Parametric estimators for stationary time series with missing observations

Three related estimators are considered for the parametrized spectral density of a discrete-time process X(n), n = 1, 2, · · ·, when observations are not available for all the values n = 1(1)N. Each of the estimators is obtained by maximizing a frequency domain approximation to a Gaussian likelihood, although they do not appear to be the most efficient estimators available because they do not fully utilize the information in the process a(n) which determines whether X(n) is observed or missed. One estimator, called M3, assumes that the second-order properties of a(n) are known; another, M2, lets these be known only up to an unknown parameter vector; the third, M1, requires no model for a(n). Under representative sets of conditions, which allow for both deterministic and stochastic a(n), the strong consistency and asymptotic normality of M1, M2, and M3 are established. The conditions needed for consistency when X(n) is an autoregressive moving-average process are discussed in more detail. It is also shown that in general M1 and M3 are equally efficient asymptotically and M2 is never more efficient, and may be less efficient, than M1 and M3.

Vector linear time series models: corrections and extensions

In this paper some theorems of Dunsmuir and Hannan [2] are corrected. The topological structure of spaces of (likelihood) equivalence classes of arma structures is more closely investigated and its relation to the statistical problem is elucidated. The formulation is also made more general.

Vector linear time series models

This paper presents proofs of the strong law of large numbers and the central limit theorem for estimators of the parameters in quite general finite-parameter linear models for vector time series. The estimators are derived from a Gaussian likelihood (although Gaussianity is not assumed) and certain spectral approximations to this. An important example of finite-parameter models for multiple time series is the class of autoregressive moving-average (ARMA) models and a general treatment is given for this case. This includes a discussion of the problems associated with identification in such models.

Can we predict recurrence of pre-eclampsia or gestational hypertension?

OBJECTIVE

To estimate the rates of recurrence of pre-eclampsia or gestational hypertension in a subsequent pregnancy and to determine factors predictive of recurrence.

DESIGN

Retrospective cohort study.

SETTING

St George Public and Private Hospitals, teaching hospitals without neonatal intensive care units.

PARTICIPANTS

A total of 1515 women with a diagnosis of pre-eclampsia or gestational hypertension between 1988 and 1998 were identified from the St George Hypertension in Pregnancy database, a system designed initially for ensuring quality outcomes of hypertensive pregnancies. Of these, 1354 women were followed up, and a further 333 records from women coded as having a normal pregnancy during that period were selected randomly as controls.

MAIN OUTCOME MEASURES

Likelihood of recurrent pre-eclampsia or gestational hypertension and clinical and routine laboratory factors in the index pregnancy predictive of recurrence of pre-eclampsia or gestational hypertension.

METHODS

The index cases from our unit's database were linked to the matched pregnancy on the State Department of Health database, allowing us to determine whether further pregnancies had occurred at any hospital in the State. The outcome of these pregnancies was determined by review of medical records, using strict criteria for diagnosis of pre-eclampsia or gestational hypertension.

RESULTS

Almost all women with a normal index pregnancy had a further normotensive pregnancy. One in 50 women hypertensive in their index pregnancy had developed essential hypertension by the time of their next pregnancy. Women with pre-eclampsia in their index pregnancy were equally likely to develop either pre-eclampsia or gestational hypertension (approximately 14% each), while women with gestational hypertension were more likely to develop gestational hypertension (26%) rather than pre-eclampsia (6%) in their next pregnancy. Multiparous women with gestational hypertension were more likely than primiparous women to develop pre-eclampsia (11 versus 4%) or gestational hypertension (45 versus 22%) in their next pregnancy. Early gestation at diagnosis in the index pregnancy, multiparity, uric acid levels in the index pregnancy and booking blood pressure parameters in the next pregnancy significantly influenced the likelihood of recurrence, predominantly for gestational hypertension and less so for pre-eclampsia. No value for these parameters was significant enough to be clinically useful as a discriminate value predictive of recurrent pre-eclampsia or gestational hypertension.

CONCLUSIONS

Approximately 70% of women with pre-eclampsia or gestational hypertension will have a normotensive next pregnancy. The highest risk group for recurrent hypertension in pregnancy in this study was multiparous women with gestational hypertension. No readily available clinical or laboratory factor in the index pregnancy reliably predicts recurrence of pre-eclampsia.