Abstract
In a time series, an observation is called a record at time «t» if its value is greater than all previous values. As «t» increases, consider the sequence of records and the sequence of indices of occurrence of the records.The stochastic properties of sequences of record values have been much studied in the case where the observations are independent and identically distributed (iid) random variables. It turns out that many of these properties are universal, i.e. they hold for any cumulative distribution function for the underlying observations. In particular, records have a tendency to become further separated in time as «t» increases. However, this is not what is observed in many real data sets. This has lead to the development of more comprehensive models to provide better prediction.One of the simplest and popular model for a series of records extracted from independent but not identically distributed observations is the linear drift model (LDM). This model has been studied by many authors and found to be in agreement with some data sets where the iid assumption does not hold. However, for its uses in practical situations, the LDM requires the specification of the drift parameter of the model and this brings the problem into the realm of statistics.There are similarities between records and censored data in e.g. survival analysis. In particular, all observations that fall between two consecutive records and beyond the last record, can be seen as censored, by the last observed record. To highlight these similarities, consider the sequence of record indicators which are 1 if the observation is a record and 0 otherwise.Another popular model is the Yang-Nevzorov model. This model is interesting because it has the structure of a proportional hazard model, which have been shown to provide good fit to many data sets in survival analysis. However, to the best of our knowledge, statistical inference for the Yang- Nevzorov model has been little developed.The goal of this work is to introduce some estimators of the parameters in LDM and Yang’s model respectively and derive their statistical properties. It is shown that the censoring mechanism is informative for certain parameters. This justifies investigating the usefulness of estimators that can be extracted from record indicators. We give some exact and asymptotic properties of these estimators. It turns out that in a Yang’s model, the behavior of these estimators is distribution-free, i.e. does not involve the underlying CDF. Note that our estimators can be used even when the exact value of the records are themselves unavailable or of poor quality and only the indicators of their occurrence are available or trustworthy. Also, it is shown that distribution-free goodness-of-fit tests for Yang’s model can be derived from these indicators. These tests even have some diagnostic capabilities that can help in suggesting corrections to the model.Still in the context of a Yang’s model, we study the stochastic behavior of the inter-record time and give its asymptotic distribution regardless of the choice of the underlying distribution. In addition, we apply our theoretical results to a previously analyzed data set.Finally, we turn to the use of all available data (record values and indices/indicators) in order to calculate, by several methods, estimators of parameters in LDM and Yang-Nevzorov’s model. In addition, we introduce statistical tests that help us to check the conformity of the choice of the underlying distribution and to choose between LDM and Yang.