Whittaker-Henderson Graduation Research Articles

Boosted Whittaker–Henderson Graduation

The Whittaker–Henderson (WH) graduation is a smoothing method for equally spaced one-dimensional data such as time series. It includes the Bohlmann filter, the Hodrick–Prescott (HP) filter, and the Whittaker graduation as special cases. Among them, the HP filter is the most prominent trend-cycle decomposition method for macroeconomic time series such as real gross domestic product. Recently, a modification of the HP filter, the boosted HP (bHP) filter, has been developed, and several studies have been conducted. The basic idea of the modification is to achieve more desirable smoothing by extracting long-term fluctuations remaining in the smoothing residuals. Inspired by the modification, this paper develops the boosted version of the WH graduation, which includes the bHP filter as a special case. Then, we establish its properties that are fundamental for applied work. To investigate the properties, we use a spectral decomposition of the penalty matrix of the WH graduation

تسوية معدلات الوفاة الخام باستخدام طريقة (whittaker - Henderson )

Abstract: This research aims to use the (Whittaker - Henderson) method, which is an (Nonparametric Method) for graduate, the crude death rates to obtain the smoothness and fitness death rates, because it is one of the most common methods for smoothing the crude death rates, for crude mortality rates. It has the capacity to explicitly balance the smoothness as well as the fitness of the smoothed curve relative to the crude rates through its parameters by minimizing a difference equation. Non-parametric methods depend only on the nature of the observed data. The crude central death rates for the single ages were smoothed and fitted using Whittaker-Henderson graduation technique by four assumptions through the program (R) and the third hypothesis was the best assumption of graduated for both males and females. The study recommended the important of graduation for the crude death rates and their smoothing to rid them of the irregularities resulting from the randomness in the 2 exposure and the number of deaths in order to facilitate dealing with these rates to complete the process of establishing death tables.

Variational Approach to Data Graduation

The Whittaker-Henderson graduation is a data smoothing method commonly used in actuarial science for mortality table computation. This method is a finite-dimensional minimization problem, which aims to give a smooth approximation of the crude data. Our goal is to treat the Whittaker-Henderson method as a minimization problem in a Sobolev space. We take advantage of some results in the study of Sobolev spaces to analyze the solution to the graduation problem. Solving the arising minimization problem is equivalent to solving a partial differential equation (PDE). To solve the PDE numerically, we use the finite element method (FEM). We apply this variational approach to the data sets found in the literature. In particular, we test our method to check for patterns in (1) mortality rates for Filipino males, (2) global average temperature anomaly, and (3) monthly eBay share price. We also compare the results with that of the Whittaker-Henderson method.

A note on Whittaker–Henderson graduation: Bisymmetry of the smoother matrix

Whittaker–Henderson (WH) graduation is a popular smoothing method that has been used for mortality table construction in the actuarial sciences and for the trend-cycle decomposition in time series econometrics. This paper proves that the smoother matrix of WH graduation is bisymmetric (i.e., symmetric centrosymmetric). This result implies, for example, that the first row of the smoother matrix is equivalent to the last row of it in reverse order. We also provide some related results.

Explicit formulas for the smoother weights of the Whittaker–Henderson graduation of order 1

The Whittaker–Henderson (WH) graduation is a widely applied smoothing method. This paper contributes to the literature by providing explicit formulas for the smoother weights of the WH graduation of order 1 along with some related results, which leads to a richer understanding of the filter.

A modification of the Whittaker–Henderson method of graduation

This paper presents a modified Whittaker–Henderson (WH) Method of Graduation. After giving a closed-form solution, we show that it is of practical use because it provides not only a smoothed series identical to that of the WH graduation, but also an extrapolation beyond the sample limit of current data. In addition, we introduce two other penalized least squares problems and show that they provide the same results as those of the modified WH graduation.

Non-Parametric Likelihood Enhancements to Parametric Graduations

ABSTRACTParametric graduation may fail to achieve satisfactory results without overparameterisation. Whittaker-Henderson graduation tends to constrain the graduated values towards a low-order polynomial. Non-parametric methods do not generally make direct use of true likelihood functions. This paper suggests a method of enhancing the likelihood of a parametric graduation by means of non-parametric methods, thus reducing the disadvantages of both methods.The parametric graduation is taken to be ideally smooth by definition and is adjusted by using constrained maximum likelihood estimation to obtain better fidelity to the experience. The constraint imposes a minimum sacrifice of smoothness, in terms of a quantitative smoothness criterion, from the initial ideal. The method is not entirely objective in that, in some cases, professional judgement is required in order to assess the degree of smoothness that can be imposed. In other cases the method provides an objective optimum. In either case, by quantifying the degrees of departure from perfect fidelity and from ideal smoothness, the suggested method provides useful and theoretically sound criteria for the purposes of the optimisation process. In particular, by inverting the parametric graduation formula for the purposes of defining the smoothness criterion, the method ensures that the smoothness criterion is consistent over the whole age range, thus resolving the main objection to non-parametric graduation.The method is applied to the 1979-82 experience for life office pensioners in the United Kingdom with positive results.

071016 (M21) Multidimensional Whittaker-Henderson graduation with constraints and mixed differences : Lowrie W.B.,Transactions of the Society of Actuaries, Vol. XLV, 1993, pp. 215–256

071016 (M21) Multidimensional Whittaker-Henderson graduation with constraints and mixed differences : Lowrie W.B.,Transactions of the Society of Actuaries, Vol. XLV, 1993, pp. 215–256

A Bayesian interpretation of Whittaker—Henderson graduation

The primary purpose of the paper is to place Whittaker—Henderson graduation in a Bayesian context and show that this determines in a precise manner the extent to which goodness-of-fit should be traded off against smoothness in the Whittaker—Henderson loss function. This is done in Section 2. Section 3 generalizes the set of admissible graduating functions to a normed linear space. A specific example of this generalization is Schoenberg graduation, which is strongly related to Whittaker—Henderson but leads to particular spline graduations. These are placed in a Bayesian context parallel to that of Section 2. The similarity to shrunken or Stein-type estimators is pointed out. Section 4 considers the practical implications of these theoretical developments. Transformations of observations under graduation are examined and shown to be natural in some circumstances. The precise trade off mentioned above is enlarged upon, and the main conclusions reached here are seen to carry over to general spline graduation. The relation between Whittaker—Henderson and spline graduation is identified.

A note on the computation in Whittaker-Henderson graduation

Abstract This note considers the computation of graduated values with the Whittaker- Henderson graduation method. It is shown that the fundamental matrices are banded and thus solution of the linear equations can be done very efficiently. Numerical timings are presented to support this contention.

Trend-seasonal decomposition of time series as whittaker-henderson graduation

The diagnostic and essentially modelfree seasonal adjustment (SA) procedures of BAYSEA by AKAIKE and of SCHLICHT-PAULY are generalized, in particular by admitting arbitrary and data-dependent weights. This admits iterative analyses such as nonlinear robustification, or minimization of the relative rather than of the absolute errors. One step of these methods rests upon the rather well-known and flexible WHITTAKER-HENDERSON graduation. The uniqueness of its minimizing solution is proved for a given set of decomposition parameters. Quick and easy but high-quality graphical representations are produced by the available packages even on PCs. They serve as important checks in the fast iterative trial and error fitting process of the trend-season decompositions produced.Qualitative as well as quantitative criteria of the fit can be checked. Artefacts, the typical defect of model misspecifications, are efficiently controlled. The tools guarantee the superiority of this type of exploratory approaches over tentati...

Cross-validatory graduation

Cross-validation is applied to the choice of the parameters that control the degree of smoothing in generalized Whittaker—Henderson graduation. The approach is compared with the Bayesian method, and is illustrated by graduation of a data set of Hickman and Miller and of data underlying the English Life Table number 13 (males).

Applications of linear and quadratic programmings to some cases of the Whittaker-Henderson graduation method

Abstract It is shown that the Whittaker-Henderson graduation method of minimizing Fp (u)+λSq (u) over u⩾0, where the fit and the smoothness Sq ≡ ∑∣Δ zux ∣q, , can be formulated as a linear programming problem when p, q=1 and ∞, and can be formulated as a quadratic programming problem when either p or q is 2. The monotone properties and Pareto properties are also obtained.

Properties and modifications of Whittaker-Henderson graduation

Absract First some properties of the Whittaker-Henderson graduation are obtained. Then modifications of the Whittaker-Henderson criterion are investigated in which u is determined (a) to minimize the fit function F d(u) subject to the restriction that the smoothness function S d(u) does not exceed a specified tolerance level and (b) to minimize S d(u) subject to a similar restriction on F d(u). Sufficient conditions for optimality are given and it is shown that the optimal solution can be obtained as the unique solution of a equation of a single variable.

The Whittaker-Henderson Graduation Process, a manual. By Kingsland Camp, F.S.A. [Pp. 47. Offset, published by the Author, 1950.

The Whittaker-Henderson Graduation Process, a manual. By Kingsland Camp, F.S.A. [Pp. 47. Offset, published by the Author, 1950.