Nonparametric Predictive Inference (An Introduction)

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1 Nonparametric Predictive Inference (An Introduction) SIPTA Summer School 200 Durham University 3 September 200

2 Hill s assumption A (n) (Hill, 968) Hill s assumption A (n) (Hill, 968) X,..., X n, X n+ are real-valued and exchangeable random quantities x < x 2 <... < x n are the ordered observed values of X,..., X n (and let x 0 = and x n+ = ) For X n+, A (n) is given by P(X n+ I j = (x j, x j )) = n +, j =,..., n +

3 Nonparametric predictive inference (NPI) Nonparametric predictive inference (NPI) NPI is based on Hill s assumption A (n) Let B be the Borel σ-field over R. For any element B B, lower probability P(.) and upper probability P(.) for the event X n+ B, based on the intervals I j = (x j, x j ) (j =, 2,..., n + ) created by n real-valued non-tied observations, and the assumption A (n), are P(X n+ B) = n + {j : I j B} P(X n+ B) = n + {j : I j B }

4 Illustration example Illustration example n = 4 0 x x 2 x 3 x 4

5 Illustration example Illustration example n = 4 0 x x 2 x 3 x 4 P(X (0, x )) = P(X (x 4, )) = P(X (x i, x i+ )) =, i =, 2, 3

6 Illustration example Illustration example n = 4 B 0 x x 2 x 3 x 4

7 Illustration example Illustration example n = 4 B 0 x x 2 x 3 x 4 P[X B] = P[X B] = 3

8 Illustration example Illustration example n = 4 B 0 x x 2 x 3 x 4 P[X B] = P[X B] = 3 Imprecision = P P = 2 = 0.4

9 NPI for m future observations NPI for m future observations We are interested in m future observations, X n+i for i =,..., m. We link the data and future observations via Hill s assumption A (n), actually via A (n+m ) (which implies A (n+k) for all k = 0,,..., m 2). Let S j = #{X n+i I j, i =,..., m}, then inferences about these m future observations, assuming A (n+m ), can be based on the following probabilities, for any (s,..., s n+ ) with non-negative integers s j with n+ j= s j = m ( n + m {S j = s j }) = n n+ P( j= )

10 NPI for Pairwise comparisons Comparing two independent groups, X and Y We have two independent groups X and Y : x < x 2 <... < x nx and y < y 2 <... < y ny The classical methods test H 0 : F X = F Y. For complete data, Coolen (996) introduced NPI to compare two independent groups depending on A (n). This is given via the lower and upper probabilities P(X nx + < Y ny +) P(X nx + < Y ny +)

11 Illustration n y + n y + n y + n y + 0 y y 2... y ny y ny n x + n x + n x + n x + 0 x x 2... x nx x nx

12 Illustration Lower Probability, P(X nx + < Y ny +) n y + n y + n y + n y + 0 y y 2... y ny y ny n x + n x + n x + n x + 0 x x 2... x nx x nx

13 Illustration Upper Probability, P(X nx + < Y ny +) n y + n y + n y + n y + 0 y y 2... y ny y ny n x + n x + n x + n x + 0 x x 2... x nx x nx

14 Illustration Example We use data on birthweights for 2 male and 2 female babies as presented by Dobson (983). Male (X ) Female (Y ) P(X 3 > Y 3 ) = = 0.09 P(X 3 > Y 3 ) = 69 = 0.67.

15 Right-censored data Right-censored data Very common in lifetime data, e.g. in survival analysis and reliability studies Actual event of interest for a unit (e.g. death or failure) is not observed, only that this event has not happened by an observed time We assume that the censoring mechanism is independent of the residual lifetime at the time of censoring, hence that every unit still in the experiment had the same probability of being the unit censored ( non-informative censoring )

16 rc-a (n) assumption (Coolen & Yan, 2004) rc-a (n) assumption (Coolen & Yan, 2004) n = 4 0 x c x 2 x 3

17 rc-a (n) assumption (Coolen & Yan, 2004) rc-a (n) assumption (Coolen & Yan, 2004) n = 4 0 x c x 2 x 3

18 rc-a (n) assumption (Coolen & Yan, 2004) rc-a (n) assumption (Coolen & Yan, 2004) n = 4 0 x c x 2 x

19 rc-a (n) assumption (Coolen & Yan, 2004) rc-a (n) assumption (Coolen & Yan, 2004) n = 4 0 x c x 2 x 3

20 rc-a (n) assumption (Coolen & Yan, 2004) rc-a (n) assumption (Coolen & Yan, 2004) n = 4 B 0 x c x 2 x 3

21 rc-a (n) assumption (Coolen & Yan, 2004) rc-a (n) assumption (Coolen & Yan, 2004) n = 4 B 0 x c x 2 x 3 P[X B] = 0 P[X B] = 7 Imprecision = P P = 7 =

22 rc-a (n) assumption (Coolen & Yan, 2004) Example Table: Cervical cancer survival data (> t indicates right-censoring at t) Control - A New therapy - B 90 > > >978 0 > > >3 > >9 >360 > >63 > > >77 827

23 rc-a (n) assumption (Coolen & Yan, 2004)

24 rc-a (n) assumption (Coolen & Yan, 2004)

25 rc-a (n) assumption (Coolen & Yan, 2004)

26 NPI for Bernoulli data NPI for Bernoulli data n + m exchangeable Bernoulli trials (each success or failure ) Yj l : the number of successes in trials j to l Data on first n trials: Y n = s Interest in Y n+m n+ FC: Low structure imprecise predictive inference for Bayes problem Stat. & Prob. Letters 36, 998,

27 NPI for Bernoulli data R t = {r,..., r t }, with t m + and 0 r < r 2 <... < r t m, (define ( s+r 0 ) s = 0). NPI upper probability: P(Y n+m n+ R t Y n = s) = ( ) t [( ) ( )] ( ) n + m s + rj s + rj n s + m rj. n s s n s j= NPI lower probability: P(Y n+m n+ R t Y n = s) = P(Y n+m n+ Rc t Y n = s).

28 NPI for Bernoulli data Centre (n i, s i ) s i /n i order Centre (n i, s i ) s i /n i order (8,43) (23,27) (200,27) (369,7) (7,26) (24,28) (42,) (84,3) (27,36) (740,67) (47,49) (268,32) Table. Heart operations mortality data.

29 NPI for Bernoulli data m : P( > ) P( > ) ( > ) P( ) P( ) ( ) P(3 > ) P(3 > ) (3 > ) P(3 ) P(3 ) (3 ) P(3 > 4) P(3 > 4) (3 > 4) P(3 4) P(3 4) (3 4) Table 2. Some pairwise comparisons between centres.

30 NPI for Bernoulli data i [P, P](i > max j i m = 0 m = 0 j) [P, P](i max j) [P, P](i > max j i j i j) [P, P](i max j i [0.77, 0.97] [0.369, 0.397] [0.426, 0.482] [0.26, 0.83] 2 [0.033, 0.039] [0.2, 0.28] [0.022, 0.032] [0.04, 0.07] 3 [0.06, 0.072] [0.73, 0.96] [0.073, 0.098] [0.4, 0.48] 4 [0.07, 0.022] [0.067, 0.082] [0.007, 0.0] [0.04, 0.022] [0.060, 0.070] [0.73, 0.93] [0.069, 0.089] [0.0, 0.39] 6 [0.02, 0.024] [0.082, 0.092] [0.008, 0.0] [0.07, 0.022] 7 [0.06, 0.020] [0.067, 0.078] [0.00, 0.008] [0.0, 0.06] 8 [0.048, 0.04] [0.48, 0.63] [0.042, 0.04] [0.073, 0.09] 9 [0.030, 0.036] [0.04, 0.20] [0.08, 0.026] [0.034, 0.048] 0 [0.064, 0.074] [0.79, 0.20] [0.077, 0.0] [0.2, 0.3] [0.009, 0.0] [0.046, 0.02] [0.00, 0.002] [0.003, 0.004] 2 [0.022, 0.027] [0.08, 0.098] [0.00, 0.04] [0.020, 0.028] Table 3. Multiple comparisons between centres. j)

31 Further comments and research challenges Further comments and research challenges NPI has been presented for many problems in Statistics, Reliability, Risk and OR NPI is never in disagreement with inferences based on empirical probabilities, so one could call NPI objective A particularly nice NPI model for multinomial data has been presented (FC and Thomas Augustin) Main challenges are to develop NPI for multi-dimensional random quantities, including use of co-variates and multivariate statistics Most importantly: NPI has helped us to get better understanding of foundations of statistics with imprecise probabilities

32 References References For more information and references:

System reliability using the survival signature

System reliability using the survival signature Frank Coolen GDRR Ireland 8-10 July 2013 (GDRR 2013) Survival signature 1 / 31 Joint work with: Tahani Coolen-Maturi (Durham University) Ahmad Aboalkhair