The generalized lognormal distribution and the Stieltjes moment problem

THE GENERALIZED LOGNORMAL DISTRIBUTION AND THE STIEL TJES MOMENT PR OBLEM Christian Kleiber This pap er studies a Stieltjes-type momen t problem defined by the generalized lognormal distribution, a heavy-tai led distribution with ap- plications i n economics, finance and related fields. It arises as the dis- tribution of the exp onen tial of a random v ariable following a general- ized error distributi on, and hence figures prominently in the EGAR CH mo del of asset price volatilit y . Compared to the classical lognormal dis- tribution it has an additional shap e parameter. It emerges that momen t (in)determinacy depends on the v alue of this parameter: for some v al- ues, the distribution does not ha v e finite moments of all orders, hence the momen t problem is not of in terest in these cases. F or other v alues, the distribution has moments of all orders, yet it is moment-indeterminate. Finally , a limiting case i s supported on a bounded in terv al, and hence de- termined by its moments. F or those generalized lognormal distributions that are moment-indeter minate Stieltjes classes of moment-equiv alent distributions are presented. Keywords : Generalized error distribution, generalized lognormal dis- tribution, lognormal distribution, momen t problem, size distribution, Stieltjes class, volatilit y mo del. AMS 2010 Ma thema tics Subject Classifica tion: Pri mary 60E05, Secondary 44A60. JEL Classifica tion: C46, C02. 1. INTR ODUCTION The momen t problem asks, for a giv en distribution with distribution function (CDF) F with finite moments m k ( F ) = R ∞ −∞ x k d F ( x ) of all orders k = 1 , 2 , . . . , whether or not F is uniquely determined b y the sequence of these moments. If F is uniquely determined b y this sequence, F or a random v ariable X follo wing this distribution are called momen t- determinate (for brevity , M-det); otherwise F or X are called momen t-indeterminate (M- indet). Cases where the supp ort of the distribution F is the p ositiv e half-axis R + = [0 , ∞ ) are called Stieltjes moment problems, cases where the support is the real line are called Ham burger momen t problems, and cases where th e supp ort is a bounded in terv al are called Hausdorff momen t problems. The probably most widely kno wn example of an M-indeterminate distribution is the log - normal distribution, first describ ed b y Stieltjes ( 1894/1895 ) in a non-probabilistic setting V ersion: June 8, 2018. Correspondence to: Christian Kleib er, F acult y of Business and Economics, Uni- v ersit ¨ at Basel, Peter Merian-W eg 6, 4002 Basel, Switzerland. christian.kleiber@unibas.ch 1 2 C. KLEIBER and further de veloped b y Heyde ( 1963 ). The logno rmal distribution is a basic model for de- scribing size phenomena in economics and related fields (see, e.g., Kleib er and Kotz , 2003 ), including distributions of p ersonal income, actuarial losses, or city sizes. It also arises in mathematical finance in the fundamental geometric Bro wnian motion mo del of asset price dynamics. Giv en the cen tral role of the lognormal distribution in Stieltjes-t yp e momen t problems it is, therefo re, of sp ecial interest to explore closely related distributions with resp ect to M-ind eterminacy . Recen tly , Lin and Stoy anov ( 2009 ) stud ied a generalization of the lognormal di stribution deriv ed from a sk ewed generalization of the normal d istribution, finding that it is M-indeterminate for every v alue of the skewness parameter. The presen t pap er explores a family of generalized lognormal distributions deriv ed from a more clas- sical symmetric generalization of the normal distribution, whic h compared to the normal distribution has an additional shap e parameter. Lik e the classical lognormal distribution, this generalized v ersion has b een employ ed in financial economics as well as in mo deling size distributions. It turns out that this family of distributions sheds new light on the classical lognormal momen t problem, in that M-determinacy no w depends on the v alue of the shap e parameter. Sp ecifically , the family incorp orates heavy-tailed distributions for which not all in teger momen ts exist, mo derately hea vy-tailed distributions for whic h all moments exist y et the distributions are M-indeterminate, and, as a limiting case, a distribution with b ounded supp ort that is, therefore, determined b y its momen ts. It also emerges that the classical lognormal di stribution does not constitute an extreme case within the family: in the setting considered here, there exist more as well as less heavy-tailed M-indet distribut ions than the lognormal. The paper is organized as follo ws: Section 2 pr ovides some bac kground on the generalized lognormal distribution. Section 3 con tains a characterization of moment (in)determinacy for the family of generalized lognormal distributions in terms of their shap e parameter, while Section 4 describ es Stieltjes classes p ertaining to the indeterminate cases. Section 5 concludes. 2. THE GENERALIZED LOGNORMAL DISTRIBUTION Being one of the basic distributions in probabilit y and statistics, the normal distribution has triggered a n umber of generalizations. One suc h generali zation is defined by the density f ( y ) = 1 2 r 1 /r σ Γ(1 + 1 /r ) exp  − 1 r σ r | y − µ | r  , −∞ < y < ∞ , (2.1) whic h includes the normal as the sp ecial case where r = 2. Here µ ∈ R is a lo cation parameter and σ ∈ R + is a scale parameter. The new parameter r ∈ R + is a shap e parameter measuring tail thickn ess, with low er v alues of r indicating hea vier tails. The parameter r pla ys a crucial role b elo w. GENERALIZED LOGNORMAL DISTRIBUTION AND STIEL TJES MOMENT PROBLEM 3 This distribution is fairly widely known; ho w ev er, it is kno wn under differen t names in differen t fields and it was (re)disco vered sev eral times in differen t contexts. Specifically , since r = 2 yiel ds the normal distribution and r = 1 the Laplace distribution, the distribu- tion ( 2.1 ) is known b oth as a generalized normal distribution, in particular in the Italian language literature ( Lunetta , 1963 ; Vianelli , 1963 ), and as a generalized Laplace distribu- tion. It is also known as the normal distribution of order r , again esp ecially in the Italian literature (e.g., Vian elli , 1983 ), and as the generalize d error distribution , not ably in econo- metrics and finance (e.g., Nelson , 1991 ). A further name is exp onen tial p o w er distribution ( Bo x and Tiao , 1973 ), the name under which this distribution is presumably b est known in the statistical literature. T o the b est of the author’s kno wledge, the generalized form ( 2.1 ) was first prop osed in a Russian journ al by Subb otin ( 1923 ), who sought an axiomatic basis for a generalized form of Gauss’s “la w of error.” Hence the name Subb otin distri- bution is also in use, notably in econoph ysics (e.g., Alfarano et al. , 2012 ). A m ultiv ariate generalization of ( 2.1 ) is the Kotz-t yp e distribution ( Kotz , 1975 ). In what follo ws w e sometimes set µ = 0, since in the context of momen t problems no extra generalit y is gained by includ ing this lo cation parameter. There exist differen t parameterizations of ( 2.1 ), notably regarding the scale parameter, but for the purposes of this pap er the relev an t parameter is r , so this complication shall b e ignored b elo w. The gener alize d lo gnormal distribution ( Vianelli , 1982 , 1983 ), or p erhaps lo garithmic gen- er alize d normal distribution , is less widely kno wn than the generalized normal distribution. In fact, most of the curren tly av ailable works are written in Italian and published in Italian journals and collected v olumes that are often not easily av ailable outside of Italy . A more accessible source ma y b e Kleib er and Kotz ( 2003 , Ch. 4.10), who summarize many basic prop erties. The distribution is defined as the distribu tion of X = exp( Y ), where Y follo ws eq. ( 2.1 ), leading to the densit y f ( x ) = 1 2 x r 1 /r σ Γ(1 + 1 /r ) exp  − 1 r σ r | ln x − µ | r  , 0 < x < ∞ . (2.2) If a random v ariable X follows eq. ( 2.2 ) this is denoted as X ∼ GLN( µ, σ , r ). The distribution will sometimes b e referred to as the generalized lognormal distribution of order r if further emphasis is needed. The case where r = 2 gives the classical lognormal distribution. In eq. ( 2.2 ), e µ is a scale parameter, while σ and r are b oth shape parameters. The effect of the new parameter r is illustrated in Figure 1 . This Figure suggests that the densit y b ecomes more and more concen trated on a bounded in terv al with increasing r . Sp ecifically , for r = 1 . 5 the densit y is muc h like the classical lognormal density , but with sligh tly heavi er tails, while for r = 15 several p oin ts of inflect ion and a more rapid decrease in the tails emerge. The limiting case where r → ∞ will also be explored b elo w, see Theorem 3 . 4 C. KLEIBER 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 x Figure 1 .— Some generalized lognormal distributions (solid grey: µ = 0, σ = 1, r = 1 . 5, dashed-dotted grey: µ = 0, σ = 1, r = 15). The dashed black curve corresp onds to the classical lognormal distribution ( r = 2, with µ = 0 , σ = 1). Lik e the classical lognormal distribution, the general ized lognormal distribution has been emplo y ed in economics and finance. As menti oned ab ov e, it has been used as a mo del for the size distribution of p ersonal incomes. In an application to Italian income data, Brunazzo and P ollastri ( 1986 ) estimate r in the vicinit y of 1.45, suggesting a model with ev en hea vier tails than the classical lognormal dist ribution for their data. It wil l emerge b elo w that their estimated mo del is not determined b y its momen ts. P erhaps more prominen tly , the distribution also arises in the widely used exp onen tial GAR CH (EGARCH) mo del of asset return dynamics ( Nelson , 1991 ), where it pro vides a more realistic specification of the inno v ation distr ibution in the v olatility equation than the normal distribu tion. Recall that, in view of the exp onenti al transformation emplo yed in the GENERALIZED LOGNORMAL DISTRIBUTION AND STIEL TJES MOMENT PROBLEM 5 EGAR CH mo del, a widely used alternativ e to the normal distribution in GAR CH mo deling, the t distribution, leads to tails that are to o hea vy , in the sense that the distribution corresp onding to the exp onen tiated random v ariable has no moments of an y order. In con trast, it will emerge b elo w that the less extreme members of the generalized lognorm al distribution p ossess momen ts of all orders, yet they are M-indeterminate. Sp ecifically , all mo dels estimated by Nelson ( 1991 ), with shap e parameters r in the vicinit y of 1.56–1.57, are not determined b y their moments. More recen t w ork (e.g., T a ylor , 2005 ) confirms that 1 < r < 2 is the empirically relev an t range of the tail thickness parameter in this mo del. All of these ob jects are M-indeterminate. 3. GENERALIZED LOGNORMAL DISTRIBUTIONS AND THE MOMENT PROBLEM Ho w can one determine whether or not a giv en distribution with CDF F is determined b y the sequence of its momen ts? Although necessary and sufficien t conditions are known (see, e.g., Shohat and T amarkin , 1950 ), they are not very practical. F or M-determinacy , a sufficien t condition is the existence of the moment generating function (MGF) m X ( t ) = E [ e tX ] = R ∞ 0 e tx d F X ( x ), | t | < t 0 , for some t 0 > 0. F rom the expression for the density ( 2.2 ) of the generalized lognormal distribution it is immediate that, for an y r ∈ R + , E [ e tX ] = ∞ for all t > 0; hence the MGF does not exist. It remains to explore the existence of the momen ts themselves. (Note that in view of X > 0 (a.s.) it is p ossible to consider moments of fractional order.) Without loss of generality , set µ = 0 since exp( µ ) is a scale parameter. Substituting z = ln x yields, for some C > 0, E [ X k ] = Z ∞ 0 x k f ( x ) d x = C Z ∞ −∞ exp { k z − | z | r / ( r σ r ) } d z . (3.1) This sho ws that con verge nce of the integral depends on the v alue of r : for r > 1 the in tegral is finite for all k , for r = 1 the condition | k | < 1 /σ is needed, while for r < 1 it do es not con v erge for any k 6 = 0. The follo wing prop osition collects these observ ations: Pr oposition 1 Supp ose X ∼ GLN ( µ, σ, r ) . (a) The moment-gener ating function of X do es not exist for any r ∈ (0 , ∞ ) . (b) The k th moment E [ X k ] exists if and only if • k = 0 , if r < 1 . • | k | < 1 /σ , if r = 1 . • k ∈ ( −∞ , ∞ ) , if r > 1 . Apart from the in tegral rep resenta tion ( 3.1 ), it is also possible to obtain series expansions of the momen ts (when they exist). F or r > 1, they are of the form 6 C. KLEIBER E [ X k ] = e kµ Γ  1 r  ∞ X i =0 ( k σ ) 2 i (2 i )! r 2 i/r Γ  2 i + 1 r  , k = 0 , 1 , 2 , . . . , see Brunazzo and Po llastri ( 1986 ) or Nelson ( 1991 ) 1 . In view of Prop osition 1 not all generalized lognormal distributions are of in terest in the con text of the momen t problem. F or r = 1, only some moments exist, for r < 1 no momen ts exist. The cases where r < 1 therefore provide examples of distributi ons without an y momen ts, in teger or fractional. An earlier example was given by Kleib er ( 2000 ). F or the remaining cases where 1 < r < ∞ all the moments are finite yet the MGF do es not exist. These are circumstances under whic h M-indeterminacy ma y arise. It remains to sho w that the distributions where 1 < r < ∞ are indeed M-indet. F or M-indeterminacy , a useful sufficien t cond ition is the Krein condition (e.g. Sto y anov , 2000 ). In a Stieltjes-t yp e momen t problem, it requires, for a densit y f that is strictly p ositiv e for all x ≥ a > 0, for some a > 0, that the normalized logarithmic integral of the densit y K S [ f ] = Z ∞ a − ln f ( x 2 ) 1 + x 2 d x (3.2) is finite. K S [ f ] is called the Krein integral of f . The following Theorem shows that generalized lognormal distri butions of orders 1 < r < ∞ are M-indeterminate: Theorem 2 Al l gener alize d lo gnormal distributions GLN ( µ, σ , r ) of or der 1 < r < ∞ ar e M-indeterminate. Pr oof. Setting without loss of generalit y µ = 0 and σ = 1, the Krein integral ( 3.2 ) is, for a > 0 and C r > 0 the normalizing constan t, K S [ f ] = Z ∞ a − ln C r + 2 ln x + 1 r | 2 ln x | r 1 + x 2 d x. Since for large x the in tegrand is ev en tually dominated b y x − 1 − δ , for an y δ ∈ (0 , 1), this in tegral is finite for all 1 < r < ∞ , whic h giv es the result.  Alternativ e pro ofs could emplo y results presen ted by Gut ( 2002 , Remark 6.2) or Pak es et al. ( 2001 , p. 110). 1 It should be noted that these w orks employ differen t parameterizations of the distribution . Also, Nelson ( 1991 ) obtains exp ectations of somewhat more general ob jects. Setting γ = 0, p = 0 and θ = 1 in his Theorem A1.2 yields the required momen ts. The resulting expressions can b e sho wn to coincide with those presen ted b y Brunazzo and Pollastri ( 1986 ). GENERALIZED LOGNORMAL DISTRIBUTION AND STIEL TJES MOMENT PROBLEM 7 F or X i ∼ GLN( µ i , σ i , r i ), i = 1 , 2, with r i > 1 and densities f i it is easily seen that lim x →∞ f 1 ( x ) /f 2 ( x ) = ∞ iff r 1 < r 2 , hence the generalized lognormal distributions are, in a sense, “more M-indeterminate” for smaller r . (Indeed, in view of Prop osition 1 for r = 1 some momen ts no longer exist.) Sp ecifically , the generalized lognormal distributions with 1 < r < 2 are even more extreme than the classical lognormal distribution ( r = 2). Also, the cases where 2 < r < ∞ are less extreme. It is also w orth noting that although the tails of the generalized lognormal distribution become lighter and ligh ter with increasing r , the distribution is M-indet no matter how large r . It is, therefore, natural to ask what happ ens in the limit, i.e., for r → ∞ . The following Theorem addresses this case: Theorem 3 F or r → ∞ , the gener alize d lo gnormal distribution GLN( µ, σ , r ) tends to a distribution supp orte d on a b ounde d interval. Henc e this limiting distribution is M-det. Pr oof. It is con v enien t to analyze the limiting case for the distribution of Y = ln X , i.e., the generalized normal distribution . Without loss of generalit y , set µ = 0 and σ = 1. A random v ariable Y follo wing a generalized normal distribu tion admits the mixture represen tation ( Devro y e , 1986 , p. 175) Y d = U Z (3.3) where U is uniform on [ − 1 , 1] and Z ∼ ( r 1 /r ) W 1 /r with W ∼ Ga(1 + 1 /r, 1), i.e, a gamma distribution with scale 1 and shap e parameter 1 + 1 /r . Hence Z follo ws a g eneralized gamma (GG) distribution, sp ecifically Z ∼ GG( r , r 1 /r , 1 + 1 /r ). The momen ts of Z are (see, e.g., Kleib er and Kotz , 2003 , p. 151) E [ Z k ] = ( r 1 /r ) 1+1 /r Γ(1 + ( k + 1) /r ) Γ(1 + 1 /r ) , k = 1 , 2 , . . . . No w lim r →∞ E [ Z k ] = 1 for all k , and it follo ws that Z = r 1 /r W 1 /r tends to a p oin t mass at 1 by F r ´ ec het-Shohat (e.g., Galam b os , 1995 , p. 81). Thus lim r →∞ Y d = U , and the density of exp( U ) is given b y f ( x ) = 1 2 x , e − 1 ≤ x ≤ e. (3.4) This distribution has compact supp ort, hence it is determined b y its momen ts.  Lunetta ( 1963 ) provides an alternative deriv ation of the limiting distribution of the gener- alized normal distribution that analyzes the limit of its c haracteristic function. Ho w ever, w e 8 C. KLEIBER prefer the approac h in vol ving a mixture representa tion presented here because it motiv ates further questions, on whic h more b elo w. In terestingly , Bomsdorf ( 1977 ) observ ed that a distribution of the t yp e describ ed b y eq. ( 3.4 ) o ccurs as the distribution of prizes in lotteries, hence he calls it the prize c omp e- tition distribution . Among other char acteristics he also provides the MGF of this ob ject. 4. STIEL TJES CLASSES FOR MOMENT-INDETERMINA TE GENERALIZED LOGNORMAL DISTRIBUTIONS The preceding section show ed that generalized lognormal distributions of orders 1 < r < ∞ are M-indeterminate, b y wa y of an existence pro of. T o round off the discussion , this section provides explicit examples of distributions that are equiv alen t, in the sense of ha ving iden tical momen ts of all orders, to these indeterminate distributions. A Stieltjes class – a term coined b y Stoy anov ( 2004 ) – corresponding to a momen t- indeterminate distribution F with density f is a set S ( f , p ) = { f ε ( x ) | f ε ( x ) := f ( x )[1 + ε p ( x )] , x ∈ supp( f ) , ε ∈ [ − 1 , 1] } , where p ( x ) is a perturbation function satisfying − 1 ≤ p ( x ) ≤ 1 and E [ X k p ( X )] = 0 for all k = 0 , 1 , 2 , . . . . It is p ossible to obtain Stieltjes classes for the generalized lognormal distributions of orders 1 < r < ∞ that generalize a recently deriv ed Stieltjes class p ertaining to the classical lognormal distribution. The construction of the required Stieltjes classes in the follo wing Theorem is adapted from a construction presen ted by Sto y ano v and T olmatz ( 2005 , Theorem 3): Theorem 4 Supp ose X ∼ GLN ( µ, σ , r ) with density f r , ( µ, σ , r ) ∈ R × R + × (1 , ∞ ) . (a) The function h r ( x ) =  sin { ( x − 1) 1 / 4 } exp  1 rσ r | ln x − µ | r + ln x − ( x − 1) 1 / 4  , x > 1 , 0 , x ≤ 1 , (4.1) is b ounde d on R + for al l ( µ, σ, r ) ∈ R × R + × (1 , ∞ ) , with E [ X k h r ( X )] = 0 for al l k = 0 , 1 , 2 , . . . . (b) p r := h r /H r , with H r := sup x | h r ( x ) | , defines a p erturb ation c orr esp onding to f r . (c) The family of functions f r,ε ( x ) = f r ( x )[1 + ε p r ( x )] , ε ∈ [ − 1 , 1] , defines a Stieltjes class c omprising distributions whose moments ar e identic al to those of f r for any ε ∈ [ − 1 , 1] . Pr oof. The function h r is contin uous on (1 , ∞ ), with lim x → 1 + h r ( x ) < ∞ and lim x →∞ h r ( x ) = 0, hence h r is b ounded on R + . GENERALIZED LOGNORMAL DISTRIBUTION AND STIEL TJES MOMENT PROBLEM 9 By construction, with C r > 0 the normalizing constan t of f r , Z ∞ 0 x k h r ( x ) f r ( x ) d x = C r Z ∞ 1 x k sin { ( x − 1) 1 / 4 } exp  − ( x − 1) 1 / 4  d x = C r Z ∞ 0 ( x + 1) k sin { x 1 / 4 } exp  − x 1 / 4  d x = C r k X j =0  k j  Z ∞ 0 x k − j sin { x 1 / 4 } exp  − x 1 / 4  d x = 0 for k = 0 , 1 , 2 , . . . in view of Lemma 1 of Stoy anov and T olmatz ( 2005 ) and the fact that Z ∞ 0 x k sin { x 1 / 4 } exp  − x 1 / 4  d x = 0 , k = 0 , 1 , 2 , . . . . This pro v es (a). Since H r := sup x | h r ( x ) | < ∞ we may set p r ( x ) = h r ( x ) /H r , assuring | p r ( x ) | ≤ 1 for all x . This gives (b). Finally , f r,ε ( x ) = f r ( x )[1 + ε p r ( x )] defines a density for an y ε ∈ [ − 1 , 1], whic h giv es (c).  It should b e noted that the construction of Stoy anov and T olmatz ( 2005 ) is somewhat more general, in that the k ernel k ( x ) := ( x − 1) 1 / 4 used here may b e generalized to a three-parameter family of k ernels defined b y k ( x ; ξ , δ, β ) := ( δ x − ξ ) β tan( π β ), where ( ξ , δ, β ) ∈ R + × R + × (0 , 1 / 2). Thus amending the k ernel in this manner defines a four- parameter family of p erturbations p r ( x ; ξ , δ , β ) leading to Stieltjes classes that generalize the three-parameter family of Stieltjes classes for the classical lognormal distribution de- riv ed b y Sto y ano v and T olmatz ( 2005 ). Ho w ev er, the Stieltjes class presen ted ab o ve already pro vides infinitely man y distributions whose moments coincide with those of the generalized lognormal distribution. In ( 4.1 ), the c hoice of β = 1 / 4 was made because it is related to one of the classical examples of an M-indeterminate distribution that dates back to the pioneering w ork of Stieltjes ( 1894/1895 ). Stieltjes considered the case where ξ = 0 and the perturbation h ( x ) = sin( x 1 / 4 ), x > 0, used in the proof of part (a) of Theorem 4 ; it p ertains to a certain generalized gamma distribution. Moreov er, a shift ξ > 0 is needed in ( 4.1 ), as otherwise the resulting ob ject would exhibit a singularit y at the origin, see also the discussion in Sto y ano v and T olmatz ( 2005 , Section 4). 5. FUR THER DISCUSSION AND CONCLUDING REMARKS The pap er exhibited a family of distributions, o ccurring in economics and finance, that generalizes the lognormal distribution, the classical example of a moment-indeterm inate distribution. It emerged that a large subfamily consists of momen t-indeterminate distri- butions, but also that not all mem b ers share this prop ert y of the lognormal, for differen t 10 C. KLEIBER reasons: some tails are so heavy that not enough momen ts exist, while a limiting case corresp onds to a ligh t-tailed distribution with compact support. It ma y , therefore, b e ask ed to what extent it is possible to c haracterize the gener- alized lognormal distributions with r = 1, i.e. the log-Laplace distributions,for which E [ X k ] < ∞ iff | k | < 1 /σ . If one lea ves the classical setting of the momen t problem c haracterizations in terms of certain moments are p ossible. First, Th. 1 of Lin ( 1992 ) im- plies that characterizations in terms of fractional moments are feasible: for a sequence { k n | 0 < k n < 1 /σ ; n ∈ N } of p ositiv e and distinct n um b ers con verging to some k 0 ∈ (0 , 1 /σ ), the sequence { E [ X k n ] | n ∈ N } of fractional momen ts c haracterizes the distribution. Second, observ e that for r = 1 the first momen t exists iff σ < 1. It is well kno wn that existence of the first moment p ermits c haracterization of the underlying distri- bution in terms of the triangular arra y of first momen ts of the asso ciated order statistics, { E [ X k : n ] | k = 1 , 2 , . . . , n ; n ∈ N } , where X 1: n ≤ X 2: n ≤ . . . ≤ X n : n are the order statis- tics in a sample of size n . In fact, certain subsets of this array are already sufficient, see Huang ( 1989 ) f or a review. Suc h c haracterizations are meanin gful in applications to incom e distribution ( Kleib er and Kotz , 2002 ), one of the fields where the generalized lognormal distribution has b een employ ed. Note also that b oth c haracterizations, via fractional mo- men ts as w ell as via momen ts of order statistics, are av ailable for all generalized logn ormal distributions with r > 1 since momen ts of arbitrary order exist in that case. It is natural to ask ab out M-determinacy of the more widely known distribution of ln X , the generalized error or Subbotin distribution ( 2.1 ). This is a Ham burger momen t problem. The answ er is already a v ailable in the literature, although not in a p robabilistic setting: the family of generalized error distributions also admits M-indet examples, namely for r < 1, and a Stieltjes class is given in Shohat and T amarkin ( 1950 , p. 22). It is also kno wn that for some M-determinate distributions pow er tran sformations lead to M-indeterminacy and vice versa (e.g. Sto yano v , 1997 ). The standard example is the general- ized gamma distribution. F or X ∼ GLN( µ, σ, r ), it is easily seen that X p ∼ GLN( pµ, pσ , r ) for all p > 0, showi ng that the distribution is closed under p o w er transformations. Hence this w ell-known prop ert y of the classical lognormal distribution extends to the generali zed v ersion ( 2.2 ). Consequen tly , consideration of p o w er transformations do es not lead to new insigh ts regarding the momen t problem here. Ho w ever , it might b e w orth while to further explore asp ects of the mixture representat ion ( 3.3 ). This representation is a sp ecial case of a general mixture representation for uni- mo dal distributions kno wn as Khinc hine’s theorem. The exp onen tiated version states that exp( Y ) = exp( U Z ), i.e. a random v ariable follo wing a generalized lognormal distribution can be obtained as the exp onen tial of the pro duct of a uniform and a transformed gamma random v ariable. It w ould be in teresting to c haracterize the set of mixing distributions F Z leading to indeterminate log-unimodal distributions. GENERALIZED LOGNORMAL DISTRIBUTION AND STIEL TJES MOMENT PROBLEM 11 REFERENCES Alf arano, S., M. Milako vi ´ c, A. Irle, and J. 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