JEREMY TAYLOR S CONTRIBUTIONS TO TRANSFORMATION MODEL

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1 1 / 25 JEREMY TAYLOR S CONTRIBUTIONS TO TRANSFORMATION MODELS DEPT. OF STATISTICS, UNIV. WISCONSIN, MADISON BIOMEDICAL STATISTICAL MODELING. CELEBRATION OF JEREMY TAYLOR S OF 60TH BIRTHDAY. UNIVERSITY OF MICHIGAN, JUNE 9 10, 2017.? WHO HAS KNOWN JEREMY THE LONGEST?

2 2 / 25 BERKELEY STATISTICS SOCCER TEAM

3 3 / 25 BACKSTORY THE BOX-COX TRANSFORMATION MODEL h(y i ; λ) = Y λ i 1 λ λ = 1: LINEAR MODEL = Xi T β + σε i, 1 i n λ = 0: ACCELERATED FAILURE TIME MODEL λ = 1: RECIPROCAL LINEAR MODEL (Y=MPG) BOX-COX 1964 JRSS: AN ANALYSIS OF TRANSFORMATIONS λ R, ( ˆβ, ˆλ, ˆσ) = MLE

4 4 / 25 BACKSTORY BICKEL-DOKSUM 1981 JASA AN ANALYSIS OF TRANSFORMATIONS REVISITED THE COST OF ADDITIONAL PARAMETER λ. BOX-COX 1982 JASA AN ANALYSIS OF TRANSFORMATIONS REVISITED, REBUTTED ALSO LINDLEY 1983 JASA HINKLEY AND RUNGER 1984 JASA

5 5 / 25 BACKSTORY THE BCLHR ARGUMENT: IF WE THINK OF λ AS UNKNOWN, THEN β AND ˆβ WILL BE ON AN UNKNOWN SCALE. WE NEED TO THINK THAT λ IS KNOWN AND EQUAL TO ˆλ. THEN REPORT THAT THE ANALYSIS IS ON THE SCALE SELECTED THAT IS, BICKEL & DOKSUM ARE WRONG TO CONSIDER COST OF ESTIMATING λ.

6 6 / 25 BACKSTORY CONSEQUENCES FOR ME: ANNALS OF STATISTICS: JASA: COLLABORATOR: WISCONSIN FRIEND: ETC : BUT JEREMY: BRILLINGER: CHI-WING WONG:

7 7 / 25 BACKSTORY 1984: JASA ASKS BICKEL & DOKSUM TO COMMENT ON HINKLEY-RUNGER PAPER. DOKSUM 1984 JASA, p REBUTTAL REBUTTED (MEAN? FUNNY?) 1. WITH BCLHR WE NO LONGER NEED STUDENT t OR F DISTRIBUTIONS. BECAUSE σ IS ALSO A SCALE PARAMETER 2. BCLHR THINK THERE IS A FREE LUNCH (PRETENDING λ = ˆλ HAS NO COST) 3. BCLHR ARE THROWING THE BABY (EFFICIENCY) OUT WITH THE BATHWATER

8 8 / 25 BACKSTORY 4. BCLHR WOULD MAKE PEARSON RIGHT IN THE FISHER-PEARSON ARGUMENT. JUST ADD ON THE SCALE SELECTED, AND PEARSON WOULD APPEAR TO BE RIGHT. 5. WOULD A BIOMEDICAL SCIENTIST WANT A CONFIDENCE INTERVAL FOR β j ON THE SCALE SELECTED, OR GENERALLY VALID INTERVAL? 6. EXAMINING BENEFITS AND COSTS OF INTRODUCING MORE COMPLEX MODELS IS TO A LARGE EXTENT WHAT STATISTICIANS DO!

9 9 / 25 BACKSTORY CONSEQUENCES, REVISITED: LINDLEY: COX: BOX:

10 10 / 25 COX & REID 1987 JRSS: PARAMETER ORTHOGONALITY = NUISANCE PARAMETER ESTIMATION HAS NO ASYMPTOTIC COST. CONJECTURE: IF β /σ = O(n 1 2 ), THEN I (σ, β, λ) = σ σ 2 E(XX T ) M EFFICIENT SCORES ORTHOGONAL = NO ASYMPTOTIC COST

11 11 / 25 BRILLINGER (1983), STOKER (1986): AVERAGE DERIVATIVE APPROACH (ADE) MODEL: E(Y X ) = g(x T β) FOR SOME UNKNOWN g. DEFINITION: δ j = REGRESSION PARAMETER = E ( X j E(Y X ) ) IN BOX-COX MODEL j = β j β, p 2. THERE IS NO λ! ˆβ j HAS A λ-free INTERPRETATION AS AN ADE CHEN-LOCKHART-STEPHENS 2002 CANADIAN J. STATISTICS Y = g(x T β + σε), ( g/ X j )/( g/ ε) = β j /σ. THERE IS NO λ!

12 12 / 25 JEREMY S TRANSFORMATION PAPERS 1985 JASA. LOCATION. SKEW DISTRIBUTIONS X 1,..., X n i.i.d. G ROBUSTNESS STUDIES ASSUMED G SYMMETRIC BY USING BOX-COX TRANSFORMATION JEREMY EXTENDED USEFULNESS OF ROBUSTNESS STUDIES BY A FACTOR OF (λ R).

13 13 / 25 TRANSFORMATION PAPERS 1985 JASA Z(λ) = X λ 1 λ = µ + σε, ε HAS A SYMMETRIC DISTRIBUTION F, VAR ε = 1. STRATEGY: 1) ESTIMATE λ BY SOLVING S ( Z(λ) ) = 0 WHERE S IS A MEASURE OF SYMMETRY. E.G. S(Z) = Z (r) + Z (n r+1) 2Z ( 1 2 n) (HINKLEY 75) 2) USE A ROBUST STATISTIC L BASED ON Z n (ˆλ),..., Z n (ˆλ)

14 14 / 25 3) TRANSFORM BACK TO X SCALE USING h 1 = ( (X λ 1)/λ ) 1 CALL ESTIMATE ˆM. THEN ˆM MEDIAN L(X ) EXAMPLE: L = Z, ( 1 ) ˆM = X ˆλ 1ˆλ n i, ˆλ 0 THEOREM. SUPPOSE ˆλ = (ΠX i ) 1 n, λ = 0 P λ, THEN ˆM P M WITH M = [ E G (X λ ) ] 1 λ, λ 0 exp [ E G {log(x )} ], λ = 0

15 15 / 25 COMPARISON OF 3 ESTIMATES 1) ˆM = INVERSE OF MEAN WITH λ ESTIMATED 2) ˆM 0 = INVERSE OF MEAN WITH λ = λ 0 =TRUE λ 3) SAMPLE MEDIAN OF Y S

16 16 / 25 CONCLUSIONS: 1) ˆM TYPICALLY MORE EFFICIENT THAN THE MEDIAN 2) USING BOX-COX ˆλ DOES NOT MAKE ESTIMATES MORE EFFICIENT 3) THE COST OF NOT KNOWING λ IS NOT SEVERE. IN SOME CASES, USING ˆλ INSTEAD OF λ 0 IS MORE EFFICIENT.

17 17 / BIOMETRIKA X 1,..., X n i.i.d. G Z i (λ) = h(x i, λ) = (X i 1) λ /λ = µ + σε i WHERE ε i F SYMMETRIC, VAR(ε i ) = 1 WHAT ARE THE BEST TRANSFORMATIONS TO SYMMETRY? WHAT ARE THE BEST ˆλ s. IF f = N(0, 1), THEN ˆλ BC = MLE IS BEST.

18 18 / 25 A GENERAL CLASS OF ˆλ S: ˆλ ψ SOLVES 1 { ψ [Zi (λ) n Z(λ)]/ˆσ ( Z(λ) )} = 0 WHERE ψ IS A MONOTONE ODD FUNCTION. THEOREM (a) n(ˆλ ψ λ) N ( 0, A ψ (λ) ) (b) A ψ (λ) IS MINIMIZED BY ψ 0 (t) = a ( f (x)/f (x) ) (x 2 1) + bx WHERE a, b ARE ARBITRARY CONSTANTS 0

19 19 / 25 WHEN F = N(0, 1), ψ(x) = x 3 IS OPTIMAL WHAT ABOUT F = HUBER LEAST FAVORABLE? JEREMY SUGGESTS ψ J (x) = x 3 x c = c sgn (x)(x 2 1) + x, x > c HUBERESQUE (NORMAL-DOUBLE EXPONENTIAL)

20 20 / 25 COMPARISONS: 1) ˆλ BC 2) ˆλ RBC : R = ROBUST = HUBERIZED L(ε) 3) ˆλ H (1): H = HINKLEY, Z (1) + Z (n) Z ( 1 2 n) = 0 4) ˆλ H (2): H = HINKLEY, MEAN MEDIAN = 0 5) ˆλ (3) : ψ(x) = x 3 6) ˆλ J : ψ = ψ J 7) ˆλ (2) : ψ(x) = sgn (x)x 2

21 21 / 25 ASYMPTOTIC AND MONTE CARLO CONCLUSIONS 1) ˆλ RBC WINNER IN MONTE CARLO 2) ˆλ BC PERFORMED WELL 3) ˆλ (3), ˆλ J and ˆλ (2) VERY SIMILAR, NOT GOOD IN MC. 4) ˆλ H (1) SO-SO 5) ˆλ (2) POOR

22 22 / JASA. THE MEAN OF Y, GIVEN X h(y ; λ) = X β + σε, ε N(0, 1) X IS A KNOWN DESIGN MATRIX h(y ; λ) = BOX-COX TRANSFORMATION GOAL: ESTIMATE E(Y X ), WHICH CAN BE INTERPRETED WHEN λ IS UNKNOWN. JEREMY COMPARES 2 ESTIMATES. SMALL θ and SMEARING

23 23 / 25 E(Y X ) IS A MESS. APPROXIMATION BASED ON SMALL θ λσ/ β E(Y X ) = g(λ, β, σ) FOR SOME g 1) Ŷ1 = Ê(Y X ) = g(ˆλ, ˆβ, ˆσ) = 1ST ESTIMATE 2ND ESTIMATE (SMEARING) Ŷ 2 = Ê(Y X ) = s(ˆλ, ˆβ, ˆσ, ε)d ˆFε, s h 1 WHERE ˆFε = EMPIRICAL OF ê i = h(y i ; ˆλ) X i ˆβ/ˆσ.

24 24 / 25 RESULTS: THE TWO METHODS HAVE SIMILAR GOOD PROPERTIES EXCEPT WHEN λ 0 AND σ IS LARGE WHEN λ 0 AND σ IS LARGE, THE SMALL θ ESTIMATE SUFFERS BIAS AND THE SMEARING ESTIMATE SUFFERS VARIANCE.

25 25 / 25 HAPPY BIRTHDAY JEREMY 60 YEARS YOUNG