Stochastic Differential
Equation Mortality Estimation Confidence Interval and
Prediction In the present paper we consider an application of stochastic differential equation to model age-specific mortalities. We use New Zealand mortality data for the period 1948–2015 to fit the model. The point predictions of mortality rates at ages 40, 60 and 80 are quite good, almost undistinguishable from the true mortality rates observed.
## 1. INTRODUCTIONHuman Lagarto and Braumann (2014) used SDE model (stochastic Malthusian) for mortality data of Portugal and considered precisely the joint evolution of the crude death rates of two age-sex groups. Lagarto (2014) has considered a study of alternative structures and has determined, among those, which ones have a good performance. These can then be used in predictions and in applications. ## 2.
STOCHASTIC MORTALITY MODEL
The random fluctuations are mostly explained by environmental stochasticity Consider a Stochastic differential equation (SDE) (1) Where is one-dimensional ‘white noise’ By (1951) calculus, the satisfies the following Stochastic integral equation
Or in differential form (2) It is the formula that is the key to the solution of many stochastic differential equations. Various models describe the pattern of human mortality, the one such model that we consider is a stochastic Gompertz model also called Malthusian growth model.
(3) Where is the arithmetic average growth rate, which
is assumed constant over time and is
the white noise. The parameter The solution of equation (3) is obtained by calculus (see also Øksendal 2003) (4) (5) Where is called the geometric average growth rate. We conclude that 1) If then as a.s 2) If then as a.s 3) If then will fluctuate between arbitrary large and small values as a.s To fit model (4), we made it age-specific by writing (6)
Where and hence, , which called as geometric average growth rate. We apply the methodology used in (Lagarto
and Braumann, 2014; Braumann,
2019; Talawar and Agadi,
2020) for parameters estimation. The value of of the rate of decay of would be negative since the tendency is for to decrease exponentially. The parameter measures
the intensity of the effect of environmental fluctuations (weather conditions,
epidemic diseases, social conditions, etc.) on the rate of change. In this case
of mortality rates are time equidistant observations with year
and that
is, the mortality rates are age-specific with single year. Denoting by and by,
the log-returns, one gets for model (6) the maximum likelihood /year and /year Therefore the 95% confidence intervals of the parameters and are and ## 3. APPLICATIONS OF THE MODEL
Figures 2-4, show
the results of using model (6) for the age-specific mortality rates of 60 year-old New Zealand males and females. Notice the
decline fromin
1948 to in 2008 and in 2023. For parameter estimation, we only use
the data of the period 1948–2008, reserving the period 2009–2023 for prediction
purposes. Therefore the /year /year The 95% confidence intervals of the parameters are and The parameter value obtained is used in (6) to get the
plots and putting and an approximate 95% confidence prediction interval From which extremes, by taking exponentials, one gets an
approximate confidence prediction interval for
## 4. CONCLUSIONSThis stochastic mortality model gives good prediction intervals and the point prediction for each age. The point predictions of mortality rates at ages 40, 60 and 80 are quite good, almost undistinguishable from the true mortality rates observed. The similar procedure can be used to estimate parameters at each age . Once all the age-specific mortalities are obtained, the different columns of a life table can be constructed using their interrelationships. ## SOURCES OF FUNDINGNone. ## CONFLICT OF INTERESTNone. ## ACKNOWLEDGMENTNone. ## REFERENCES[1] Aït-Sahalia, Y. (2002). Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes: Comment. Journal of Business and Economic Statistics. 20(3):317-21 DOI: 10.1198/073500102288618405 [2] Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 (2008), no. 2, 906--937. doi:10.1214/009053607000000622. https://projecteuclid.org/euclid.aos/ [3] 1205420523 [4] Braumann, C.A. (1993) Model fitting and prediction in stochastic population growth models in random environments. Bulletin of the International Statistical Institute, LV (CP1), 163–164. Braumann, C.A. (1999a) Comparison of geometric Brownian motions and applications to population growth and finance. Bulletin of the International Statistical Institute, LVIII (CP1), 125–126. [5] Braumann, C. A., Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance, John Wiley & Sons Ltd, 2019. [6] Itô, K. (1951) On Stochastic Differential Equations, American Mathematical Society Memoirs, No. 4. [7] Kessler,M., Lindner A. and, Sorensen, M. Statistical Methods for Stochastic Differential Equations, Chapman and Hall/CRC, 2012. [8] Lagarto, S. and Braumann, C.A. Modeling human population death rates: A bi-dimensional stochastic Gompertz model with correlated Wiener processes, in New Advances in Statistical Modeling and Applications (eds, A. Pacheo, R. Santos, M.R. Oliveira, and C.D. Paulino), Springer, Heidelberg, pp. 95–103, doi:10.1007/978-3-319-05323-3_9, 2014. [9] Øksendal, B. Stochastic Differential Equations. An Introduction with Applications, 6th edition, Springer, New York, 2003. [10] Talawar, A. S. and Agadi, R.P. (2020). Novel corona virus pandemic disease (covid-19): Some applications to south asian countries. International Journal of Research and Analytical Reviews, 7(2), 905-912.
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