Multivariate Generalized Spatial Signed-Rank Methods

Transcription

1 Multivariate Generalized Spatial Signed-Rank Methods Jyrki Möttönen University of Tampere Hannu Oja University of Tampere and University of Jyvaskyla Robert Serfling 3 University of Texas at Dallas June 005 Final preprint version for Special Issue of Journal of Statistical Research 004, Volume 39, Number, pages 9-4 celebrating A. K. E. Saleh s tenure as Chief-Editor Department of Mathematics, Statistics and Philosophy, Statistics Unit, FIN-3304 University of Tampere, Finland. jyrki.mottonen@uta.fi. Tampere School of Public Health, FIN-3304 University of Tampere, Finland, and Department of Mathematics and Statistics, P.O. Box 35, MaD, FIN-4004 University of Jyvaskyla, Finland. Hannu.Oja@uta.fi. 3 Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas , USA. serfling@utdallas.edu. Website: serfling. Support by National Science Foundation Grants DMS and CCF is gratefully acknowledged.

2 Abstract New multivariate generalized signed-rank tests for the one sample location model having favorable efficiency and robustness properties are introduced and studied. Limiting distributions of the tests and related estimates as well as formulae for asymptotic relative efficiencies are found. Relative efficiencies with respect to the classical Hotelling T test and the mean vector are evaluated for the multivariate normal, t, and Tukey models. While the tests estimates are only rotation invariant equivariant, versions that are affine invariant equivariant are discussed as well. AMS 000 Subject Classification: Primary 6G35 Secondary 6H05. Key words and phrases: Nonparametric; Multivariate; Spatial signed-ranks; One-sample location; Hodges-Lehmann estimation; Hypothesis tests; Asymptotic distributions; Asymptotic relative efficiency.

3 Introduction The purpose of this paper is to consider the robustness and efficiency properties of new multivariate generalized signed-rank tests that we introduce here and related multivariate generalized Hodges-Lehmann estimators that were proposed and studied by Chaudhuri 99. These tests and estimates provide more or less robust and efficient competitors of the classical Hotelling T test and the mean vector. Let Y {y,..., y n } be a random sample from a k-variate distribution with cdf F and pdf f symmetric around θ. We wish to estimate the unknown location centre θ and test the null hypothesis H 0 : θ 0. Möttönen and Oja 995 and later Möttönen, Oja, and Tienari 997 developed and studied multivariate spatial sign and signed-rank tests for this problem. The two test statistics they considered may be written in the form W mn n m n i n Sy i + + y im, i m for m giving the spatial sign test statistic or m giving the spatial signed-rank test statistic. Here the function { y Sy y if y 0 0 if y 0, where denotes the Euclidean norm in R k, denotes the spatial sign function of y R k. It is invariant under affine transformations except for those involving heterogeneous scale changes. In this paper, we consider the above statistics W mn for arbitrary m,,... and call them generalized signed-rank test statistics. Unlike the univariate case with m and, the test statistics W mn in general are only conditionally and asymptotically, but not strictly, distribution-free under the null hypothesis. The conditional sign-change tests are then based on the n under H 0 and symmetry equiprobable cases ±y,...,±y n. These test statistics are not affine invariant, which means that the obtained p-values depend on the chosen coordinate system. Related location estimators generalized Hodges-Lehmann estimators are obtained by a standard approach as follows. First, the observations are centered with respect to a candidate θ by setting z i y i θ, i,..., n. Next one calculates the value of the W mn statistic, say W mn θ, for the centered data. Finally, one obtains the estimator θ mn via the equation W mn θ mn 0. this yields the class of estimators studied by Chaudhuri 99. Two special cases are the multivariate spatial median m and multivariate spatial Hodges-Lehmann estimator m ; for these see also Möttönen and Oja 995. While the estimators ˆθ mn are found to have favorable efficiency and robustness properties, they are not fully affine equivariant, however.

4 In the paper, the efficiency and robustness properties of generalized spatial signed-rank tests and generalized Hodges-Lehmann estimates are considered in general, so as to include the cases m 3. It will be seen, for example, that efficiency at normal models increases with m, while robustness decreases. It is important, therefore, to establish benchmarks by which m may be selected so as to obtain a suitable trade-off between efficiency and robustness. The desired balance between these criteria depends, of course, upon the given context of application and the particular losses associated with lack of efficiency and lack of robustness. Numerical asymptotic relative efficiencies are provided for some specific models in Section 4. For m,,...,5, and for k,, 3, 4, 6 and 0, we compare the generalized spatial signed-rank test procedures with the classical Hotelling T test for the k-variate standard normal model and for k-variate t models with 3, 6, and 0 degrees of freedom see Tables -4. For example, for 6-variate standard normal, this ARE increases from 0.90 for m to for m 5. On the other hand, for the t distributions, the ARE s are all greater than, but do not follow a strict monotonicity pattern. We note, though, that for the 6-variate t distribution with 3 degrees of freedom, the ARE s decrease from.344 for m to.697 for m 5. Also, for the same range of m and k, we examine both the ARE of the tests and the bias of the estimators, for a family of Tukey models, i.e., contaminated multivariate normal models, with a range of contamination and spread parameters. Favorable robustness properties of our generalized signed-rank methods are indicated by their bounded influence functions and nonzero breakdown points see Section 3.. It is of interest that, independently of the dimension, the breakdown point decreases from 0.5 for m to 0.9 for m 5. The theoretical foundation underlying the above practical results is developed as follows. In Section we treat the asymptotic properties of our generalized signed-rank test procedures. These are obtained via an asymptotic equivalence Lemma with certain statistics of technically more convenient form expressed in terms of a multivariate generalized signed rank function, which we define and discuss. In this fashion we obtain chi-square limit distributions for Wald-type generalized signed-rank test statistics, both under the null hypothesis Theorem and under contiguous alternatives Theorem. The related estimators are discussed in Section 3: asymptotic normality of the estimators is provided by Theorem 3, and natural V-statistic estimators of the limiting covariance matrix are introduced. Proofs are relegated to the Appendix, which contains numerous technical lemmas taken from a more complete source, Möttönen 00. In Section 5 we discuss efficiency-robustness trade-offs, we describe the construction of fully affine invariant/equivariant versions, and we briefly make comparisons with selected other procedures. For the latter, an interesting finding is that the comparisons of competing procedures tend to depend upon the dimension.

5 Tests. Generalized spatial signed-ranks We first define what we mean by the multivariate generalized spatial signed-rank function: Definition Assume that y,...,y m, m,,..., is a random sample from a k-variate symmetrical distribution with symmetry centre 0 and cdf F. Then the population generalized spatial signed-rank function of order m is R m y E F {Sy y... y m }. If y,...,y n is a random sample from F, the corresponding empirical spatial signed-rank function of order m is R mn y ave{sy y i... y im }, where the average is taken over all possible m -sets of observations, that is, over all cases i <...<i m n. Note that, R y reduces to the spatial sign function and R y is the regular population spatial rank function F symmetric around the origin. Note also that R m y is the theoretical rank function for the distribution of y y m. See Möttönen and Oja 995. We could symmetrize R mn y, if desired, by taking the average over cases ±y,...,±y n, which would be useful when constructing sign-change tests. For spherical distributions, the generalized spatial signed-rank function at y r y and u y y is R m y q m ru, where q m r ϱ mr and ϱ m r E F y y... y m the expectation does not depend on u. As in Möttönen, Oja and Tienari 997, we can establish Proposition Assume that f is continuous and bounded. Then. Test statistics sup R mn y R m y P 0 y Definition The generalized signed-rank test statistic of order m for testing H 0 : θ 0 is the vector-valued V-statistic n n W mn n m... Sy i y im. i i m 3

6 An asymptotically equivalent version of this test statistic can be constructed using the generalized spatial ranks. Lemma Under the null hypothesis, W mn is asymptotically equivalent to T mn n n R m y i, i that is, nw mn T mn P 0. This in fact holds uniformly over all alternatives, since the model is a location model and the statistics are location equivariant. Under the null hypothesis, the expectation and variance of T mn are respectively, where E 0 T mn 0 and Var 0 T mn n B m, B m B m F E F [R m yr T my] is the generalized rank covariance matrix. For m and m, the regular spatial sign and rank covariance matrices are obtained. See Visuri et al The generalized rank covariance matrix B m may be consistently estimated by the corresponding sample counterpart sample rank covariance matrix, B mn ave{r mn y i R T mny i }. Note that related sign-change tests can be constructed, and that asymptotic tests may be constructed where the p-value is calculated using the limiting distribution. For brevity, we omit the details. Lemma Under the null hypothesis, B mn P B m, n. Theorem Under the null hypothesis, the limiting distribution of nt T mnb mnt mn is a chi-square distribution with k degrees of freedom. 4

7 .3 Formulae for Pitman efficiency For limiting efficiencies, we next consider a sequence of contiguous alternatives H n : θ n / δ. Write Ly logfy for the optimal location score function and assume that, under the null hypothesis, L n n [logfy i n / δ log fy i ] n / δ T U n δt I 0 δ + o P, i where U n n Ly i and I 0 E 0 LyL T y. i Note that L n is the log-likelihood test statistic H 0 vs. H, U n the optimal score statistic, and I 0 the expected Fisher information matrix for a single observation. Then, under the sequence of contiguous alternatives H n, the limiting distribution of n / T mn is N k A m δ,b m where A m E 0 R m yl T y and B m E 0 R m yr T m y. This can be derived as in Möttönen and Oja 995. Finally, the limiting distribution of the squared test statistic nt T mn B mn T mn follows. Theorem Under the sequence of alternatives H n, the limiting distribution of nt T mnb mnt mn is a noncentral chi-square distribution with k degrees of freedom and noncentrality parameter δ T A T mb m A m δ. Asymptotic Pitman efficiencies with respect to the optimal likelihood ratio test test are then simply δ T A T mbm A m δ. δ T I 0 δ This may be used in a standard way, as discussed, for example, in Hettmansperger and McKean

8 3 Estimators 3. Generalized Hodges-Lehmann estimators For completeness, we briefly treat the generalized Hodges-Lehmann estimators defined in Section. Lemma 3 Under general assumptions, nθmn na m W mn + o P Theorem 3 Under general assumptions, θ mn AN θ, n A m B m A m. The limiting variance can be estimated by estimating A m and B m separately. In Section. we gave a natural estimate of the generalized rank covariance matrix, namely the empirical rank covariance matrix B mn. To estimate A m, note that with ȳ m /my +...+y m we have [ ] A m E F ȳ m ȳm ȳ T m I k E. ȳ m 3 A natural estimate is then the V -statistic n n { [ȳiȳi A mn n m ȳi T I k ȳi 3 i i m where I i,...,i k and ȳi /my i y im. 3. Comparison of efficiency and robustness Comparison of asymptotic covariance matrices A m B ma m ]}, has been carried out in the multivariate normal case by Chaudhuri 99, in which case it is found that efficiency increases with m. The influence function of the functional θ m is found to be IFy; θ m,fa m R m y, which is bounded since R m y is bounded R m y. The breakdown point of ˆθ mn decreases, although slowly, as m increases, and is independent of the dimension. Specifically, BPˆθ mn / /m, which takes values 0.5, 0.93, 0.06, 0.59, 0.9 for m,, 3, 4, 5. 6

9 4 Asymptotic relative efficiency results for some specific models 4. Multivariate normal distributions The multivariate normal case has been considered earlier by Chaudhuri 99. We assume that y,...,y n is a random sample from the N0,I k distribution. If y r and u y y, then R m y q m ru where r q m r q m and qr r exp r / k+ Γ k + Γ k+ F ; k + ; r is the regular spatial signed-rank function Möttönen and Oja, 995. We then obtain see the Appendix for details A m Γ k+ k m Γ k I k, B m Γ k+ mk Γ k F, ; k + ; I m k, [ A T m B m A m F, ; k + ] ; I m k, [ ] ARE m, F, ; k + ; where ARE m is the Pitman efficiency of the generalized spatial signed-rank test with respect to Hotelling s T test. See Table for values of ARE m for selected values of k and m,,...,5. The notation F denotes the Gauss hypergeometric function. 4. Multivariate t distributions Let us now assume that y,...,y n is a random sample from a k-variate t-distribution with ν degrees of freedom, denoted by t k,ν. See the Appendix. The Pitman asymptotic relative efficiency of the generalized signed-rank test with respect to Hotelling s T is then ARE m µν + k kν [ { E q m r 7 r ν + r m }] [ E{q m r}],

10 Table : Asymptotic relative efficiencies of W mn with respect to T in the k-variate standard normal model, for selected values of k and m,,...,5. m k where r /k has a F k,ν distribution. See the Appendix for q m. Tables, 3 and 4 list the Pitman efficiencies of W mn with respect to T for t k,ν distributions with selected dimensions k and degrees of freedom ν, for m,,...,5. Table : Asymptotic relative efficiencies of W mn with respect to T in the case of the k- variate t distribution with ν 3 degrees of freedom, for selected values of k and m,,...,5. m k Multivariate Tukey distributions In this section we consider contaminated multivariate normal distributions, called here multivariate Tukey distributions. We say that the distribution of y is k-variate Tukeyɛ, µ, σ if its p.d.f. is fy ɛφy+ɛσ k φσ y µ, 8

11 Table 3: Asymptotic relative efficiencies of W mn with respect to T in the case of the k- variate t distribution with ν 6 degrees of freedom, for selected values of k and m,,...,5. m k Table 4: Asymptotic relative efficiencies of W mn with respect to T in the case of the k- variate t distribution with ν 0 degrees of freedom, for selected values of k and m,,...,5. m k where φy is the p.d.f. of N0,I k and ɛ [0, ]. This model proves useful in considering the asymptotic efficiency and asymptotic bias in the case that part of the data comes from a population of outliers. The problem is then to estimate the location vector of the population associated with the majority of the data. Now where q m r m h0 qr m h r exp r / ɛ h ɛ m h q k+ Γ k + Γ k+ F 9 r h µ m +σ h ; k + ; r,

12 is as in Section 4.. The optimal score function is then Ly ɛφy+ɛσ k φσ y ɛφy+ɛσ k φσ y y. See the Appendix for details on A m and B m. In Figure we exhibit the Pitman efficiencies of W mn with respect to T under k-variate Tukeyɛ, 0,σ models with various σ, ɛ, and k, for m,,...,5. It is seen that the ARE increases with dimension k and σ and exceeds for m. Figure exhibits the bias of the functional θ m, under k-variate Tukeyɛ, µ,σ models with various µ, σ, ɛ, and k, for m,,...,5. As expected, bias increases with µ. 5 Comparisons and further remarks 5. Trade-offs between efficiency and robustness We see that for low degrees of freedom, say ν 3, the ARE at t distributions becomes decreasing as m increases as soon as the dimension k is greater than. Thus the gain in ARE at the normal for higher values of m is paid for by a decrease in ARE at a heavy-tailed distribution such as the multivariate t with 3 degrees of freedom. For degrees of freedom ν 0, however, the dimension needs to be much higher for the case m to overpower the case m. This approach leads to a broad spectrum of trade-offs between efficiency at the normal and protection against heavier-tailed models. Depending on one s utility for efficiency versus protection, one might choose any one of the cases m,, 3, 4, or 5. Tables 4 provide some guidance for users to make such selections. For example, W mn for m 4 or 5 proves to be very attractive in comparison with the cases m or, in that a significant gain in efficiency at the normal model is achieved while also maintaining good efficiency at heaviertailed t distributions along with a reasonably high breakdown point. 5. Affine invariant/equivariant versions As mentioned earlier, the generalized signed-rank tests and related estimators are not fully affine invariant/equivariant, due to the use of the spatial sign function and spatial quantile function see Serfling, 004, for some discussion. Affine invariant/equivariant tests/estimates, however, may be constructed for the one-sample location problem using the well-known transformation-retransformation technique, as follows. Write C mn n m n i n Sy i y im S T y i y im. i m 0

13 a.3 k, epsilon0. b k, epsilon ARE ARE Sigma Sigma c.4 m m m3 m4 m5 k4, epsilon0. d.8 m m m3 m4 m5 k4, epsilon ARE.. ARE Sigma Sigma m m m3 m4 m5 m m m3 m4 m5 Figure : Asymptotic relative efficiencies of W mn with respect to T under k-variate Tukeyɛ, 0,σ models with various σ, ɛ, and k, for m,,...,5.

14 a 5 k, epsilon0., sigma b 5 k, epsilon0., sigma m5 m m3 bias m5 bias m4 m m3 m m m c 5 mu k4, epsilon0., sigma d 5 mu k4, epsilon0., sigma m5 m m3 bias m5 bias m4 m m3 m m m mu mu Figure : The bias of the functional θ m under k-variate Tukeyɛ, µ,σ models with various µ, σ, ɛ, and k, for m,,...,5.

15 For m, the spatial sign covariance matrix is obtained. See Visuri et al For invariant tests consider a symmetric positive definite k k-matrix V and standardize the observations by setting z i V / y i V / also symmetric. Then calculate C mn for the transformed data, and finally solve C mn V /ki k. The estimate ˆV mn is then unique up to a multiplicative constant and we can make the choice with Trace ˆV mn k. Finally, the invariant test statistic for the location problem is Ŵ mn, calculated from the ˆV / y i. For m, ˆV n is Tyler s scatter matrix estimate 987, and W n was introduced by Randles 000. Finally, for the affine equivariant location estimate, consider the standardized observations z i V / y i θ with location vector and scatter matrix candidates θ and V. Calculate the values of the W mn and C mn statistics for the standardized data and solve simultaneous equations W m,n θ,v0 and C mn θ,v k I k. The location estimate is then the generalized transformation-retransformation Hodges-Lehmann estimate. For m, the transformation-retransformation spatial median is obtained; Tyler s scatter matrix estimate is used in the transformation. See Chakraborty and Chaudhuri 998, Chakraborty, Chaudhuri, and Oja 998, and Hettmansperger and Randles Selected comparisons with other estimators Restricting to elliptically symmetric multivariate distributions, Hallin and Paindaveine 00 develop linear signed-rank type test statistics based on the interdirections of Randles 989. Model-based scores yield locally asymptotically maximin tests, which include the following: V: a new van der Wearden type statistic S: a sign statistic of Randles 989 W: a Wilcoxon statistic of Peters and Randles 990 t 3 : a statistic optimal for t with 3 degrees of freedom t 6 : a statistic optimal for t with 6 degrees of freedom t 5 : a statistic optimal for t with 5 degrees of freedom Let us staying within the elliptically symmetric framework compare V, S, W, t 3, t 6, and t 5 with our generalized spatial signed-rank tests GSSR,..., GSSR 5 corresponding to m,,...,5, respectively. Let us also include an affine invariant signed-rank test HMO of Hettmansperger, Möttönen, and Oja 997. As a quick-and-dirty criterion for comparison, we have compared ARE values for each test under the k-variate normal, t 3, and t 6 models and eliminate less competitive tests. We omit details and just indicate our findings. Interestingly, the comparisons depend upon the dimension k, as follows. 3

16 V, t 5, GSSR, GSSR 3, and GSSR 4 : good for all k HMO: good for k and k 6 GSSR 5 : good for k W: good for k GSSR : good for k 0 t 3 and t 6 : poor for all k Of course, many of these statistics including the GSSR statistics remain defined and attractive under broader distributional assumptions, such as central symmetry. It would be desirable to develop a more extensive comparison study to include other statistics, for example, the simpler, affine-invariant, multivariate, distribution-free sign test of Randles 000, to include other models such as the multivariate Tukey types, and to conclude robustness criteria such as breakdown points and influence functions. This is beyond the scope of the present paper and deferred to future work. 6 Acknowledgements Support of the third author by National Science Foundation Grants DMS and CCF is gratefully acknowledged. References [] Bose, A Bahadur representation of M m estimates. Ann. Statist [] Chakraborty, B., and Chaudhuri, P On an adaptive transformation retransformation estimate of multivariate location. J. Roy. Statist. Soc. B [3] Chakraborty, B., Chaudhuri, P., and Oja, H Operating transformation and retransformation on spatial median and angle test. Statist. Sinica [4] Chaudhuri, P. 99. Multivariate location estimation using extension of R-estimates through U-statistics type approach. Ann. Statist [5] Hallin, M. and Paindaveine, D. 00. Optimal tests for multivariate location based on interdirections and pseudo-mahalanobis ranks. Ann. Statist [6] Hettmansperger, T. P. and McKean, J. W Robust Nonparametric Statistical Methods. Arnold, London. 4

17 [7] Hettmansperger, T. P., M ott onen, J., and Oja, H Affine-invariant multivariate one-sample signed-rank tests J. Amer. Statist. Assoc [8] Hettmansperger, T. P., Nyblom, J. and Oja, H Affine invariant multivariate one-sample sign tests. J. R. Statist. Soc. B [9] Hettmansperger, T. P. and Randles, R. 00. A practical affine equivariant multivariate median. Biometrika [0] Möttönen, J. 00. Efficiency of the spatial rank test. Unpublished manuscript. [] Möttönen, J. and Oja, H Multivariate spatial sign and rank methods. J. Nonpar. Statist [] Möttönen, J., Oja, H. and Tienari, J On the efficiency of multivariate spatial sign and rank tests. Ann. Statist [3] Möttönen, J., Hettmansperger, T. P., Oja, H. and Tienari, J On the efficiency of affine invariant multivariate rank tests. J. Mult. Analy [4] Peters, D. and Randles, R. H A multivariate signed-rank test for the onesampled location problem. J. Amer. Statist. Assoc [5] Randles, R. H A distribution-free multivariate sign test based on interdirections. J. Amer. Statist. Assoc [6] Randles, R.000. A simpler, affine invariant, multivariate, distribution-free sign test. J. Amer. Statist. Assoc [7] Serfling, R Nonparametric multivariate descriptive measures based on spatial quantiles. J. Statist. Plann. and Inf [8] Visuri, S., Koivunen, V. and Oja, H Sign and rank covariance matrices. J. Statist. Plann. and Inf A Asymptotic relative efficiency evaluations for selected models Here we provide some of the basic lemmas underlying the ARE computations given in the paper. Due to space limitations, many details of proof and some further lemmas are omitted. An expanded version is available on the website serfling. Complete details are available in Möttönen 00. 5

18 A. Multivariate normal distribution Lemma 4 Let y,...,y m be independent N k 0,I k distributed random vectors and y 0,...,0,r T a fixed k-variate vector. Then r y s m χ k, m where s y y m. Proof. we have With s N k 0, m I k, [ s y s m m s k m + r s ] k m r m χ k. m Lemma 5 We have where ϱ m r r m ϱ, m ϱr / exp r k+ Γ k + Γ k F ; k ; r. Proof. ϱ m r E{ } r m χ k m m E{ } r χ k m r m ϱ m See Lemma of Möttönen 00. 6

19 Lemma 6 where and qr r exp r / R m ru q m ru r q m r q m k+ Γ k + Γ k+ F ; k + ; r Proof. q m r d dr ϱ mr m ϱ r m m r q m Lemma 7 Eq m rr k+ Γ /m Γ k 7

20 Proof. Eq m rr r E q r m E / r r exp m m / i0 Γ k+ + i Γ k+ + i i i! / Γ k+ + i Γ k+ + i i0 i i! k+ m Γ k k+ m m k+ Γ k k+ m m k+ /m m k+ Γ k+ Γ k i0 m E /+i Γ k+ + i Γ k+ + ii! r +i exp r m r Γ k ++i+i m /+i Γ k Γ k+ + i i i! m i0 k + k + Γ F 0 ; m Γ k+ Γ k m k+ m +k/++i m i Lemma 8 Eqmr Γ k+ mk Γ k F, ; k + ; m 8

21 Proof. Eqmr E q r m Γ k+ + iγ k+ [ + j r i+j+ ] Γ k+ + iγ k+ + j i0 j0 i+j+ i!j! E exp r m m Γ k+ + iγ k+ + j i+j+ Γ k + i + j + i+j+ Γ k+ + iγ k+ + j i0 j0 i+j+ i!j! m Γ k m +k/+i+j+ m Γ k m +k/+ Γ k+ + i + jγ k+ + iγ k+ + j i j Γ k+ + iγ k+ + ji!j! m + m + i0 j0 m Γ k F k + m +k/+ Γ k+, k + m Γ k F k + Γ k+ m kγ k, k + ; k + Γ k+ Γ k+, k + ; Γ k+ m +k/+ k Γ k, k + m+ m k+ m+ Γ k+ m k/ kγ k m k+ Γ k+ mk Γ k F ; k + ; m k/+ F m+ k + m+ m +,, k + m k+ k+ k+, ; k + ; m m + m + k+ ; k + ; m + m F, ; k + ; m k+ 9

22 Lemma 9 A m k Γ k+ m Γ k I k B m Γ k+ mk Γ k F A T mb m A m ARE m [ [ F, ; k + ; F, ; k + ;, ; k + ; ] I k m ] m I m k Proof. A m k Eq mrri k k m B m k Eq mri k Γ k+ mk Γ k F Γ k+ Γ k I k, ; k + ; I m k A T m B m A m k m Γ k+ [ Γ k k m F, ; k + ; kγ k [ Γ k+ ] I m k F, ; k + ; ] I m k ARE δt A T mbm A m δ δ T Σ δ δt A T mbm A m δ δ T Iδ [ ] δ T δ F, ; k+ ; m [ δ T δ F, ; k + ; ] m 0

23 A. Multivariate t distribution Definition 3 If x χ ν, z N k 0,I k, and x and z are independent, then y x/ν / z has k-variate t distribution with ν degrees of freedom y t ν,k. Lemma 0 Let y x /ν / z,...,y m x m /ν / z m be independent t ν,k distributed random vectors where x,...,x m,z,...,z m are independent with x i χ ν and z i N k 0,I k.ify 0,...,0,r T is a fixed k-variate vector and s y y m, then r y s x,...,x m σ χ k σ where σ σ x ν/x ν/x m. Proof. Suppose that x,...,x m are given. Then Es 0 and Vars σ I k where σ x /ν +...+x m /ν. Since the components of s are independent and normally distributed we get the result σ y s s σ s k σ + r s k r χ k σ σ Lemma ϱ m r / Γ k+ i0 + i r Γ k + ii! i i / ] E[ exp r σ σ Proof. E y s x,...,x m E σ i0 χ k r σ r exp r i Γ k+ + i σ σ σ i!γ k + i / i0 / Γ k+ + i r i Γ k + ii! exp r σ ϱ m r E[E y s x,...,x m ] Lemma R m ru q m ru where q m r d dr ϱ mr σ i /

24 A.3 Multivariate Tukey distribution Definition 4 We say that the distribution of y is k-variate T ukeyɛ, µ,σ distribution if the p.d.f. of y is gy ɛfy+ɛσ k fσ y µ, where fy is the p.d.f. of N k 0,I k -distribution and ɛ [0, ]. The c.d.f. of y is then Gy ɛf y+ɛf σ y µ. Lemma 3 Let v Bin,ɛ and x N k 0,I k be independent random variables. Then is T ukeyɛ, µ, σ distributed. y vx + vµ + σx vµ ++σ vx Lemma 4 Let y i v i µ++σ v i x i, i,...,m, be independent T ukeyɛ, µ,σ distributed random vectors. Then the conditional distribution of s m i y i given v,...,v m is N k hµ,a hi k where h m i v i and a h m +σ h. Proof. s m i m v i µ + +σ v i x i Es v,...,v m i m v i µ hµ i Vars v,...,v m m + σ v i I k i m + σ v i +σ vi I k i m + σ v i +σ v i I k i m + σ v i I k i m +σ hi k

25 Lemma 5 Let y 0,...,0,r T and µ 0,...,0,µ k T be fixed k-variate vectors. Then r y s v,...,v m a hχ hµk k ah Proof. a h y s s a h + + s k a h + r s k a h r sk E ah si E ah v,...,v m v,...,v m r hµ k ah 0 when i,...,k r a h y s v,...,v m χ hµk k ah Lemma 6 E y s v,...,v m / Γ k+ Γ k i0 + i exp r hµ k r hµk i ah + ii! a h a h Proof. Using Lemma 4 of Möttönen 00 we get exp r hµ k r hµ k i a E y s v,...,v m ah h a h Γ k+ + i i! Γ k / + i i0 / Γ k+ + i exp r hµ k r hµk i ah + ii! a h a h Lemma 7 i0 Γ k where ϱ m r E y s m m r ɛ h ɛ m h hµk ahϱ h ah h0 ϱr / exp r k+ Γ k + Γ k F ; k ; r 3

26 Proof. Since h Binm,ɛ we have ϱ m r E y s / Γ k+ + i Γ k + ii! E exp r hµ k r hµk i ah a h a h i0 / Γ k+ m + i Γ k + ii! exp r hµ k r hµk i ah a h a h i0 h0 m ɛ h ɛ m h h m m exp r hµ k r hµ k i ɛ h ɛ m h a ah h a h h i! h0 i0 m m r ɛ h ɛ m h hµk ahϱ h ah where h0 ϱr i0 exp r r i Γ k+ + i / i! + i / exp r See Lemma of Möttönen 00. Lemma 8 where Proof. h0 Γ k Γ k+ Γ k F k + ; k ; r m m r q m r ɛ h ɛ m h hµk q h ah qr r exp r / k+ Γ k + Γ k+ F ; k + ; r q m r d dr ϱ mr m m ɛ h ɛ m h ah d r h dr ϱ hµk ah h0 m m r ɛ h ɛ m h hµk q h ah h0 4 Γ k+ Γ k + i / + i

27 where qr / r exp r r See Lemma 3 of Möttönen 00. exp r / i0 Γ k+ + i + ii! r Γ k+ Γ k+ Γ k+ F k + Lemma 9 Let µ 0. The optimal score function is then Ly αy gy y i ; k + ; r where αy ɛfy+ɛσ k fσ y Proof. Ly d dy loggy d dy log ɛfy+ɛσ k fσ y ɛfy+ɛσ k fσ y [ ɛ d dy fy+ɛσ k d dy fσ y] ɛfy+ɛσ k fσ y [ ɛfyy ɛσ k fσ yy] ɛfy+ɛσ k fσ y ɛfy+ɛσ k fσ y y αy gy y Lemma 0 σr E q r ah σ σ + a h / Γ k+ Γ k 5

28 Proof. σr E q r ah [ σr E r exp σ r Γ k+ + i σ r i ] ah a h Γ k+ + ii! a h i0 Γ k+ + i i+ [ ] σ E r i+ exp σ r Γ k+ + ii! i0 ah a h Γ k+ + i i+ σ Γ k + i + i+ Γ k+ + ii! i0 ah Γ k σ a h +k/+i+ σ Γ k+ + i σ ah Γ k σ a h +k/+ i! σ + a h i0 σ Γ k+ k+ i σ i ah Γ k σ a h +k/+ i! σ + a h i0 σ Γ k+ k + σ ah Γ k σ a h +k/+ F 0 ; σ + a h σ ah Γ k σ Γ k+ ah Γ k σ σ + a h Γ k+ σ a h +k/+ / Γ k+ Γ k a h σ + a h σ k+ σ + a h k+ a k+ h σ + a h i See Lemma 8 of Möttönen 00. Lemma Let µ 0. Then A E[R m yl T y] ɛσ Eq m σrr+ ɛeq m rr k Γ k+ m { m ɛ h k Γ k ɛ m h h where r χ k 0. h0 I k ɛ σ + a h + ɛ / + a h / } I k 6

29 Proof. A E[R m yl T y] αy R gy R myy T gydy...dy k k αyr m yy T dy...dy k R k ɛ R m yy T fydy...dy k + ɛσ R m yy T σ k fσ ydy...dy k R k R k ɛ R m yy T fydy...dy k + ɛσ R k R m σyσy T fydy...dy k R k ɛer m yy T +ɛσ ER m σyy T ɛeq m ruru T +ɛσ Eq m σruru T ɛeq m rreuu T +ɛσ Eq m σrreuu T ɛ k Eq mrri k + ɛ k σ Eq m σrri k ɛeq mrr+ɛσ Eq m σrr I k k where r χ k 0. ɛσ Eq m σrr m m σr ɛ h+ ɛ m h E q r h ah σ h0 m m ɛ h+ ɛ m h σ / Γ k+ h σ + a h Γ k σ h0 m / m ɛ h+ ɛ m h Γ k+ h σ + a h Γ k h0 ɛeq m rr m m ɛ h ɛ m h E h h0 m m ɛ h ɛ m h h h0 r q ah +a h r / Γ k+ Γ k 7

30 A m m k+ ɛ h m h Γ ɛ { k h Γ k ɛ h0 Γ k+ m { m ɛ h k Γ k ɛ m h h h0 σ + a h Lemma The p.d.f of r is ft ɛgt+ɛ t σ g σ where gt is the p.d.f. of the χ k 0 distribution. Proof. where u Bin,ɛ and x T x χ k 0. r y T y +σ u x T x +σ ux T x / + ɛ +a h } I k ɛ σ + a h + ɛ / + a h / F t P r t P r t u P u +P r t u 0P u 0 ɛp σ χ k0 t+ ɛp χ k0 t t ɛg + ɛgt σ ft d dt F t ɛ t σ g + ɛgt σ Lemma 3 Let x be a random variable with the p.d.f. ft ɛgt+ɛ t σ g σ where gt is the p.d.f. of the χ k 0 distribution. Then where d> σ. Ex i Γ k exp dx ɛ + ii Γ k d + Γ k + ɛσi ii +k/+i Γ k dσ + k/+i 8 / } I k

31 Proof. Ex i exp dx 0 ɛ ɛ ɛ Lemma 4 When µ 0 Proof. B E[R m yr T my] x i exp dx ɛgx+ɛ σ xσ g dx x i exp dxgxdx + ɛ σ 0 x x i exp dxg dx σ x i exp dxgxdx + ɛ σ i y i exp dσ ygyσ dy σ 0 x i exp dxgxdx + ɛσ i y i exp dσ ygydy Γ k ɛ + ii Γ kd + Γ k + ɛσi ii +k/+i Γ k dσ + k/+i k E[q mr]i k m m m m ɛ h +h ɛ m h h k h h ah ah h 0 h 0 [ ɛ k+ x k+ y k+ + + a h a h k + F, k + ; k + ; x y + x y ɛσ k+ x k+ y k+ σ + σ + a h a h F k +, k + ; k + ; x y x y B E[R m yr T m y] E[q m ruq m ru T ] E[qmr]Euu T k E[q mr]i k 9 0 ] I k Γ k Γ k+ Γ k+

32 m qm r m h 0 h 0 m m h 0 h 0 m m h 0 h 0 m m h h m m h r ah ah exp i0 j0 m i0 h j0 ɛ h +h ɛ m h h q ɛ h +h ɛ m h h h r r ah exp r a h a h r i r Γ k+ + iγ k+ + j Γ k+ + iγ k+ m h + ji!j! a h ɛ h +h ɛ m h h a h ah ah Γ k+ + iγ k+ + j i Γ k+ + iγ k+ + ji!j! a h a h r i+j+ exp a h + r a h r q ah j j Eq mr m m h 0 h 0 m m i0 h j0 h ɛ h +h ɛ m h h Γ k+ + iγ k+ + j + iγ k+ + ji!j! Γ k+ [ ɛ ɛσ i+j+ ah ah i j a h a h Γ k + i + j + i+j+ k/+i+j+ + Γ k ah + ah + Γ k + i + j + ] i+j+ k/+i+j+ Γ k σ ah + σ ah + 30

33 where m Eqm r m h 0 h 0 m m h h [ ɛ k+ + + a h a h i0 j0 σ a h + i0 j0 ɛ h +h ɛ m h h Γ k+ + iγ k+ + jγ k+ ɛσ Γ k+ + iγ k+ + ji!j! k+ σ + a h + i + j x i y j + Γ k+ + iγ k+ + jγ k+ ] + i + j x i y j Γ k+ + iγ k+ + ji!j! ah ah Γ k x y x y a h a h + ah + a h a h + ah + σ a h a h + σ a h + σ a h a h + σ a h + 3

34 m Eqm r m h 0 h 0 m m [ h h ɛ k+ + + a h a h k + F, k + ɛσ σ + a h ɛ h +h ɛ m h h, k + σ +k+ a h k + F, k +, k + Γ k+ k+ Γ Γ k+ ; k +, k + ; x ; y + Γ k+ k+ Γ Γ k+ ; k +, k + ] ; x ; y ah ah Γ k m Eqm r m h 0 h 0 m m [ h h ɛ h +h ɛ m h h ah ah ɛ k+ x k+ y k+ + + a h a h k + F, k + ɛσ σ + a h F k + ; k + ; k+ σ + a h, k + ; k + ; x y + x y x k+ y k+ ] x y x y Γ k Γ k+ Γ k+ 3