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SUMMARYIn this paper we suggest a distribution-free state space model to be used with the Kalman filter in run-off triangles. It works with original incremental amounts and relates the triangle with a column of observed values, which can be chosen in order to describe better the risk volume in each year. On the traditional application of run-off triangles (the paid claims run-off), this model relates the amount paid j years after the accident year with a column of observed values, that can be the claims paid on the first year, the number of claims, premiums, number of risks, etc. Two advantages of this model are the perfect split between observed values and random variables and the capacity to incorporate the changes in the speed of the company's reality into the model and in its projections.Particular care is taken on the evaluation of the final forecast mean square error as well as on the estimation of the model parameters, specially the error variances. Also, two sets of claims data are analysed. In comparison with other methods, namely, the chain ladder, the analysis of variance, the Hoerl curves and the state space modelling with the chain ladder linear model, the proposed model gave a final reserve with a mean square error within the smallest.

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In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.linear regression, autoregressive stationary process, maximum likelihood, least squares, trend, seasonality, linear difference equation,

SUMMARYThis paper examines monthly average temperature series in two widely separated European cities, Lisbon (1856Lisbon ( -1999 and Prague (1841Prague ( -2000. The statistical methodology used begins by fitting a straight line to the temperature measurements in each month of the year. Hence, the 12 intercepts describe the seasonal variation of temperature and the 12 slopes correspond to the rise in temperature in each month of the year. Both cities show large variations in the monthly slopes. In view of this, an overall model is constructed to integrate the data of each city. Sine/cosine waves were included as independent variables to describe the seasonal pattern of temperature, and sine/cosine waves multiplied by time were used to describe the increase in temperature corresponding to the different months. The model also takes into account the autoregressive, AR(1), structure that was found in the residuals. A test of the significance of the variables that describe the variation of the increase in temperature shows that both Lisbon and Prague had an increase in temperature that is different according to the month. The winter months show a higher increase than the summer months.