Piet de Jong

Smoothing and Interpolation with the State-Space Model

Abstract A new result dealing with smoothing and interpolation in the state-space model is developed and explored. The result simplifies the derivation of existing smoothing algorithms and provides alternate forms that have analytic and practical computing advantages. The connection to signal extraction and interpolation is explored, and diffuse specifications are considered.

A statistical approach to Saaty's scaling method for priorities

Abstract This paper investigates the logarithmic least squares (LLSM) approach to Saatys ( Journal of Mathematical Psychology , 1977, 5 , 234–281) scaling method for priorities in hierarchical structures. It is argued that statistical criteria are important in deciding the scaling method controversy. It is shown that LLSM is statistically optimal under a number of realistic and practical models. Variances and covariances of parameter estimates are derived. The covariance matrix associated with overall priority differences is also developed. These results allow for a significance analysis of apparent priority differences.

Diagnosing Shocks in Time Series

Abstract Efficient means of modeling aberrant behavior in times series are developed. Our methods are based on state-space forms and allow test statistics for various interventions to be computed from a single run of the Kalman filter smoother. The approach encompasses existing detection methodologies. Departures commonly observed in practice, such as outlying values, level shifts, and switches, are readily dealt with. New diagnostic statistics are proposed. Implications for structural models, autoregressive integrated moving average models, and models with explanatory variables are given.

Claims reserving, state-space models and the Kalman filter

1.1. This paper describes a consistent and justifiable means of establishing adequate claims provisions in General Insurance. The topic has created widespread interest amongst actuaries, accountants and regulatory authorities. The issue of adequate provisions is of utmost importance to policyholders, whose justifiable claims must be paid, insurance companies who must be able to satisfy shareholders and make proper assessments of premiums, and regulatory authorities who must be satisfied that adequate provision has been made for all liabilities.

The Health Impact of Mandatory Bicycle Helmet Laws

This article seeks to answer the question whether mandatory bicycle helmet laws deliver a net societal health benefit. The question is addressed using a simple model. The model recognizes a single health benefit--reduced head injuries--and a single health cost-increased morbidity due to foregone exercise from reduced cycling. Using estimates suggested in the literature on the effectiveness of helmets, the health benefits of cycling, head injury rates, and reductions in cycling leads to the following conclusions. In jurisdictions where cycling is safe, a helmet law is likely to have a large unintended negative health impact. In jurisdictions where cycling is relatively unsafe, helmets will do little to make it safer and a helmet law, under relatively extreme assumptions, may make a small positive contribution to net societal health. The model serves to focus the mandatory bicycle helmet law debate on overall health.

Forecasting Runoff Triangles

Abstract This paper deals with the methodology of liability forecasting using the runoff triangle data. Techniques are based on time series models and methods that facilitate the calculation of forecast distributions and the assessment of model fit. The models deal with correlation within triangles. Correlations are critical to proper reserving. The output of the methodology is the complete shape of the liability distribution. Methods are applied to a well-known runoff triangle and results compared to those from previous studies.Abstract This paper deals with the methodology of liability forecasting using the runoff triangle data. Techniques are based on time series models and methods that facilitate the calculation of forecast distributions and the assessment of model fit. The models deal with correlation within triangles. Correlations are critical to proper reserving. The output of the methodology is the complete shape of the liability distribution. Methods are applied to a well-known runoff triangle and results compared to those from previous studies.

Smoothing with an unknown initial condition

The smoothing filter is appropriately modified for state space models with an unknown initial condition. Modifications are confined to an initial stretch of the data. An application illustrates procedures. Copyright 2003 Blackwell Publishing Ltd.

Credibility theory and the Kalman filter

Abstract Following Mehra (1975) we indicate how some of the well known credibility models may be formulated as Kalman filters. The formulation yields recursive premium forecasts including recursive predictions errors which are of importance to practitioners.

Rational economic data revisions

The efficient handling of provisional economic time series data is considered. Procedures are justified in terms of a testable hypothesis regarding the nature of data revisions. The hypothesis leads to explicit and practical variance expressions for measurement errors. Testing and estimation issues are dealt with. An efficient dual filter is developed for recursive signal estimation. Techniques are applied to the Canadian index of production data.

Modeling Dependence Between Loss Triangles

A critical problem in property and casualty insurance is forecasting incurred but as yet unpaid losses. Forecasts and risk margins are often based on individual loss triangles with each triangle corresponding to a different line of business. However lines of business are often related and an overall risk margins must reflect dependence between triangles. This article develops, implements and applies a model for loss triangle dependence. The model relates payments in different triangles in the same calendar year. Dependence is modeled with a Gaussian copula correlation matrix. Correlations are moderated by quantities called communalities which measure the relative impact of cross dependence in each triangle. Correlations can be structured in terms of factor models. Methods reduce to relatively simple calculations in the case of marginal normal distribution. Procedures are applied to US loss triangle data and the impact of loss triangle dependence on risk margins is considered.