GENERALIZED FIDUCIAL INFERENCE VIA DISCRETIZATION

Transcription

1 Statistica Sinica ), oi: GENERALIZED FIDUCIAL INFERENCE VIA DISCRETIZATION Jan Hannig The University of North Carolina at Chapel Hill Abstract: In aition to the usual sources of error that have been long stuie by statisticians, many ata sets have been roune off in some manner, either by the measuring evice or storage on a computer. In this paper we investigate theoretical properties of generalize fiucial istribution introuce in Hannig 2009) for iscretize ata. Limit theorems are provie for both fixe sample size with increasing precision of the iscretization, an increasing sample size with fixe precision of the iscretization. The former provies an attractive efinition of generalize fiucial istribution for certain types of exactly observe ata overcoming a previous non-uniqueness ue to Borel paraox. The latter establishes asymptotic correctness of generalize fiucial inference, in the frequentist, repeate sampling sense, for i.i.. iscretize ata uner very mil conitions. Key wors an phrases: Asymptotic properties, Bernstein-von Mises theorem, Dempster-Shafer calculus, generalize fiucial inference. 1. Introuction Fisher 1930) introuce the iea of fiucial probability an fiucial inference as an attempt to overcome what he saw as a serious eficiency of the Bayesian approach to inference the use of a prior istribution on moel parameters even when no prior information is available. In the case of a one-parameter family of istributions, Fisher gave the following efinition for a fiucial ensity rθ) of the parameter base on a single observation x 0 for the case where the istribution function F x θ) is a function of θ ecreasing from 1 to 0: rθ) = F x 0 θ). 1.1) θ Fiucial inference create some controversy once Fisher s contemporaries realize that, unlike earlier simple applications involving a single parameter, fiucial inference often le to proceures that were not exact in the frequentist sense an i not possess other properties claime by Fisher Linley 1958); Zabell 1992)). More positively, Fraser 1968) evelope a rigorous framework for making inferences along the lines of Fisher s fiucial inference assuming that

2 490 JAN HANNIG the statistical moel was couple with an aitional group structure, e.g., the location-scale moel. Wilkinson 1977) argue that for complicate problems the fiucial istribution is not unique an shoul epen on the parameter of interest. Dawi an Stone 1982) provie further insight by stuying certain situations where fiucial inference leas to exact confience statements. Barnar 1995) propose a view of fiucial istributions base on the pivotal approach that seems to eschew some of the problems reporte in earlier literature. Dempster 2008) an Shafer 2011) iscusse Dempster-Shafer calculus, which is closely relate to fiucial inference. An intereste reaer can consult Section 2 of Hannig 2009) for a more thorough iscussion of the history of fiucial inference an a more complete list of references. Tsui an Weerahani 1989) an Weerahani 1993) propose a new approach for constructing hypothesis tests using the concept of generalize P -values an generalize confience intervals. These generalize confience intervals have been foun in many simulation stuies to have goo empirical frequentist properties, see Hannig, Iyer, an Patterson 2006) for references. Hannig, Iyer, an Patterson 2006) establishe a irect connection between fiucial intervals an generalize confience intervals an prove the asymptotic frequentist correctness of such intervals. These ieas were unifie for parametric problems in Hannig 2009) without requiring any group structure relate to the moel. This unification is terme generalize fiucial inference an has been foun to have goo theoretical an empirical properties for a number of practical applications E, Hannig, an Iyer 2008, 2009); Hannig an Lee 2009); Wanler an Hannig 2011, 2012a,b)). Traitionally, the goal of fiucial inference was to formulate clear principles that woul guie a statistician to a unique fiucial istribution. Generalize fiucial inference oes not have such a goal. It treats the mechanics of generalize fiucial inference as a tool to efine a istribution on the parameter space an uses this istribution to propose statistical proceures, e.g. approximate confience intervals. The quality of the propose proceures is then evaluate on their own merit using theoretical large sample properties an simulations. Generalize fiucial inference begins with expressing the relationship between the ata, X, an the parameters, ξ, as X = Gξ, U), 1.2) where G, ) is terme a structural equation, an U is the ranom component of the structural equation, a ranom variable or vector whose istribution is completely known an inepenent of any parameters. We intentionally leave the efinition of the structural equation as general as possible. We offer some comments an suggestions on how to select a structural equation in Section 5.

3 FIDUCIAL INFERENCE VIA DISCRETIZATION 491 A formal efinition of a generalize fiucial istribution will be presente in Section 2. The purpose of the rest of this section is to give the reaer a heuristic unerstaning of the ieas evelope in this manuscript. Let x 0 be the fixe realize value of X generate using some fixe unobserve parameter ξ 0. To explain the iea behin the formal efinition of generalize fiucial istribution, suppose first that the structural relation 1.2) can be inverte an the inverse G 1 x 0, u) always exists. That is, for observe x 0 an all u, there is the unique ξ solving x 0 = Gξ, u). As example of such a situation consier x 0 a sample of size n = 1 from a location parameter family, X = ξ + U, or x 0 a sample of size n = 2 generate using X 1 = ξ 1 + ξ 2 U 1, X 2 = ξ 1 + ξ 2 U 2 with ξ = ξ 1, ξ 2 ) R 2. Since the istribution of U is completely known, one can always generate a ranom sample ũ 1,..., ũ M from it. This ranom sample of U is transforme into a ranom sample { ξ 1 = G 1 x 0, ũ 1 ),..., ξ M = G 1 x 0, ũ M )}, which is calle the fiucial sample. The fiucial sample ξ 1,..., ξ M is a sample from the fiucial istribution an can be use to obtain estimates an approximate confience intervals for ξ. The inverse G 1, ) often oes not exist. This can happen uner two situations: for some value of x 0 an u, either there is more than one ξ, or there is no ξ satisfying x 0 = Gξ, u). The first situation can be ealt with by using the mechanics of Dempster- Shafer calculus Dempster 2008)), see Section 4 of Hannig 2009) for more etail. A more practical solution is to select one of the several solutions using some possibly ranom mechanism. We provie some guiance on how this selection can be mae in Section 2. see also Section 6 of Hannig 2009). In any case, we show in this paper that in many problems of practical interest the metho of selection has only a secon orer effect on statistical inference. For the secon situation, Hannig 2009) suggests removing the values of u for which there is no solution from the sample space an then re-normalizing the probabilities, i.e., using the istribution of U conitional on the event that the there is at least one ξ solving the equation x 0 = Gξ, U). The rationale for this choice is that we know that the observe ata x 0 were generate using some fixe unknown ξ 0 an u 0, i.e., x 0 = Gξ 0, u 0 ). The information that the solution of the equation x 0 = Gξ, U) exists for the true U = u 0 is available to us in aition to knowing the istribution of U. The values of u for which x 0 = G, u) oes not have a solution coul not be the true u 0 hence only the values of u for which there is a solution shoul be consiere in the efinition of the generalize fiucial istribution, which leas to the conitioning. However, the set of u for which the solution exists has probability zero in many practical situations, e.g., most problems involving absolutely continuous ranom variables. Conitioning on such a set of probability zero will therefore lea to non-uniqueness ue to the Borel paraox Casella an Berger 2002, Section 4.9.3).

4 492 JAN HANNIG Careful evaluation of the event there is at least one ξ solving the equation x 0 = Gξ, u) reveals that it has probability zero only if the probability of generating the realize ata is zero. Hence, the Borel paraox is not an issue when efining generalize fiucial istribution for iscrete moels. Taking this observation a step further, notice that any ata that a statistician can come into contact with has been roune off in some manner, e.g., by a measuring instrument or by storage on a computer. Mathematically speaking, we o not know the exact realize value X = x 0. Instea we only observe an occurrence of an event {X A x0 }, for some multivariate interval A x0 = [a, b) containing x 0 an satisfying P ξ0 X A x0 ) > 0, where X is an inepenent copy of X. To emonstrate this, if the exact unobserve value of the ranom vector X were x 0 = π, e, 1.28) an, ue to instrument precision, all the values were roune own to one ecimal place, our observation woul be the knowlege that the event {X [3.1, 3.2) [2.7, 2.8) [1.2, 1.3)} happene. Replacing the event of probability zero {X = x 0 } with the event of positive probability {X A x0 } removes non-uniqueness ue to the Borel paraox in the efinition of generalize fiucial istribution. This is one without any loss of information as only the occurrence of the event {X A x0 } is known to us an the interval A x0 is a member of some fixe partition of R n etermine by the measuring instrument or computer precision. Using the language of σ-algebras, iscretization is accommoate by restriction of the the Borel σ-algebra to a subσ-algebra generate by the countable partition {A i } whose events have positive P ξ0 probability. In situations where exact, non-iscretize ata is available, we propose to efine the generalize fiucial istribution as a limit offering an attractive resolution of the Borel paraox. To this en, we first stuy the limit of the generalize fiucial istribution for a fixe sample size of jointly absolutely continuous ranom variables uner general conitions as the precision of the iscretization increases, b a) 0. We erive the limit in a close form an show that it oes not suffer from non-uniqueness ue to multiple solutions of x 0 = Gξ, u). Inee the limiting istribution is the conitional istribution conitione on the limit of the σ-algebras generate by the iscretizations. Secon, we stuy the limit of the generalize fiucial istribution for i.i.. ata as the sample size goes to infinity an the iscretization of the ata remains fixe. We show that uner very mil conitions the generalize fiucial istribution always leas to asymptotically correct inference. Here we evaluate the quality of an inference proceure in the repeate sample frequentist sense. To o this we effectively prove a Bernstein-von Mises theorem for generalize fiucial istributions an show that the effect of the particular selection of one of the ξ solving Gξ, u) A x0 is of a secon orer as the sample size increases. Our

5 FIDUCIAL INFERENCE VIA DISCRETIZATION 493 result greatly relaxes the conitions uner which the asymptotic correctness of generalize fiucial istribution has been previously prove. The thir source of non-uniqueness in the efinition of the generalize fiucial istribution is ue to the choice of structural equation 1.2). In particular, two ifferent structural equations resulting in the same sampling istribution for ata can lea to a ifferent generalize fiucial istribution. While we o not fully resolve this issue, we offer some practical suggestions an comments on this topic. The rest of the paper is organize as follows. In Section 2 we provie a rigorous efinition of the generalize fiucial istribution. Section 3 stuies the limit of the fiucial istribution as the precision of the ata increases. Section 4 explores large sample asymptotics for the generalize fiucial inference uner the presence of iscretize ata. Thoughts on the choice of structural equation are offere in Section 5. Section 5 conclues. 2. Generalize Fiucial Inference We closely follow the efinition of generalize fiucial istribution foun in Hannig 2009) with a small moification to allow for iscretize ata. In orer to avoi repeating the same arguments, we refer the reaer to Section 4 of Hannig 2009) for a more etaile evelopment. Let X R n be a ranom vector with a istribution inexe by a parameter ξ Ξ. Assume that the ata generating mechanism for X is expresse by 1.2) where G is a jointly measurable function an U is a ranom variable or vector with a completely known istribution inepenent of any parameters. We efine for each measurable set A R n an all u a set-value function QA, u) = {ξ : Gξ, u) A}. 2.1) The function QA, u) is the inverse image of the function G for fixe u. Next, we select a possibly ranom point out of each inverse image QA, u). Following Section 4 of Hannig 2009), let {V S)} S B p be a collection of ranom elements each with support S. Since we will use V QA, u)) in the efinition of the generalize fiucial istribution, the ranom elements {V S)} S B p shoul be selecte to be as uninformative as possible. A goo efault choice is a selection that maximizes the ispersion of the parameters of interest. For example if S = a, b) R, V a, b)) selects one of the enpoint a, b at ranom maximizing the variance of V a, b)), or if S is a polyheron an the parameters of interest are a subset of all parameters, V S) first projects the polyheron on the subspace of the parameters of interest an then selects one of the vertices of the projection at ranom maximizing the eterminant of the relevant covariance matrix. Simulation stuies in Section 6 of Hannig 2009) an E, Hannig, an Iyer

6 494 JAN HANNIG 2009) examine the effects of the choice of {V S)} S B p on frequentist behavior of generalize fiucial istribution arriving to a similar recommenation. Assume that our ata were generate by 1.2) using some true unknown parameter value ξ 0 an, instea of observing the exact realize value X = x 0, we only observe the event that the sample values lie in some measurable set {X A x0 }, where x 0 A x0 is a member of a partition of R n. In aition to the information given to us by observing {X A x0 }, we also know that the true values of ξ 0 an u 0 satisfy Gξ 0, u 0 ) A x0. Using the argument immeiately preceing Equation 4.3) in Hannig 2009), we efine a generalize fiucial istribution for ξ as V QA x0, U )) {QA x0, U ) }, 2.2) where U is an inepenent copy of U. The conitional istribution in 2.2) is well-efine provie that P QA x0, U ) ) > 0, which is the case as soon as P X = Gξ 0, U ) A x0 ) > 0, since {u : Gξ 0, u) A x0 } {u : QA x0, u) }. Otherwise, aitional care is neee to interpret the conitional istribution. We provie such an interpretation in Section 3. A generalize fiucial istribution for a subset θ of the parameter vector ξ is obtaine through marginalization of the istributions in 2.2) Hannig 2009, Equation 4.4)). Finally notice that for exactly observe ata we have A x0 = {x 0 } an 2.2) is the same as 4.3) in Hannig 2009). We prove that the effect of the particular choice of {V S)} S B p isappears asymptotically. In orer to simplify some of the notation in the proofs we moify the generalize fiucial istribution by having it efine as a probability istribution on the set of all subsets 2 Θ ; QA x0, U ) {QA x0, U ) }. 2.3) The object efine in 2.3) is a ranom set of parameters such as an interval or a polygon) with istribution conitione on the set being non-empty. If there is no anger of misunerstaning, we call the moifie generalize fiucial istribution of 2.3) also a generalize fiucial istribution. Examples 1 an 2 of Section 4 of Hannig 2009) provie simple illustrations of the efinition of generalize fiucial istribution for exactly observe ata. An example provies a slightly more complicate illustration of the efinition of a generalize fiucial istribution for iscretize continuous ata. Example 1. Suppose U = U 1,..., U n ), where U i are i.i.. N0, 1) an X = X 1,..., X n ) = Gµ, U) = µ + U 1,..., µ + U n )

7 FIDUCIAL INFERENCE VIA DISCRETIZATION 495 for some µ R, so the X i s are i.i.. Nµ, 1). We observe a iscretize realization of X, i.e., the event {X a, b)}, where a = a 1,..., a n ), b = b 1,..., b n ) an a, b) is an n-imensional cube etermine by the way the ata is roune off at the measuring evice. If n = 1 then Qa, b), u) = a u, b u) an P ξ0 Qa, b), U ) ) = 1. Thus following 2.3), the moifie generalize fiucial istribution is the istribution of the ranom interval a U, b U ) where U N0, 1), inepenent of the ata. If n > 1, take La, u) = max i {a i u i } an Rb, u) = min j {b j u j }. The inverse image is { La, u), Rb, u)) if La, u) < Rb, u), Qa, b), u) = otherwise. Using Φx) an φx) for the istribution function an ensity of N0, 1), respectively, we compute for constants l, r, P La, U ) l, r < Rb, U )) = P a i l Ui < b i r, for all i) n = Φb i r) Φa i l)) + 2.4) Notice that the probability in 2.4) is not zero if an only if b i r > a i l, for all i = 1,..., n, which is equivalent to > r l with = min i {b i a i }. The joint ensity La, U ), Rb, U ) is compute by taking erivatives, an the moifie generalize fiucial istribution for µ, 2.3), is the istribution of the ranom interval L, R), where the joint ensity f LR l, r) is 0 i j +l l i j i=1 φa i l)φb j r) ) k / {i,j} Φb k r) Φa k l)) φa i l )φb j r ) k / {i,j} Φb k r ) Φa k l )) I {l<r<l+ } ). r l The generalize fiucial istribution for µ, 2.2), is obtaine by selecting a point insie of the interval [L, R]. A reasonable efault is to take V L, R)) = L with probability.5 an V L, R)) = R with probability.5, inepenently of everything else. 3. Increasing Precision Asymptotics In this section we iscuss the behavior of the generalize fiucial istribution as we increase the precision of the measurements. This provies a efinition of the generalize fiucial istribution for exactly observe observations. Such

8 496 JAN HANNIG asymptotic consierations are not relevant for iscrete istributions, an therefore we turn our attention to istributions that are absolutely continuous with respect to Lebesgue measure. Let us now state the assumptions of Theorem 1; these are weaker than the assumptions state in Section 4.1 of Hannig 2009). In particular the current assumptions apply to a wier selection of moels than just the i.i.. sequences covere in Hannig 2009). For examples, see the en of this section. Assume that the realize value of X, generate using some true unknown parameter value ξ 0, is x 0. Suppose that the parameter of interest ξ 0 Ξ R p is p-imensional. Recall 1.2), an assume that U R n is an absolutely continuous ranom vector with a joint ensity f U u), efine with respect to Lebesgue measure on R n, continuous on its support. Write G = g 1,..., g n ) so that X i = g i ξ, U) for i = 1,..., n. Assume that for each fixe ξ Ξ the function Gξ, ) is one-to-one an continuously ifferentiable, enoting the inverse by G 1 x, ξ). Using the Jacobian transformation, the ensity of X is f X x ξ) = f U G 1 x, ξ) ) ) et x G 1 x, ξ). 3.1) For all p-tuples of inexes i = 1 i 1 < < i p n) {1,..., n} we enote the list of unuse inexes by i = {1,..., n} \ i, the collection of variables inexe by i by x i = x i1,..., x ip ), an its complement by x i = x i : i i ). Assume that there is an open neighborhoo Bx 0 ) an a measurable sets U i, P U i U) > 0, such that, for all x = x 1,..., x n ) Bx 0 ) an for all p-tuples of inexes i, the function G 1 x i, ), ), viewe as a function of ξ an x i, is one-to-one an ifferentiable onto U i. Thus, the ensity of ξ, X i ) is f ξ, x ξxi i x i ) = f U G 1 x, ξ) ) ) et ξ, x i ) G 1 x, ξ). 3.2) Here ξ,x i )G 1 x, ξ) stans for the Jacobian matrix compute with respect to all parameters ξ an all observations x i. It follows that for any fixe ξ the function f ξ, x ξxi i x i ) is continuous in x = x i, x i ). Assume aitionally that the marginal ensity Ξ f ξx ξ, x i i x i ) ξ is continuous in x = x i, x i ). Finally, consier a sequence of iscretizations of x 0 = x 1,0,..., x n,0 ). In particular, for each m = 1, 2,..., each coorinate x 0,i a i,m, b i,m ) for all i = 1,..., n. Let a m = a 1,m,..., a n,m ), b m = b 1,m,..., b n,m ), an assume that for all m = 1, 2,... the probability P ξ0 X a m, b m )) > 0, so that the conitional istributions in 2.3) are uniquely efine. Theorem 1. Uner the assumptions of this section, consier a sequence of p- imensional intervals a 1, b 1 ) a 2, b 2 ) an numbers c m such that

9 FIDUCIAL INFERENCE VIA DISCRETIZATION 497 m a m, b m ) = {x 0 } an c m b m,i a m,i ) w i > 0 for all i = 1,..., n. Then the moifie generalize fiucial istribution Qa m, b m ), U ) {Qa m, b m ), U ) } 3.3) converges weakly to a singleton that has an absolutely continuous istribution with ensity f X x 0 ξ)jx 0, ξ) rξ) = Ξ f Xx 0 ξ )Jx 0, ξ ) ξ, 3.4) where f X x 0 ξ) is the likelihoo function an 1 et Jx, ξ) = w i1... w ip i=i 1,...,i p ) 1 i 1 < <i p n ξ,x i )G 1 x, ξ) et x G 1 x, ξ) ) ). 3.5) As iscusse in Hannig 2009), Section 4, Remark 1, there are three sources of non-uniqueness in the efinition of the fiucial istribution: the choice of structural equation, the Borel paraox if P ξ0 QA, U ) ) = 0, an the choice of a particular value in QA, U ) if it contains more than one element. Theorem 1 gives a reasonable, consistent way of resolving the non-uniqueness ue to the last two issues for large class of moels. In particular, the limit of conitional istributions 3.3) contains only one element with probability 1, an the non-uniqueness ue to the choice of a particular value in QA, U ) is therefore not present in the limit. Moreover, since lim m a m, b m ) = {x 0 }, the limiting probability ensity 3.4) can be taken as an appealing implementation of the conitional istribution 2.2) with A = {x 0 }, resolving non-uniqueness ue to the Borel paraox. Finally, the w i are fully etermine by the relative limiting size of the iscretization. For example we have w 1 = = w n = 1 if the observe ata is iscretize to the same precision an recore on the same scale, such as in the case of i.i.. observations measure by the same instrument. A proposition shows that the limiting generalize fiucial istribution in 3.4) an 3.5) is invariant uner smooth reparametrizations. This is a esirable property similar to the invariance of the posterior compute using the Jeffreys prior. Proposition 1. Let ξ = ϕη) be a one-to-one, continuously ifferentiable function onto the parameter space Ξ. Let rξ) be the generalize fiucial istribution compute from X = Gξ, U) using 3.4) an 3.5), an rη) the generalize fiucial istribution compute from X = Gϕη), U) using 3.4) an 3.5). Then for each measurable set A Ξ rξ) ξ = rη) η. A ϕ 1 A)

10 498 JAN HANNIG Proof. The multivariate chain rule reveals, after simplification of the secon eterminant that ) ) ) et η, x i ) G 1 x, ϕη)) = et ϕη), x i ) G 1 x, ϕη)) et η ϕη). By the usual Jacobian transformation ) et ξ,x f X x 0 ξ) i )G 1 x, ξ) A et x G 1 x, ξ) ) ξ ) et ϕη),x = f X x 0 ϕη)) i )G 1 x, ϕη)) ) ϕ 1 A) et x G 1 x, ϕη)) ) et η ϕη) η ) et η,x = f X x 0 ϕη)) i )G 1 x, ϕη)) ϕ 1 A) et x G 1 x, ϕη)) ) η. The statement now follows by simple algebra. If the observations are from an i.i.. univariate absolutely continuous istribution, we can choose a particular structural equation 1.2) that recovers Fisher s original efinition of fiucial istribution. To this en, with F x, ξ) an fx, ξ) the istribution an ensity functions, respectively, efine the usual pseuo-inverse F 1 ξ, u) = inf x {F x, ξ) u) an use the structural equation X i = F 1 ξ, U i ), i = 1,..., n, 3.6) where U i are i.i.. U0, 1). If, aitionally, the assumptions of Theorem 1 are satisfie, the inverse of the structural equation u = G 1 x, ξ) is u i = F x i, ξ), i = 1,..., n, an the generalize fiucial istribution is 3.4), with 3.5) simplifie to et ξ F xi1, ξ),..., F x ip, ξ) )) Jx, ξ) = fx i1, ξ) fx i,p, ξ). 3.7) i=i 1,...,i p ) 1 i 1 < <i p n If n = p = 1, 3.4) an 3.7) become 1.1). Similarly if n p = 1, 3.4) an 3.7) agree with the proposal of Dempster 1963). We remark that 3.4) an 3.5) agree with, valiate, an generalize a heuristically motivate proposal of Hannig 2009), Section 4.1, which uses a particularly

11 FIDUCIAL INFERENCE VIA DISCRETIZATION 499 simple iea to implement the conitional istribution in the efinition of generalize fiucial istribution. This is an unexpecte result because the heuristically motivate proposal is base on picking p equations at ranom, solving for the parameters using the selecte equations, an conitioning on the rest of the equations, while the result presente here is a consequence of the geometry of the ranom sets use in the efinition of the generalize fiucial istribution. Hannig 2009) relates, in Section 4.2, 3.4) an 3.5) to Linley 1958), see also Dempster 1963). It is also of interest that, in the same section, Hannig 2009) notices that the function Jx, ξ) can be viewe as a U-statistic estimator of πξ) = E ξ0 JX, ξ), where X is an inepenent copy of the ata, giving the generalize fiucial istribution an empirical Bayes interpretation. We remark that the Jx, ξ) of 3.7) is relate to the ata epenent priors propose by Fraser, Rei, Marras, an Yi 2010). Consier a matrix V x, ξ) = ξ F x 1,ξ) fx 1,ξ). ξ F x n,ξ) fx n,ξ) The solution in 3.7), erive as the limit of the generalize fiucial istribution for iscretize ata, obtains the ata epenent efault prior Jx, ξ) as a sum of all possible absolute values of eterminants of p p matrices obtaine by selecting p rows from V x, ξ). Alternatively, Fraser, Rei, Marras, an Yi 2010) consier their ata epenent prior as etax, ξ)v x, ξ)) q, where q > 0 an Ax, ξ) is a suitable matrix. Fraser, Rei, Marras, an Yi 2010) propose several choices of Ax, ξ) but, as a reasonable efault motivate by maximum likelihoo ieas, recommen q = 1 an Ax, ξ) = 2 ξξ lx, ξ)) 1/2 2 ξxlx, ξ), where lx, ξ) is the log likelihoo of the ata. The rawback of this proposal is that it requires the existence of a secon erivative of the log-likelihoo. If the log-likelihoo is not ifferentiable, they recommen q = 1/2 an Ax, ξ) = V x, ξ). They o not provie any simulation stuies that woul exhibit small sample performance. To conclue this section we consier two examples. Example 2. Let X 1,..., X n be i.i.. Uθ, θ 2 ) ranom variables, θ > 1. Using the inverse istribution function for a structural equation we get. X i = θθ 1)U i + θ, i = 1,..., n with U i i.i.. U0, 1). Using the limit in 3.3) an 3.7) we get the generalize fiucial ensity rθ) I 1/2 x n),x 1)) θ) n θθ 1)) n i=1 x i2θ 1) nθ 2, 3.8) θθ 1)

12 500 JAN HANNIG where the first term on the right sie of 3.8) is the likelihoo an the secon term is the Jacobian factor in 3.7). We performe a limite simulation stuy to valiate the frequentist performance of the confience intervals base on the generalize fiucial istribution 3.8). We use θ = 1.01, 1.5, 2, 10, 50, 250 an sample sizes n = 1, 2, 3, 5, 10, 20, 100. The simulation results show that confience intervals base on the generalize fiucial istribution have nearly exact frequentist coverage for all parameter combinations an all confience levels. Moreover, the expecte length of the propose 95% equal taile confience intervals base on 3.8) was slightly shorter than the 95% intervals base on the reference prior solution of Berger, Bernaro, an Sun 2009) an the proposal of Fraser, Rei, Marras, an Yi 2010). The etails of the simulation stuy are available from the author upon request. Example 3. Consier the Gaussian AR1) moel. The usual moel formulation X i = ax i 1 + Z i, with Z i i.i.. N0, σ 2 ), can be reexpresse as the structural equation i X i = a i x 0 + σ a i j U j, i = 1,..., n, j=1 with parameters ξ = a, σ, x 0 ) an ranom component U = U 1,..., U n ), where the U i are i.i.. N0, 1). The inverse of the structural equation u = G 1 x, ξ) is u i = x i ax i 1, i = 1,..., n. σ The generalize fiucial istribution in 3.4) is n rξ) 2πσ 2 ) n/2 i=1 exp x i ax i 1 ) 2 ) 2σ 2 Jx, ξ), with Jx, ξ) given by 3.5). To compute Jx, ξ), notice that the Jacobian matrix in the enominator is triangular an therefore the Jacobian is ) et x G 1 x, ξ) = σ n. The Jacobians in the numerator are more complicate but careful algebra reveals that if i = i, j, k), 1 i < j < k n, ) et ξ, x i ) G 1 x, ξ) = k i 1 l=1 1) i+l 1 a l+i 1 x max{j l,i} x min{k,k+j l i} x j x k l ) σ n+1.

13 FIDUCIAL INFERENCE VIA DISCRETIZATION 501 Recall that ξ,x i )G 1 x, ξ) stans for the Jacobian matrix compute by taking erivatives with respect to the parameters a, σ, x 0 ) an x l, l i, j, k. Again, w i = 1 an Jx, ξ) is obtaine by simple algebra. Numerical stuies reveal an interesting property of the generalize fiucial istribution for the Gaussian AR1) moel. If the observe X i are stationary, the marginal generalize fiucial istribution for a, σ) is bimoal with one moe near the true values a 0, σ 0 ) an the other near a 1 0, σ 0 a 0 1 ). The existence an location of the secon moe is intriguing, given that the secon moe is near the parameters of the same time series run backwars in time X i = X n i. To explain this, recall that the istribution of a stationary Gaussian AR1) series is the same as the istribution of the time reverse stationary time series, as both have the same covariance function. The existence of the two moes in the generalize fiucial ensity therefore correctly reflects the fact that we cannot istinguish causal an non-causal stationary time series base on observations only. Since the time series is stationary, we might feel at the first glance that this non-causal bump is superfluous. However, the irection of the time series is important for preicting the starting value ˆX an the joint generalize fiucial istribution correctly recognizes the non-ientifiability in the time irection. This is all the more exciting because the likelihoo function itself has only one moe near the causal values, a 0 < 1. We remark that base on our simulations, if the true a 0 < 1 an we assume that the starting value ˆX is observe, the corresponing generalize fiucial istribution oes not have the secon non-causal moe. Similarly, if the observe time series is far from stationary, both the likelihoo an the marginal generalize fiucial istribution for a, σ) is unimoal with moe near the true value a 0, σ 0 ), regarless of whether the time series is causal or not. 4. Increasing Sample Size Asymptotics In this section we look at the behavior of the generalize fiucial istribution for i.i.. ranom variables as the number of observations increases an observational iscretization remains fixe. The conitions state here are weak an easy to verify. They are formulate in terms of the istribution function, an only the existence an continuity of the first partial erivative with respect to the parameters is assume. Also, unlike in Section 3 where only the intervals incluing the fixe realize ata x 0 are consiere, here we are investigating repeate sampling performance an nee to know all the members of the partition iscretizing the real line. Assume the structural equation 3.6), X i = F 1 ξ, U i ), i = 1,..., n,

14 502 JAN HANNIG where the X i are ranom variables, ξ Ξ R p is a p-imensional parameter, the U i are i.i.. U0, 1) an F 1 ξ, u) = inf x {F x, ξ) u). We choose this structural equation, because it fits naturally into the structure of our proof an oes not require introuction of aitional assumptions. If another structural equation generating the same sampling istribution of the ata were chosen, aitional assumptions on this structural equation woul be require. Assume that R is partitione into the fixe intervals, a 1 ], a 1, a 2 ],..., a k, ) with a 0 =, a k+1 =. The values of X i are observe only up to the resolution of the gri, i.e., we observe k = k 1,..., k n ) so that x i a ki, a ki +1], or equivalently x a k, a k+1 ] with a k = a k1,..., a kn ). Assume that k p, p j ξ) = P X a j, a j+1 ]) > 0 for all j = 0,..., k an all ξ. Assume F x, ξ) is continuously ifferentiable in ξ for all x {a 1,..., a k }. Aitionally, assume for all j = j 1 < < j p ) {1,..., k} that F a j1, ξ),..., F a jp, ξ)) = u 1,..., u p ) as a function in ξ with u 1 < < u p, is one-to-one an the Jacobian ) F aj1, ξ),..., F a jp, ξ)) et 0. ξ Finally, let R ξ be a ranom variable having the generalize fiucial istribution 2.2). Theorem 2. Uner the assumptions of this section, R ξ has an asymptotically normal istribution an any confience set base on R ξ of a shape satisfying Assumption 3 of Hannig 2009) has asymptotically correct coverage regarless of the choice of V ). The proof of the theorem is relegate to Appenix B. We remark that, as a consequence of this result, the o-not-know probability in Dempster-Shafer calculus Dempster 2008)) vanishes an oes not influence inference for large n. The nee to properly account for uncertainty ue to iscretization of observations moele by continuous ranom variables is of particular importance in the fiel of metrology. The problem of inference for the mean of iscretize normal ata has obtaine some attention in the last ecae. Frenkel an Kirkup 2005) an Corero, Seckmeyer, an Labbe 2006) propose a maximum likelihoo base approach, Willink 2007) use an a-hoc moification of the sample mean, an Hannig, Iyer, an Wang 2007) propose a generalize fiucial solution. Witkowsky an Wimmer 2009) report a thorough simulation stuy comparing the coverage an expecte length of approximate confience intervals for various sample sizes an levels of iscretization. They compare approximate confience intervals base on the stanar Stuent t, asymptotic istribution of the maximum likelihoo, the proposal of Willink 2007), an the generalize

15 FIDUCIAL INFERENCE VIA DISCRETIZATION 503 fiucial istribution. Among them, the Stuent t interval ignores the iscretization while the rest accounts for it. Witkowsky an Wimmer 2009) report that the Stuent t intervals work poorly if the iscretization is coarse; this is not surprising as it ignores the iscretization. The maximum likelihoo base confience intervals are reporte not to maintain the state coverage for small sample sizes making them unreliable. Both Willink 2007) an the generalize fiucial solution performe aequately in terms of maintaining the state coverage with the generalize fiucial intervals having uniformly shorter average length than the interval of Willink 2007). Hannig, Iyer, an Wang 2007) report a smaller simulation stuy that also shows goo small sample performance of the generalize fiucial inference base intervals. Theoreom 2 therefore complements the goo small sample properties by proviing the necessary theoretical backing for the use of the generalize fiucial inference in practice. 5. Comments on the Choice of Structural Equation The efinition 1.2) is kept very general in orer to make it applicable to many statistical moels. This also means that the same sampling istribution can be generate by ifferent structural equations. For example, it is well-known that one can always fin a function G so that the ranom vector X = Gξ, U) where U is a single U0, 1). However, such a choice can lea to generalize fiucial istributions that are mathematically an computationally intractable. If X is absolutely continuous, we recommen choosing a structural equation so that the limiting generalize fiucial istribution 3.4) an 3.5) in Theorem 1 can be use. In particular the imension of X shoul be the same as the imension of the ranom vector U, the inverse of the function u = G 1 x, ξ) shoul exist an be continuously ifferentiable in ξ an x, an the istribution of U shoul be absolutely continuous with a known simple ensity, e.g., U = U 1,..., U n ) with U i i.i.. U0, 1) or N0, 1). This recommenation is base on the fact that we fin these assumptions necessary to erive a tractable expression for the generalize fiucial istribution an to prove its asymptotic properties, c.f., Section 5 of Hannig 2009). Moreover, Proposition 1 establishes that the generalize fiucial istribution given by 3.4) an 3.5) is invariant uner one-to-one continuously ifferentiable reparametrizations of the parameter vector ξ. Ientifiability consierations imply that the structural equation shoul be chosen so that, for all isjoint A 1, A 2 an all u, the sets QA 1, u) an QA 2, u) at 2.1) are isjoint. If X = X 1,..., X n ) are i.i.. with a istribution function ifferentiable in the parameter vector ξ, an the number of ata points n is much larger than the number of parameters p, then the result in Section 4 together with its proof

16 504 JAN HANNIG strongly suggest using a structural equation base on the inverse istribution function 3.6) as a reasonable efault. Aitionally, if X is absolutely continuous the generalize fiucial istribution obtaine 3.4) an 3.7) coul be viewe as a irect generalization of Fisher s original efinition 1.1). It is known that ifferent structural equations generating ata with the same sampling istribution can lea to ifferent generalize fiucial istributions. We assert that this non-uniqueness is to be welcome rather than eschewe. Wilkinson 1977) argues that the fiucial istribution shoul epen on the choice of parameter of interest, an the same argument has also been mae in connection with the choice of an objective prior in Bayesian inference Efron 1986); Berger, Bernaro, an Sun 2009)). Similarly, we conjecture that any general theory on the choice of the structural equation cannot ignore the parameter of interest. We emonstrate this conjecture with an example. Example 4. Consier the sequence of inepenent ranom variables X i Nµ i, 1), i = 1,..., n. The parameter of interest is θ = n ) 1/2. i=1 µ2 i The nuisance parameter is a point on the unit sphere η = µ/θ, where µ = µ 1,..., µ n ) = θη. The naïve structural equation X i = µ i + Z i, Z i i.i.. N0,1), i = 1,..., n, 5.1) leas to the fiucial istribution that is the same as the Bayesian posterior compute with respect to the flat prior that is known to have exact frequentist properties for every iniviual µ i but very ba frequentist properties for the parameter of interest θ. Guie by our interest in θ, we propose another structural equation that isolates θ in a part of the structural equation. Write X = X 1,..., X n ). We moel ) 1/2 an X/ X separately. First, X = F 1 n X = n i=1 X2 i θ, U 1 ), where an U 1 U0, 1) an Fn 1 is the square root of the inverse of the non-central chi-square istribution function with n egrees of freeom an non-centrality parameter θ 2 /2. Next, X/ X = η H n θ, U 1, U 2 ), where H n θ, U 1, U 2 ) is an appropriate function generating X/ X if η = 1, 0..., 0) were the truth, is the rotation group operator on the unit sphere an U 2 U0, 1) inepenent of U 1. By combining these two expressions we get the structural equation X = F 1 n θ, U 1 ) η H n θ, U 1, U 2 )). 5.2) Notice, that for any observe x 0 an any fixe u 1, u 2 0, 1), there is unique θ, η solving 5.2). Moreover, θ is the solution to x = Fn 1 θ, u 1 ) only, an the resulting generalize fiucial istribution is base entirely on the non-central chi-square portion of the structural equation. It is well-known that the generalize fiucial istribution for θ, erive from the structural equation

17 FIDUCIAL INFERENCE VIA DISCRETIZATION 505 x = Fn 1 θ, U1 ), leas to confience intervals with very goo frequentist properties, see for example the Example 5 of Hannig, Iyer, an Patterson 2006). We conclue that if θ is the parameter of interest, the structural equation 5.2) is preferable to the naïve structural equation 5.1). A similar issue arises in the objective Bayes literature. If the parameter of interest is θ the efault prior is πµ) = µ p 1) = θ p 1) an not the naïve efault prior πµ) = 1 Stein 1985); Tibshirani 1989)). It woul be esirable if one coul start with the naïve structural equation an obtain the correct structural equation by some well efine process. Search for such a process is a topic of our future research. Some promising ieas in that irection can be foun in Zhang an Liu 2011). 6. Conclusions In this paper we have stuie asymptotic properties of generalize fiucial istribution for iscretely observe ata. The use of iscretize ata is natural because all ata is iscretize ue to instrument precision an computer storage. The limiting istribution of the generalize fiucial istribution of iscretely observe ata as the precision of the iscretization increases is obtaine an use to resolve an ambiguity in the efinition of generalize fiucial istribution for exactly observe ata. We also show that, uner some mil conitions on the parametric moel, the generalize fiucial istribution for iscretize ata leas to asymptotically correct inference. This paper i not eal with the computational issues surrouning generalize fiucial inference. Typically, a numerical scheme, such as MCMC or Sequential MC, nees to be employe. For example, Hannig, Iyer, an Wang 2007) implement a moifie Gibbs sampler an show that generalize fiucial inference for iscretize normal ata inee has very goo small sample statistical properties. More complicate computational schemes for generalize fiucial inference can be also foun in Hannig an Lee 2009), Wanler an Hannig 2011), an elsewhere. Finally, we remark that there are interesting connections between generalize fiucial istribution, the asymptotic theory of likelihoo Davison, Fraser, an Rei 2006), the theory of secon orer ancillaries Fraser, Fraser, an Staicu 2010) an as iscusse in Section 3, the ata epenent prior of Fraser, Rei, Marras, an Yi 2010). We plan to investigate these connections in future work. Acknowlegement The author thanks anonymous referees for comments that greatly improve the manuscript. We also thank Jessi Cisewski, Hari Iyer, Thomas Lee, Fuchang Gao an Marco Ferreira for helpful conversations an suggestions. Part of this

18 506 JAN HANNIG work has been carrie out while the author was visiting the Royal University of Phnom Penh, Camboia. The author is thankful to Herb Clemens an the National Research Council for funing this trip. Jan Hannig s research was supporte in part by the National Science Founation uner Grant No an Appenix A: Proof of Theorem 1. The exact form of the generalize fiucial istribution 3.3) appears to be rather ifficult to erive explicitly. Fortunately, one can fin an explicit formula for the istribution of certain extremal points of the set Qa, b), U ). We restrict our attention to x Bx 0 ). The assumptions guarantee that for every u U i an x i the function Q i x i, u) = ξ if G i ξ, u) = x i, u) is a homeomorphism an, for each fixe x i, the function Q i x i, ) is continuous. Moreover, for any a, b), Qa, b), u) = Q i a i, b i ), u). i Thus any point on the bounary of Qa, b), u) is also on the bounary of Q i a i, b i ), u) for some i. Let C i enote the set of 2 p vertices of a i, b i ), Ca, b), u) = i Q ic i, u) an Q E a, b), u) = Ca, b), u) Q[a, b], u). Because of our assumptions on uniqueness of inverses, the set Q E a, b), u) = if an only if Qa, b), u) =. Moreover, the points in Q E a, b), u) lie on the bounary of Qa, b), u). In fact, Q E a, b), u) coul be viewe as the set of extremal points of Qa, b), u). Let = { 1,..., p } R p be a collection of orthonormal basis vectors. Take the furthest point in Q E a, b), u) along the irection 1. If there are ties, select among the tie points the one furthest along 2, etc. Eventually a unique value in Q E a, b), u) is selecte. We enote it by Q a, b), u). Similarly, for each i consier the furthest point in Q i C i, u) along an enote the vertex in C i that maps to this extreme c i u). is well efine. Moreover, for each fixe u, the function Q i Lemma A.1. Uner the assumptions of Theorem 1 the istribution of Q a, b), U ) {Qa, b), U ) } is absolutely continuous with ensity r ξ) f ξ, s ξxi i a i,b i ) i s i )I {c i G 1 s i,s i ),ξ))=s i} s i, s i C i where f ξxi ξ, s i s i ) is given by 3.2). A.1)

19 FIDUCIAL INFERENCE VIA DISCRETIZATION 507 Proof. The conitional istribution A.1) is well-efine, because the conition P Qa, b), U ) ) P ξ0 X a, b)) > 0. The assumptions imply Q a, b), U ) is equal to exactly one of the c i U ) with probability 1. Denote by Y i u, s i ) the unique solution ξ, s i ) to the equation s i, s i ) = G ξ, u). By assumptions, Y i U, s i ) is a ranom variable with ensity given by 3.2). Compute ) P {Q a, b), U ) z} {Qa, b), U ) } = P {c i U ) = s i } {Y i U, s i ), z) a i, b i )}) i s i C i = f ξ, s ξxi i,z) a i,b i s i )I {c i G 1 s i,s i ) i ),ξ))=s i} ξs i s i C i The last step follows from 3.2) an the fact that for each s i, there is a one-to-one map between s i, ξ) an u. The proof now follows by ifferentiation. Proof of Theorem 1. The assumptions of the theorem guarantee that C i {x}. Thus Qa m, b m ), U ), if non empty, converges to a singleton. To fin the istribution of the limit it is enough to fin the limiting istribution of Q a m, b m ), U ) {Qa m, b m ), U ) } for any fixe. Fix ξ an recall that x 0, the observe value of our ata, is also fixe. The continuity of f ξxi ξ, y i y i ) implies that lim sup m y [a m,b m ] an a simple calculation shows that for each i, c n p m s i C i f ξxi ξ, y i y i ) f ξxi ξ, x 0,i x 0,i ) = 0, a i,b i ) f ξxi ξ, s i s i )I {c i G 1 s i,s i ),ξ))=s i} s i f ξxi ξ, x 0,i x 0,i ) j i w j. Similarly the assumption on the continuity of the integral implies f ξ, s ξxi Ξ s i C a i i,b i ) i s i )I {c i G 1 s i,s i ),ξ))=s i} s i ξ f ξ, x ξxi 0,i x 0,i ) w j ξ. j i c n p m Ξ

20 508 JAN HANNIG The statement of the Theorem follows immeiately. Appenix B: Proof of Theorem 2 Let S j, j = 0,..., k, be the number of observations in a j a j+1 ]. The istribution of S is multinomialn, p 0 ξ 0 ),..., p k ξ 0 )). Just as in Appenix A, let = { 1,..., p } R p be a collection of orthonormal basis vectors. For j {1,..., k} an t {0, 1} p, efine c j,t u j) as the the vertex in a i+t 1, a i+t ) that maps to the furthest point in Q j {a j+t 1, a j+t }, u j ) along. Lemma B.1. Uner the assumptions of Theorem 2 the istribution of A.1) is absolutely continuous with ensity r ξ) 2π/n)k p)/2 Γ k i=0 S i) k i=0 ΓS i) Ja j, ξ) I {c j,t G 1 j t {0,1} p k p i ξ) S i 1 i=0 j a j,ξ))=a j } j j+t 1 n 1 S j j {0,...,k}\j+t 1 where the Jacobian Jx 1,..., x p ), ξ) = et F x 1, ξ),..., F x p, ξ)) ξ. ) p j ξ), B.1) Proof. If F F 1 ξ, u), ξ) = u for each fixe ξ, then the lemma follows immeiately from a calculation analogous to the proof of Lemma A.1 by simply rearranging the non-zero terms an multiplying both numerator an enominator by a suitable constant. Otherwise, we can fin F s, ξ) so that F a i, ξ) = F a, ξ) for all i = 1,..., k an F F 1 ξ, u), ξ) = u. This is achieve by reistributing jumps continuously over the intervals a i, a i+1 ). Define X i = F 1 U i, ξ) an enote the corresponing inverse 2.1) by Q. For a, b {a 0,..., a k+1 }, the inverse Qa, b), u) = Qa, b), u). Since we only observe X a, b), the generalize fiucial istributions 2.3) compute base on the structural equation for X an X are the same. Proof Theorem 2. We prove the theorem in two steps. First, we prove Bersteinvon Mises for some special points in Qa, b), u) an verify the conitions of Theorem 1 of Hannig 2009) for them. We only nee to verify Assumptions 1 an 2, as Assumption 3, relate to the shape of the confience set, is assume. Secon, we show that the same is true for all the other points in Qa, b), u). Take p = p 0 ξ 0 ),..., p k ξ 0 )) an Σ as the variance matrix of the Multinomial 1, p) istribution. By the Skorokho Representation Theorem we can fin S Multinomialn, p) an H Normal0, Σ) such that S = np + n 1/2 H + o as n 1/2 ),

21 FIDUCIAL INFERENCE VIA DISCRETIZATION 509 n. Recall that S 0 = n k j=1 S j, p 0 ξ) = 1 k j=1 p jξ), an H 0 = k j=1 H j. Let R ξ have the generalize fiucial istribution given by B.1). The ensity of n 1/2 R ξ ξ 0) is gz) = r ξ 0 + n 1/2 z)n p/2. We investigate the behavior of gz) as n. Set g 2,j ξ) = Ja j, ξ) I {c j,t G 1 t {0,1} p j a j+t 1,a j+t ),ξ))=a j } ) n 1 S j p j ξ). j j+t 1 j {0,...,n}\j+t 1 By our assumptions, g 2,j ξ 0 + n 1/2 z) g 2,j ξ 0 ) a.s.. Compute, using Taylor series an Stirling s formula, logg 1 z)) = log n p/2 2π/n) k p)/2 Γ k i=1 S i) k i=1 ΓS i) = p k 2 log2π) S j logn 1 S j ) By our assumptions ) k p i ξ 0 + n 1/2 z) S i 1 i=0 k logn 1 S j ) k k S j logp j ξ 0 +n 1/2 z)) logp j ξ 0 +n 1/2 z))+o as 1).B.2) 1 2 k logn 1 S j ) k logp j ξ 0 + n 1/2 z)) 1 2 k logp j ξ 0 )) a.s.. Using S j = np j ξ 0 ) + n 1/2 H j + o as n 1/2 ), we compute k S j logn 1 S j ) = = k S j logp j ξ 0 )) + n 1 S j p j ξ 0 ) 1 n 1 S j p j ξ 0 ) ) 2 + oas n )) 1 p j ξ 0 ) 2 p j ξ 0 ) k S j logp j ξ 0 )) k H 2 j p j ξ 0 ) + o as1).

22 510 JAN HANNIG Using p j ξ 0 + n 1/2 z) = p j ξ 0 ) + n 1/2 p j ξ 0 ) z + on 1/2 ), we analogously compute k S j logp j ξ 0 + n 1/2 z)) = k S j logp j ξ 0 )) + k H j p j ξ 0 ) z) p j ξ 0 ) 1 p j ξ 0 ) z) 2 + o as 1). 2 p j ξ 0 ) By plugging back into B.2) we get exp ) k p jξ 0 ) z H j ) 2 /2p j ξ 0 ) g 1 z) k ) 1/2 2π) p/2 p jξ 0 ) a.s.. B.3) Denote the function on the right sie of B.3) by ñz). We show this function is, up to a constant, a ensity of a non-egenerate, multivariate normal istribution. The ranom vector H = H 1,..., H k ) is a non-egenarate Normal0, Σ). Define the p k Jacobi matrix ) pj ξ 0 ) A = ξ r, r=1,...,p, j=1,...,k the iagonal k k-matrix D = iagp 1 ξ 0 ),..., p k ξ 0 )) 1, an the p p-matrix V = A D + 1 ) k j=1 p jξ 0 )) A. By our assumptions, A is full rank an V is non-singular, hence positive efinite. A simple multiplication reveals that Σ 1 = D + 1 k j=1 p jξ 0 )) 1 1 1, so that V = A Σ 1 A. Recall that properties of multinomial istribution imply et Σ = k i=0 p iξ 0 ). After some slightly teious algebra we obtain that the function nz) = Cñz), C = et V k 1/2 1 p j ξ 0 )) exp 2 H Σ 1 Σ 1 A V 1 A Σ 1) ) H, is the ensity of a multivariate normal istribution with mean V 1 A Σ 1 H an variance matrix V 1. In particular, if k = p, et A = Ja 1,..., a p ), ξ 0 ), an consequently C = Ja 1,..., a p ), ξ 0 ). Thus g 1 z)ja 1,..., a p ), ξ 0 + n 3/2 z) nz) a.s.. B.4) However, since the function g 1 z)ja 1,..., a p ), ξ 0 +n z) is a transformation of Dirichlet ensity, it integrates to 1. Hence the convergence in B.4) is also in L 1.