Penalized least squares smoothing of two-dimensional mortality tables with imposed smoothness

Transcription

1 Penlized lest squres smoothing of two-dimensionl mortlit tbles with imposed smoothness Eliud Silv 1 Víctor M. Guerrero (speker) 1 Acturil School, Universidd Anáhuc, México Deprtment of Sttistics, Instituto Tecnológico Autónomo de México (ITAM)

2 Introduction Smoothed estimtes of mortlit rtes (or grduted rtes), re of prmount importnce for mking strtegic decisions nd plnning in popultion councils, insurnce compnies nd reserch centers. There re severl proposls for the nlsis nd estimtion of mortlit rtes, e. g. Whittker grdution, Clton nd Schifflers (1987) Age-Period-Cohort model, Brouhns et l. () pproch to work with life tbles, Broffitt s (1996) contribution to bidimensionl grdution nd tht of Currie et l. (4) to smooth bidimensionl tbles. None of those works hve considered the possibilit of controlling the smoothness chieved b the estimtes, to llow for vlid comprisons of mortlit trends.

3 Introduction PLS smoothing uses smoothing prmeter λ, tpicll selected b optimizing criterion such s AIC or BIC (with no control of the smoothness chieved). A specific vlue of λ implies specific mount of smoothness nd we should, t lest, quntif it. Our proposl goes one step further: t the outset of the stud we impose desired mount of smoothness to the smoothed mortlit rtes. B so doing, we cn enhnce comprbilit of smoothed estimtes. This ide is similr to tht of prmeter estimtion b confidence intervls: we usull fix the sme confidence level (s t 95%) to estblish vlid comprisons between intervls. 3

4 One-dimensionl smoothing PLS smoothing nd Whittker grdution seek to minimize the function i M m i1 m i i τi) λ( τi) i3 w ( with mortlit indictor ( rte or crude force), its smoothed or trend vlue nd w i weight, for ech ge i = 1,, m. The constnt λ trdes off fit ginst smoothness. Let ( 1,...,m)' nd τ ( τ1,...,τm)' be the vectors of observtions nd smoothed vlues, W the corresponding (digonl) mtrix of weights nd K the (m-) x m mtrix representtion of the second order difference opertor. τ i B minimizing M ( τ)' W( τ) λτ' K' Kτ with respect to, with λ fixed, we obtin (e. g. Hstie nd Tibshirni, 1999) 1 τˆ (W λk'k) W τ 4

5 One-dimensionl smoothing From nother perspective, let us consider the unobserved component model τ η where η ( η1,...,ηm)' is rndom error vector with E( η) nd -1 Vr( η) σ η W. Then, to induce smoothness, we let Kτ ε with E( ε) nd Vr( ε) σ ε Im-. We further ssume tht E( εη' ). Then, b GLS we obtin 1 τˆ [W (σ /σ )K' K] W i. e., the sme result s before, with λ σ η /σε nd Γ Vr( τˆ) (σ η W σε K' K) η ε 5

6 Two-dimensionl smoothing We now consider the dimensions of ge (i=1,,m) nd ers (j=1,,n). Let the following vectors be: number of deths d d11,...,dm1,...,d1n,...,dmn ', forces of mortlit μ μ 11,...,μ m1,...,μ 1n,...,μ mn ' nd exposed to the risk of ding e e,...,e,...,e,...,e '. 11 m1 1n mn We ssume d is reliztion of Poisson distribution with men e ij μ. ij ij We look for smoothed estimtes of the vector Y, whose elements re clled crude forces of mortlit (in logs), Y log ij dij/eij 6

7 Proposed method: Two-dimensionl smoothing Unobserved component model where Y Τ Ν with E( Ν) Complementr informtion (to induce smoothness) We obtin the liner model representtion Y I mn I n K K I nd Ν Ν with E θ, Vr - θ ψ - ψ Vr( Ν) dig(w 1,..., W n ), with Wi dig(wi1,..., w im ) for i 1,,n. (In K ) Τ θ, E( θ) nd Vr( θ) σθi(m)n (K Im) Τ ψ, E( ψ) nd Vr( ψ) σψim(n) m Ν Τ θ ψ with E( θν' ) nd E( ψν' ) σ η -1 σ η σ θ I -1 (m )n σ ψ I m(n ) 7

8 Proposed method: Two-dimensionl smoothing GLS produces the Best Liner Unbised Estimtor Τˆ 1 ( λ In K' K λ K' K Im) Y with λ nd λ ση/σψ. ση/σθ Its MSE mtrix is Vr( Τˆ ) (σ η σθ In K' K σψ K' K Im) 1 The inverse mtrix is the sum of three precision mtrices σ - η, σ - θ I - n K' K nd σψ K' K I, linked to the unobserved component model nd to the smoothness m in ges nd ers, respectivel. As in Guerrero (8), we mke use of n index to mesure precision shres in order to impose some desired smoothness (precision shre) to the smoothed estimtes. 8

9 Proposed method: Two-dimensionl smoothing Proposition 1. Let P, Q nd Q be three mn mn smmetric, positive definite or semidefinite mtrices. A sclr index tht mesures the proportion of P in P Q 1 Q is given b Λ(P;P Q This mesure: (i) stisfies n dd-up criterion, tht is, Q ) tr[p(p Q Q ) 1 ]/mn Λ P;P Q Q ΛQ Q ;P Q Q 1 (ii) tkes on vlues between zero nd one; (iii) is invrint under liner nonsingulr trnsformtions of the vrible involved; nd (iv) behves linerl. 9

10 Proposed method: Two-dimensionl smoothing The two-dimensionl index of smoothness we propose is given b S (λ,λ ;m, n) 1 tr[(i mn λ I n W -1 K' K λ K' K W -1 ) 1 ]/mn It is reprmeteriztion of the degrees of freedom (df) of the model df tr[(i mn λ I n W -1 K' W Our proposl is in line with Hstie nd Tibshirni s comment (1999): In fct it is resonble to select the vlue of smoothing prmeter simpl b specifing the df of the smooth. K λ K' K -1 ) 1 ] The min dvntge of our reprmeteriztion for the df is its sensible interprettion s proportion of smoothness. Here we ssume the - - weights do not chnge with time, so tht σ σ I W. η η n 1

11 Proposed method: Two-dimensionl smoothing Since we work in two-dimensionl setting we propose using the following set of indices, in order to impose the desired smoothness for ech dimension (ge nd er). Smoothness Index Attributble to er, for given ge i S i (,λ ;1, n) 1 tr[(i n λ w -1 i K' K ) 1 ]/n Attributble to ge, -1 1 S for given er j j(λ,;m,1) 1 tr[(im λw K' K) As n id to fix the desired smoothness we should be wre of the smoothness limits tht cn be ttined Mrginl, prctice, ttributble s indicted to er b the following result. m -1 S (,λ ;m, n) 1 tr[(i λ w K' K ]/m 1 i 1 n i ) ]/mn Mrginl, ttributble to ge S (λ,;m, n) 1 tr[(i m λ -1 W K' K ) 1 ]/m 11

12 Proposed method: Two-dimensionl smoothing Proposition. Let us consider the two-dimensionl index of smoothness 1 1 nd let γ nd γ be the eigenvlues of I W K' K nd K' K W,,i, j respectivel. Then, if we interpret λ s n λ, we hve i) S (,;m, n) ; ii) S iii) S iv) S (,λ ;m, n) 1 m j1 (, ;m, n) 1 /m; (λ,;m, n) 1 n i1 1 1 λ γ 1 1 λ γ, j, i v) S /m; / n; vii) S (λ, ;m, n) 1 mn vi) S (,;m, n) 1 viii) S (,λ ;m, n) 1 mn (, ;m, n) 1 n i1 /n; m j1 4/mn 1 1 λ γ 1 1 λ γ, i, j ; ; 1

13 Log(mortlit) Log(mortlit) Illustrtive pplictions Mortlit dt for ges 11-1 nd ers from the Continuous Mortlit Investigtion Bureu (CMIB) of the United Kingdom (previousl nlzed b Currie et l., 4). One-dimensionl smoothing: (i) single er nd different ges, nd (ii) different ers nd single ge Age Yer Observed nd fitted log-mortlit with 75% smoothness nd 3 stndrd devition intervls. Left pnel: er 1955 ( λ = 85) nd right pnel: ge 65 ( λ = ). 13

14 Illustrtive pplictions Two-dimensionl smoothing Currie et l. (4) smoothed dt within penlized B-splines context with smoothing prmeters λ =.6 nd λ = 15. Since we do not use B-splines we cnnot compre our results with theirs. In cse we use those prmeters the smoothness to be ttined would be 61.8% The ctul smoothness chieved b B-spline smoothing is higher becuse the use of splines induces some dditionl smoothness. In the following figure we pprecite the mortlit surfces corresponding to 9% smoothness obtined with different vlues of λ nd λ. 14

15 Log(mortlit) Log(mortlit) Log(mortlit) Log(mortlit) Illustrtive pplictions Yer Age Yer Age Yer Age Yer Age Observed log-mortlit dt (top left) nd smoothed b both ges nd ers (top right, λ = λ = 59); b ges onl (bottom left, λ = 9985 nd λ =.1) nd b ers onl (bottom right, =.1 nd = 47989). λ λ 15

16 Conclusions This proposl serves to estimte mortlit trends with desired percentge of smoothness fixed t the outset of the stud. Since it is not strictl vlid to compre smoothed estimtes with different degrees of smoothness, fixing the percentge of smoothness enhnces the comprbilit of trends in mortlit rtes. Our min contribution lies in defining the index of smoothness, which cn be clculted from the vilble dt, nd showing tht it hs some desirble properties. We cn estimte mortlit trends in two dimensions using well-known PLS smoothing methods. 16

17 References Broffitt, J. D. (1996) On smoothness terms in multidimensionl Whittker grdution. Insurnce: Mthemtics nd Economics 18, Brouhns, N., Denuit, M. nd Vermunt, J. K.,. A Poisson log-biliner regression pproch to the construction of projected life tbles. Insurnce: Mthemtics nd Economics 31, Clton, D. nd Schifflers, E., Models for temporl vrition in cncer rtes. II: Age-period-cohort models. Sttistics in Medicine 6, Currie, I., Durbn, M. nd Eilers, P. (4) Smoothing nd Forecsting mortlit rtes. Sttisticl Modelling, 4, Eilers, P. nd Mrx, B. (1996) Flexible smoothing with B-splines nd penlties. Sttisticl Science, 11, Guerrero, V. M. (8) Estimting trends with percentge of smoothness chosen b the user. Interntionl Sttisticl Review, 76, Hstie, T. nd Tibshirni, R. (199) Generlized dditive models. London, Chpmn & Hll. 17

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