sont faibles

Studies of seasonal variation are valuable in biomedical research because they can help to discover the etiology of diseases that are not well understood. Generally in these studies the data have certain characteristics that require specialized tests and methods for the statistical analysis. But the effectiveness of these specialized tests is variable, especially according to the seasonal variation, the dimension of the amplitude in the seasonal variation, and the sample size. The purpose of this paper is to present a test and methods appropriate for the analysis and modeling of data whose seasonal variation has small amplitude and whose sample size is small. This test can detect different kinds of seasonal variation. The results from a simulation study show that the test performs very well. The application of these methods is illustrated by two examples.     

Perfect aggregation of Bayesian analysis on compositional data

Sufficiency is a widely used concept for reducing the dimensionality of a data set. Collecting data for a sufficient statistic is generally much easier and less expensive than collecting all of the available data. When the posterior distributions of a quantity of interest given the aggregate and disaggregate data are identical, perfect aggregation is said to hold, and in this case the aggregate data is a sufficient statistic for the quantity of interest. In this paper, the conditions for perfect aggregation are shown to depend on the functional form of the prior distribution. When the quantity of interest is the sum of some parameters in a vector having either a generalized Dirichlet or a Liouville distribution for analyzing compositional data, necessary and sufficient conditions for perfect aggregation are also established.     

Testing nested and non-nested periodically integrated autoregressive models

Periodically integrated time series require a periodic differencing filter to remove the stochastic trend. A non-periodic integrated time series needs the first-difference filter for similar reasons. When the changing sea- sonal fluctuations for the non-periodic integrated series can be described by seasonal dummy variables for which the corresponding parameters are not constant within the sampie, such a series may not be easily & stinguished from a periodically integrated time series. In this paper, nested and non-nested testing procedures are proposed to distinguish between these two alternative stochastic and non-stochastic seasonal processes, When it is assumed there is a single unknown structural break in the seasonal dummy parameters. Several empirical examples using quarterly real macroeconomic time series for the United Kingdom illustrate the nested and non-nested approaches.     

Beta seasonal autoregressive moving average models

In this paper, we introduce the class of beta seasonal autoregressive moving average (βSARMA) models for modelling and forecasting time series data that assume values in the standard unit interval. It generalizes the class of beta autoregressive moving average models [Rocha AV and Cribari-Neto F. Beta autoregressive moving average models. Test. 2009;18(3):529–545] by incorporating seasonal dynamics to the model dynamic structure. Besides introducing the new class of models, we develop parameter estimation, hypothesis testing inference, and diagnostic analysis tools. We also discuss out-of-sample forecasting. In particular, we provide closed-form expressions for the conditional score vector and for the conditional Fisher information matrix. We also evaluate the finite sample performances of conditional maximum likelihood estimators and white noise tests using Monte Carlo simulations. An empirical application is presented and discussed.     

Extended Poisson INAR(1) processes with equidispersion,underdispersion and overdispersion

Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion, and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule–Walker, and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of the weekly number of syphilis cases and monthly counts of family violence illustrating its capabilities in challenging the overdispersed and underdispersed count data.     

Zero-modified power series distribution and its Hurdle distribution version

This paper presents a new family of distributions for count data, the so called zero-modified power series (ZMPS), which is an extension of the power series (PS) distribution family, whose support starts at zero. This extension consists in modifying the probability of observing zero of each PS distribution, enabling the new zero-modified distribution to appropriately accommodate data which have any amount of zero observations (for instance, zero-inflated or zero-deflated data). The Hurdle distribution version of the ZMPS distribution is presented. PS distributions included in the proposed ZMPS family are the Poisson, Generalized Poisson, Geometric, Binomial, Negative Binomial and Generalized Negative Binomial distributions. The paper also describes the properties and particularities of the new distribution family for count data. The distribution parameters are estimated via maximum likelihood method and the use of the new family is illustrated in three real data sets. We emphasize that the new distribution family can accommodate sets of count data without any previous knowledge on the characteristic of zero-inflation or zero-deflation present in the data.     

On Fan's adaptive Neyman tests for comparing two spectral densities

In this paper, we consider tests for assessing whether two stationary and independent time series have the same spectral densities (or same autocovariance functions). Both frequency domain and time domain test statistics for this purpose are reviewed. The adaptive Neyman tests are then introduced and their performances are investigated. Our tests are adaptive, that is, they are constructed completely by the data and do not involve any unknown smoothing parameters. Simulation studies show that our proposed tests are at least comparable to the current tests in most cases. Furthermore, our tests are much more powerful in some cases, such as against the long orders of autoregressive moving average (ARMA) models such as seasonal ARMA series.     

Time Series Models for Count or Qualitative Observations

Time series sometimes consist of count data in which the number of events occurring in a given time interval is recorded. Such data are necessarily nonnegative integers, and an assumption of a Poisson or negative binomial distribution is often appropriate. This article sets ups a model in which the level of the process generating the observations changes over time. A recursion analogous to the Kalman filter is used to construct the likelihood function and to make predictions of future observations. Qualitative variables, based on a binomial or multinomial distribution, may be handled in a similar way. The model for count data may be extended to include explanatory variables. This enables nonstochastic slope and seasonal components to be included in the model, as well as permitting intervention analysis. The techniques are illustrated with a number of applications, and an attempt is made to develop a model-selection strategy along the lines of that used for Gaussian structural time series models. The applications include an analysis of the results of international football matches played between England and Scotland and an assessment of the effect of the British seat-belt law on the drivers of light-goods vehicles.     

Reliability estimation in Maxwell distribution with progressively Type-II censored data

There may be situations in which either the reliability data do not fit to popular lifetime models or the estimation of the parameters is not easy, while there may be other distributions which are not popular but either they provide better goodness-of-fit or have a smaller number of parameters to be estimated, or they have both the advantages. This paper proposes the Maxwell distribution as a lifetime model and supports its usefulness in the reliability theory through real data examples. Important distributional properties and reliability characteristics of this model are elucidated. Estimation procedures for the parameter, mean life, reliability and failure-rate functions are developed. In view of cost constraints and convenience of intermediate removals, the progressively Type-II censored sample information is used in the estimation. The efficiencies of the estimates are studied through simulation. Apart from researchers and practitioners in the reliability theory, the study is also useful for scientists in physics and chemistry, where the Maxwell distribution is widely used.     

Compounded inverse Weibull distributions: Properties,inference and applications

ABSTRACT

In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be used to model lifetime of series system where the lifetimes follow inverse Weibull distribution and the subgroup size being random follows either geometric or zero-truncated Poisson distribution. Some of the important statistical and reliability properties of each of the distributions are derived. The distributions are found to exhibit both monotone and non-monotone failure rates. The parameters of the distributions are estimated using the expectation-maximization algorithm and the method of minimum distance estimation. The potentials of the distributions are explored through three real life data sets and are compared with similar compounded distributions, viz. Weibull-geometric, Weibull-Poisson, exponential-geometric and exponential-Poisson distributions.     

The effects of additive outliers on the seasonal KPSS test: a Monte Carlo analysis

This article builds on the test proposed by Lyhagen [The seasonal KPSS statistic, Econom. Bull. 3 (2006), pp. 1–9] for seasonal time series and having the null hypothesis of level stationarity against the alternative of unit root behaviour at some or all of the zero and seasonal frequencies. This new test is qualified as seasonal-frequency Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test and it is not originally supported by a regression framework.

The purpose of this paper is twofold. Firstly, we propose a model-based regression method and provide a clear illustration of Lyhagen's test and we establish its asymptotic theory in the time domain. Secondly, we use the Monte Carlo method to study the finite-sample performance of the seasonal KPSS test in the presence of additive outliers. Our simulation analysis shows that this test is robust to the magnitude and the number of outliers and the statistical results obtained cast an overall good performance of the test finite-sample properties.     

THE BEHAVIOR OF HEGY TESTS FOR QUARTERLY TIME SERIES WITH SEASONAL MEAN SHIFTS

ABSTRACT

This paper studies the behavior of the HEGY statistics for quarterly data, for seasonal autoregressive unit roots, when the analyzed time series is deterministic seasonal stationary but exhibits a change in the seasonal pattern. We analyze also the HEGY test for the nonseasonal unit root. the data generation process being trend stationary too. Our results show that when the break magnitudes are finite, the HEGY test statistics are not asymptotically biased toward the nonrejection of the seasonal and nonseasonal unit root hypotheses. However, the finite sample power properties may be substantially affected, the behavior of the tests depending on the type of the break.     

Panel Seasonal Unit Root Test: Further Simulation Results and An Application to Unemployment Data

Summary: In this paper the seasonal unit root test of Hylleberg et al. (1990) is generalized to cover a heterogenous panel. The procedure follows the work of Im, Pesaran and Shin (2002) and is independently proposed by Otero et al. (2004). Test statistics are given and critical values are obtained by simulation. Moreover, the properties of the tests are analyzed for different deterministic and dynamic specifications. Evidence is presented that for a small time series dimension the power is low even for increasing cross section dimension. Therefore, it seems necessary to have a higher time series dimension than cross section dimension. The test is applied to unemployment data in industrialized countries. In some cases seasonal unit roots are detected. However, the null hypotheses of panel seasonal unit roots are rejected. The null hypothesis of a unit root at the zero frequency is not rejected, thereby supporting the presence of hysteresis effects. * The research of this paper was supported by the Deutsche Forschungsgemeinschaft. The paper was presented at the workshop “Unit roots and cointegration in panel data” in Frankfurt, October 2004 and in the poster-session at the EC2 meeting in Marseille, December 2004. We are grateful to the participants of the workshops and an anonymous referee for their helpful comments.     

Using Analysis of Variance and Factor Analysis for the Reduction of High Dimensional Variation in Time Series of Energy Consumption

Summary: In this paper the complexity of high dimensional data with cyclical variation is reduced using analysis of variance and factor analysis. It is shown that the prediction of a small number of main cyclical factors is more useful than forecasting all the time-points separately as it is usually done by seasonal time series models. To give an example for this approach we analyze the electricity demand per quarter of an hour of industrial customers in Germany. The necessity of such predictions results from the liberalization of the German electricity market in 1998 due to legal requirements of the EC in 1996.