Local or Global Smoothing? A Bandwidth Selector for Dependent Data - PowerPoint PPT Presentation

The plug-in optimal local bandwidth It is derived by minimizing the asymptotic mean square error h opt σ 2 ( x ; h ) } L ( x ) = arg min h AMSE { ˆ � 1 / ( 2 p + 3 ) � C p , K × v ( x ) = n × d 2 ( x ) × f X ( x ) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 26 / 115

The plug-in optimal local bandwidth It is derived by minimizing the asymptotic mean square error h opt σ 2 ( x ; h ) } L ( x ) = arg min h AMSE { ˆ � 1 / ( 2 p + 3 ) � C p , K × v ( x ) = n × d 2 ( x ) × f X ( x ) The only unknown components (to estimate and plug-in) are ◮ v ( x ) , the conditional variance of X 2 t ; ◮ d ( x ) , the ( p + 1 ) -th derivative of σ 2 ( x ) ; ◮ f X ( x ) , the design density. F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 26 / 115

An illustrative example: local (variable) bandwidth true regression function LP estimate Kernel weights phi(X) x x+h X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 27 / 115

An illustrative example: local (variable) bandwidth true regression function LP estimate Kernel weights phi(X) x-h x x+h X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 31 / 115

An illustrative example: local (variable) bandwidth true regression function LP estimate Kernel weights phi(X) x-h x X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 42 / 115

The plug-in optimal global bandwidth It is derived by minimizing the integrated AMSE � h opt σ 2 ( x ; h ) } f X ( x ) dx = arg min AMSE { ˆ G h � 1 / ( 2 p + 3 ) � C p , K × R var = n × R f , bias F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 47 / 115

The plug-in optimal global bandwidth It is derived by minimizing the integrated AMSE � h opt σ 2 ( x ; h ) } f X ( x ) dx = arg min AMSE { ˆ G h � 1 / ( 2 p + 3 ) � C p , K × R var = n × R f , bias The only unknown components (to estimate and plug-in) are: ◮ R var = � v ( x ) dx ; ◮ R f , bias = d 2 ( x ) f X ( x ) dx . � F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 47 / 115

An illustrative example: global (fixed) bandwidth true regression function LP estimate Kernel weights phi(X) x x+h X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 48 / 115

An illustrative example: global (fixed) bandwidth true regression function LP estimate Kernel weights phi(X) x-h x x+h X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 51 / 115

An illustrative example: global (fixed) bandwidth true regression function LP estimate Kernel weights phi(X) x-h x X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 65 / 115

Local vs global bandwidths The following relation holds: � � � � x ; h opt x ; h opt AMSE L ( x ) ≤ AMSE . G F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 68 / 115

Local vs global bandwidths The following relation holds: � � � � x ; h opt x ; h opt AMSE L ( x ) ≤ AMSE . G Anyway, the estimation of h opt L ( x ) (=function) is less efficient than the estimation of h opt (=mean value) G F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 68 / 115

Local vs global bandwidths The following relation holds: � � � � x ; h opt x ; h opt AMSE L ( x ) ≤ AMSE . G Anyway, the estimation of h opt L ( x ) (=function) is less efficient than the estimation of h opt (=mean value) G Some questions follow : ◮ what is the effective gain in using the local bandwidth? ◮ Is it always convenient to perform a local smoothing instead of a global smoothing? ◮ How to deal with pilot bandwidths? F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 68 / 115

Our proposal: a two stage procedure Suppose we want to estimate the function σ 2 ( x ) on a support I X ⊂ R . F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

Our proposal: a two stage procedure Suppose we want to estimate the function σ 2 ( x ) on a support I X ⊂ R . First stage: evaluating the “homogeneity” on the support 1 Given the estimates of h opt L ( x ) and h opt G , use some relative indicator in order to evaluate the potential gain in using the local bandwidth instead of the global bandwidth on I X . F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

Our proposal: a two stage procedure Suppose we want to estimate the function σ 2 ( x ) on a support I X ⊂ R . First stage: evaluating the “homogeneity” on the support 1 Given the estimates of h opt L ( x ) and h opt G , use some relative indicator in order to evaluate the potential gain in using the local bandwidth instead of the global bandwidth on I X . Second stage: deriving the locally global bandwidth 2 Derive the optimal global bandwidths h opt on “homogeneous” G subsets of I X . F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

Our proposal: a two stage procedure Suppose we want to estimate the function σ 2 ( x ) on a support I X ⊂ R . First stage: evaluating the “homogeneity” on the support 1 Given the estimates of h opt L ( x ) and h opt G , use some relative indicator in order to evaluate the potential gain in using the local bandwidth instead of the global bandwidth on I X . Second stage: deriving the locally global bandwidth 2 Derive the optimal global bandwidths h opt on “homogeneous” G subsets of I X . ◮ How to estimate h opt and h opt L ( x ) on I X ? G ◮ Which relative indicator to use? ◮ How to smooth such bandwidths on I X ? F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 69 / 115

First stage : deriving the relative indicator Consider the relative increment of the AMSE, ∀ x ∈ I X , σ 2 ( x ; h opt σ 2 ( x ; h opt ∆ AMSE ( x ) = AMSE { ˆ G ) } − AMSE { ˆ L ( x )) } . σ 2 ( x ; h opt AMSE { ˆ L ( x )) } F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 70 / 115

First stage : deriving the relative indicator Consider the relative increment of the AMSE, ∀ x ∈ I X , σ 2 ( x ; h opt σ 2 ( x ; h opt ∆ AMSE ( x ) = AMSE { ˆ G ) } − AMSE { ˆ L ( x )) } . σ 2 ( x ; h opt AMSE { ˆ L ( x )) } We managed to express the ∆ AMSE ( x ) of the estimator in a model free form: 2 p + 3 [ π h ( x )] − 2 ( p + 1 ) + 2 p + 2 1 ∆ AMSE ( π h ( x )) = 2 p + 3 [ π h ( x )] − 1 where π h ( x ) = h opt L ( x ) ≥ 0. h opt G F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 70 / 115

First stage : deriving the relative indicator 2 p + 3 [ π h ( x )] − 2 ( p + 1 ) + 2 p + 2 1 ∆ AMSE ( π h ( x )) = 2 p + 3 [ π h ( x )] − 1 2.0 p=0 p=1 p=2 1.5 Delta_AMSE Delta_AMSE 1.0 0.5 beta 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 a 1 b pi(x) pi(x) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 71 / 115

First stage : deriving the relative indicator 2 p + 3 [ π h ( x )] − 2 ( p + 1 ) + 2 p + 2 1 ∆ AMSE ( π h ( x )) = 2 p + 3 [ π h ( x )] − 1 2.0 p=0 p=1 p=2 1.5 Delta_AMSE Delta_AMSE 1.0 0.5 beta 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 a 1 b pi(x) pi(x) Remark 1 : as a function of π h ( x ) , the relative indicator ∆ AMSE ( x ) is completely model free. Remark 2 : ∆ AMSE < β iif π h ( x ) ∈ [ a β , b β ] , ∀ β > 0. F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 71 / 115

A global measure of ∆ AMSE on the interval I X A global measure of ∆ AMSE would indicate, eventually, the need to use on the interval I X a local bandwidth. For example, the mean value of ∆ AMSE on the interval I X could be considered... � ∆ AMSE ( x ) f X ( x ) dx . I X ... but we get some advantages if we consider the median value of ∆ AMSE on the interval I X , or some other quantile. F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 72 / 115

The median of ∆ AMSE on the interval I X Delta_AMSE * * * * beta o o o o o o o o o o o o o o o o o a 1 b pi(x) For a fixed threshold β , consider the two set of points S β = { x : x ∈ I X , π ( x ) ∈ [ a β , b β ] } ; S β = { x : x ∈ I X , π ( x ) / ∈ [ a β , b β ] Given the measure of the process µ X , the median of ∆ AMSE is the threshold value β ∗ for which µ X ( S β ∗ ) = µ X ( S β ∗ ) . F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 73 / 115

The algorithm of the first stage Consider an estimation of both the global bandwidth h G and the 1 local bandwidth function h L ( X t ) , such as ◮ the neural network method proposed in the second stage; ◮ some other local method proposed in the literature. Estimate π h ( X t ) , for all X t ∈ I X , then search for β ∗ such that 2 � � I { ˆ π h ( X t ) ∈ [ a β ∗ , b β ∗ ] } ≈ I { ˆ π h ( X t ) / ∈ [ a β ∗ , b β ∗ ] } . X t ∈ I X X t ∈ I X where I ( · ) is the indicator function. Split the subset I X if β ∗ > τ , where τ represents the max relative 3 increment tolerated for ∆ AMSE when using a global bandwidth. F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 74 / 115

Second stage : the locally global bandwidth Given the “homogeneous” subset I X ⊂ R , we want to estimate � 1 / ( 2 p + 3 ) � C p , K × R I X h opt var = G n × R I X f , bias where: ◮ R I X � var = I X v ( x ) d ω I X ; ◮ R I X � I X d 2 ( x ) f X ( x ) d ω I X . f , bias = dx How to derive the measure ω I X ? (we used d ω I X = µ ( I X ) ) 1 How to estimate v ( x ) and d 2 ( x ) on I X ? 2 F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 75 / 115

Estimating the functions v ( x ) and d 2 ( x ) Note that: Var { X 2 t | X t − 1 = x } ≡ m 4 ( x ) − m 2 v ( x ) = 2 ( x ) ∂ p + 1 σ 2 ( x ) ≡ m ( p + 1 ) d ( x ) = ( x ) ∂ x p + 1 2 F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 76 / 115

Estimating the functions v ( x ) and d 2 ( x ) Note that: Var { X 2 t | X t − 1 = x } ≡ m 4 ( x ) − m 2 v ( x ) = 2 ( x ) ∂ p + 1 σ 2 ( x ) ≡ m ( p + 1 ) d ( x ) = ( x ) ∂ x p + 1 2 As usual, we should have to estimate separately the three functions m ( p + 1 ) m 4 ( x ) , m 2 ( x ) , ( x ) 2 (Fan & Yao, 1998; Franke & Diagne, 2006) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 76 / 115

Estimating the functions v ( x ) and d 2 ( x ) Note that: Var { X 2 t | X t − 1 = x } ≡ m 4 ( x ) − m 2 v ( x ) = 2 ( x ) ∂ p + 1 σ 2 ( x ) ≡ m ( p + 1 ) d ( x ) = ( x ) ∂ x p + 1 2 As usual, we should have to estimate separately the three functions m ( p + 1 ) m 4 ( x ) , m 2 ( x ) , ( x ) 2 (Fan & Yao, 1998; Franke & Diagne, 2006) An interesting result here is that we estimate the three functions by only one estimator! F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 76 / 115

Estimating the functions v ( x ) and d 2 ( x ) We use only the following neural network estimator n � 2 � � X 2 η = arg min t − q ( X t − 1 ; η ) ˆ η t = 2 F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 77 / 115

Estimating the functions v ( x ) and d 2 ( x ) We use only the following neural network estimator n � 2 � � X 2 η = arg min t − q ( X t − 1 ; η ) ˆ η t = 2 where ◮ q ( X t − 1 ; η ) = � d k = 1 c k Γ ( a k X t − 1 + b k ) + c 0 is the neural network function; ◮ η = ( c 0 , c 1 , . . . , c d , a 1 , . . . , a d , b 1 . . . b d ) is the vector of parameters to be estimated; ◮ d is the number of nodes in the hidden layer; ◮ Γ ( · ) is the logistic activation function . F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 77 / 115

Estimating the functions v ( x ) and d 2 ( x ) By defining m 4 ε = E ( ǫ 4 t ) , we reparameterize as follows: m 4 ( x ) − m 2 2 ( x ) = m 2 v ( x ) = 2 ( x ) [ m 4 ε − 1 ] m ( p + 1 ) d ( x ) = ( x ) 2 then we propose the following estimators ˆ m 2 ( x ) = q ( x , ˆ η ) � n t = 2 X 4 t ˆ m 4 ε = � n η )] 2 t = 2 [ q ( X t , ˆ m ( p + 1 ) q ( p + 1 ) ( x ; ˆ ˆ ( x ) = η ) 2 F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 78 / 115

Estimating the locally global bandwidth Finally, given the ergodicity of the process, the locally global bandwidth is estimated by: � 2 � m ( p + 1 ) ˆ � ( X t ) � n ∗ i = 1 ˆ v ( x i ) X t ∈ I X 2 ˆ R I X ˆ R I X var = . f , bias = , � n n t = 1 I ( X t ∈ I X ) � 1 / ( 2 p + 3 ) � C p , K × ˆ R I X ˆ var h I X = n × ˆ R I X f , bias The points { x 1 , x 2 , . . . , x n ∗ } in ˆ R I X var are uniformly spaced in the interval I X , and I ( · ) is the indicator function. F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 79 / 115

Assumptions (a1) the errors ε t have continuous and positive density function and, for some δ > 4, E | ε t | δ < ∞ ; E ( ε 2 E ( ε t ) = E ( ε 3 t ) = 1 , t ) = 0 , (a2) the functions m ( · ) and σ ( · ) have continuous second derivative. Moreover, the function σ ( · ) is positive; (a3) there exist the constants M 1 > 0 and M 2 > 0 such that, for all y ∈ R , E | ε t | δ � 1 /δ � | m ( y ) | ≤ M 1 ( 1 + | y | ) , | σ ( y ) | ≤ M 2 ( 1 + | y | ) , M 1 + M 2 < 1 ; (a4) the density function f X ( · ) of the (stationary) measure of the process µ X exists, it is bounded, continuous and positive on every compact set in R . (b1) � d k = 1 | c k | ≤ ∆ n . (b2) d ≡ d n , d n → ∞ , ∆ n → ∞ as n → ∞ . ∆ 2 n d n log ( ∆ 2 n d n ) ∆ 4 (b3) Let K 1 ( n ) := and K 2 ( n ) := n 1 − δ . n √ n F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 80 / 115

The consistency of the bandwidth selector Using the framework in Franke and Diagne (2006), we can state the following results. Lemma 1 Under assumptions (a1)-(a4) and (b1)-(b3), the volatility function estimator is consistent in the sense that: If K 1 ( n ) → 0 as n → ∞ , then � � � 2 η ) − σ 2 ( x ) E q ( x ; ˆ d µ X ( x ) → 0 n → ∞ if, additionally, K 2 ( n ) → 0 for some δ > 0, then � � � 2 a . s . η ) − σ 2 ( x ) q ( x ; ˆ d µ X ( x ) − → 0 n → ∞ F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 81 / 115

The consistency of the bandwidth selector Lemma 2 Under assumptions (a1)-(a4) and (b1)-(b3), the estimator of the second derivative of σ 2 ( x ) is consistent in the sense that: If K 1 ( n ) → 0 as n → ∞ , then � � η ) − [ σ 2 ( x )] ′′ � 2 q ′′ ( x ; ˆ E d µ X ( x ) → 0 n → ∞ if, additionally, K 2 ( n ) → 0 for some δ > 0, then � � η ) − [ σ 2 ( x )] ′′ � 2 q ′′ ( x ; ˆ a . s . d µ X ( x ) − → 0 n → ∞ F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 82 / 115

The consistency of the bandwidth selector Theorem 1 Under the conditions (a1)-(a4) and (b1)-(b3), ˆ R I X f , bias , with I X ⊆ R , is consistent in the sense that: If K 1 ( n ) → 0 as n → ∞ , then p R I X ˆ → R I X − n → ∞ f , bias f , bias if, additionally, K 2 ( n ) → 0 for some δ > 0, then ˆ a . s . R I X → R I X − n → ∞ f , bias f , bias Corollary Using the same conditions as in theorem 1, then ˆ m 4 ε is consistent in the sense that: p If K 1 ( n ) → 0 as n → ∞ , then ˆ m 4 ε − → m 4 ε n → ∞ a . s . if, additionally, K 2 ( n ) → 0 for some δ > 0, then ˆ m 4 ε − → m 4 ε n → ∞ F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 83 / 115

The consistency of the bandwidth selector Theorem 2 Using the same conditions as in theorem 1, then ˆ R I X var with I X ⊂ R and n ∗ = O ( n ) , is consistent in the sense that: If K 1 ( n ) → 0 as n → ∞ , then p R I X ˆ → R I X − n → ∞ var var if, additionally, K 2 ( n ) → 0 for some δ > 0, then R I X ˆ a . s . → R I X − n → ∞ var var F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 84 / 115

The consistency of the bandwidth selector For a fixed z ∈ R , let I z be a non null measure set which contains the point z . Let n z be the number of observed values in I z such that n z → ∞ when n → ∞ . Theorem 3 Using the same conditions as in Lemma 2, if n z = o ( n ) , for a set I X ⊂ R , then ˆ h I z is consistent in the sense that: If K 1 ( n ) → 0 as n → ∞ , then � � p � ˆ h I z − h opt sup L ( z ) − → 0 n → ∞ � � � z ∈ I X if, additionally, K 2 ( n ) → 0 for some δ > 0, then � � a . s . � ˆ h I z − h opt sup L ( z ) − → 0 n → ∞ � � � z ∈ I X F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 85 / 115

The consistency of the bandwidth selector The previous results imply that: 1 p a . s . ˆ → h opt ˆ → h opt h G − h G − G . or G For all z ∈ I X 2 p a . s . ˆ → h opt ˆ → h opt h I z − L ( z ) h I z − L ( z ) , or Uniformly for each z ∈ I X 3 p a . s . π h ( z ) ˆ − → π h ( z ) ˆ π h ( z ) − → π h ( z ) or F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 86 / 115

The simulation study: setting the models Model 1: X t = [ ψ ( X t − 1 + 1 . 2 ) + 1 . 5 ψ ( X t − 1 − 1 . 2 )] ǫ t ǫ t ∼ N ( 0 , 1 ) ψ ( z ) = d.f. of N ( 0 , 1 ) � 0 . 1 + 0 . 3 X 2 Model 2: X t = t − 1 ǫ t ǫ t ∼ N ( 0 , 1 ) � 0 . 01 + 0 . 1 X 2 t − 1 + 0 . 2 X 2 Model 3: X t = t − 1 I X t − 1 < 0 ǫ t ǫ t ∼ N ( 0 , 1 ) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 87 / 115

Model 1: from Härdle and Tsybakov (1997) X t = [ ψ ( X t − 1 + 1 . 2 ) + 1 . 5 ψ ( X t − 1 − 1 . 2 )] ǫ t ; ǫ t ∼ N ( 0 , 1 ); ψ ( z ) = d . f . of N ( 0 , 1 ) TIME PLOT VOLATILITY FUNCTION 2 0.35 1 volatility 0.30 0 0.25 -1 0.20 -2 0 200 400 600 800 1000 -1.0 -0.5 0.0 0.5 1.0 X(-1) F. Giordano – M.L. Parrella (UNISA) Local or Global Smoothing? COMPSTAT 2010 - Paris 88 / 115

Local or Global Smoothing? A Bandwidth Selector for Dependent Data - PowerPoint PPT Presentation

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Local or Global Smoothing? A Bandwidth Selector for Dependent Data - PowerPoint PPT Presentation

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Smoothing of Variable Bandwidth Kernel Estimate of Heavy-Tailed Density Function Natalia M.

The Dantzig selector in Coxs proportional hazards model A. Antoniadis 1 , P . Fryzlewicz 2 , F

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