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 Applied Mathematics, 2013, 4, 1340-1346 http://dx.doi.org/10.4236/am.2013.49181 Published Online September 2013 (http://www.scirp.org/journal/am) Randomly Weighted Averages on Order Statistics Homei Hajir, Hasanzadeh Leila, Mina Ghasemi Department of Statistics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Email: homei@tabrizu.ac.ir. Received July 23, 2012; revised January 4, 2013; accepted January 11, 2013 Copyright © 2013 Homei Hajir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We study a well-known problem concerning a random variable uniformly distributed between two independent random variables. Two different extensions, randomly weighted average on independent random variables and randomly weighted average on order statistics, have been introduced for this problem. For the second method, two-sided power random variables have been defined. By using classic method and power technical method, we study some properties for these random variables. Keywords: Two-Sided Power; Moment; Weighted Averages; Power Distribution 1. Introduction Van Asch [1] introduced the notion of a random variable Z uniformly distributed between two independent ran- dom variables X1 and X2 which arose in studying the dis- tribution of products of random 2 × 2 matrices for sto- chastic search of global maxima. By letting X1 and X2 to have identical distribution, he derived that: 1) for X1 and X2 on [−1, 1], Z is uniform on [−1, 1] if and only if X1 and X2 have an Arcsine distribution; and 2) Z possesses the same distribution as X1 and X2 if and only if X1 and X2 are degenerated or have a Cauchy distribution. Soltani and Homei [2] following Johnson and Kotz [3] extended Van Asch’s results. They put 1,,nXX to be inde- pendent, and considered 112211, 2nnnSRXRXRX RXn−−=+ +++≥nnU. where random proportions ()( )1, 1,,1,iiiRU Uin−=−= − ()( )1101, ,,nni niRRU−==− are order statistics from a uniform distribution on [0, 1], and ()0. These random proportions are uniformly distributed over the unit simplex. They employed Stielt- jes transform and that: 1) n possesses the same distri- bution as 10U=S,,nXX if and only if 1,,nXX are de- generated or have a Cauchy distribution; and 2) Van Asch’s result for Arcsine holds for Z only. In this paper, we introduce two families of distribu- tions, suggested by an anonymous referee of the article, to whom the author expresses his deepest gratitude. We say that Z1 is a random variable between two independ- ent random variables with power distribution, if the con- ditionally distribution of Z1 given at 112, 2XxX x== is ()()()11 21211221,12121121max,1,1max,,nZxx znzxxzx xzxxxFzx xzxxxzx≥−<<− = −−<<−≥,.x (1.1) The distribution ()11 2,Zx xFz2 will be said to follow a conditionally directed power distribution, When n is an integer. For n = 1, the distribution given by (1.1) simpli- fies to the distribution Z that was introduced before. Also we used Stieltjes methods, for more on the Stieltjes transform, see Zayed [4]. For n = 2, we call Z1 directed triangular random vari- able. For further generalizing Van Asch results, we in- troduce a seemingly more natural conditionally power distribution. We call Z2 two-sided power (TSP) random variable if the conditionally distribution of Z2 given at 112,XxX x== is ()21 2211,2111,.0.nZxx zzyzy2Fyzyyyzy≥−=<−≤< (1.2) Copyright © 2013 SciRes. AM H. HAJIR ET AL. 1341The distribution 212,Zxx will be said to follow a con- ditionally undirected power distribution, when y1 = min F()11min,2yxx=, (),212maxyxx= and n is an integer. Again for n = 1, the distribution given by (1.1) simpli- fies to the distribution Z that was introduced by Van Asch. The main aim of this article is providing a generalization of notion to the results of Van Asch for some other values of n (other than n = 1). This article is organized as follows. We introduce preliminaries and previous works in Section 2. In Section 3, we give some Characterizations for dis- tribution of Z1 given in (1.1), when n = 2. In Section 4, we find distribution of Z2 given in (1.2) by direct and power method, and give some examples and Characterizations of such distributions by use of Soltani and Homei’ results [5]. 2. Preliminaries and Previous Works In this section, we first review some results of Van Asch [1] and then modify them a little Bit to fit in our frame- work, to be introduced in the forthcoming sections. Us- ing the Heaviside function , we conclude that for any given distinct values X1 and X2 the conditional distribution ()()0, 0,1, 0Uxxx x=<=≥()11 2,Zx xFz in (1.1) is () ()()11 211,2112121.nZxxinizxFz Uzxxxnzx Uzxixx=−=−− −−− − (2.1) Lemma 2.1. For distinct real’s x1, x2, z and integer n, we have ()( )()() ()()()()111212 2121211d11!d1.nnnnzxx xnzxx xxxzxz−−−−+⋅−− −−−=−−1 Proof. It follows from the Leibniz formula. Let , where is an interval, and is the set of all real functions f that are - ()hDIα∈)II⊆(DααTimes differentiable on I. If () ()1gz zx=−. for some constants c and {}1, ,k∈n. Then ()()()() (){}() (){}11ddddd.dkkkkkkPMgzhz kgzhzgzPzzhzgzz−−=− +−= We use the Leibnitz formula for the th deriva- tive of a product, namely (1k−)() (){}() ()1110dd1dd.ddkkikikikiihzgzzkNhzizz−−−−−=−== Let () (){}() (){}11ddd .dddkkkkhzgzhzgzM Nzzz−−==+ where () ()1111101dd,ddikikikiikMhz gzizz+−−−+−−=− =  () ()101ddddikikikiikNhzizz−−−=−gz = . Since () ()1d.drr!rgzzzx+=−r It follows that () ()()11ddddrrrrgzrgz gzzz−−=. Consequently, ()() (){}11ddkkN kgzhzgzPz−−=− where after some al- gebraic work () ()dkPMgz hz=− dkz. Therefore, () () () (){}11dd .ddkkkkMNgzhzkhzgzzz−−+= + This completes the proof. Another tool for proving our main theorem is the fol- lowing formula taken from the Schwartz Distribution theory, namely, ()[]() () ()1dd!dnnnnxxnxϕϕ∞∞−∞ −∞−Λ= Λxx (2.2) where is a distribution Function and is the n-th distributional derivative of . Λ[]nΛΛThe conditional distribution ()11 2,ZxxFz given by (1.1) leads us to a linear functional on complex Valued func- tions f: , defined on the set of real numbers : →()()()()( )()11 2,12121 211d .d!ZxxninniniFffxifxzxx nixx−−==−−−− It easily follows that () () ()11 211 2112,, .ZxxZxx Zxx,Faf bgaFfbFg+= + (2.3) For any complex-valued functions f, g and complex constants a, b. We note that () ()11 211 2,,,zZxx ZxxFzF f= Whenever ()()(nz)fxzxUzx=− − and ()gz ()()()( )()11 2,12121 211d .d!ZxxninzzniniifxFf fxzxx nixx−−==−−−−Copyright © 2013 SciRes. AM H. HAJIR ET AL. 1342 Also we note that ()() ()1d.!dnnznUz xfxnx−−= Thus ()()()()()111 2212,1dd,iZXiZxx iPZzUz xFxFzFx=≤= −=∏ can be viewed as: () () ()()()111 222,11dd!dd.innzZnzXiZxx ifxFxnxFfFx=−∏= (2.4) Therefore by using (2.3) along with (2.4) and a standard argument in the integration theory, we obtain that () () ()()()111222,11dd!dd.innZnXiZxx ifxFxnxFfFx=−=∏ (2.5) For any infinitely differentiable functions f for which the corresponding integrals are finite. Now (2.5) together with (2.2) lead us to ()()() ()()111 222,1dind.ZXiZxx ifxFxFfF x==∏ (2.6) For the above mentioned functions f, where 1()nZF is the (n)-th distributional derivative of the distribution of Z1. Let us denote the Stieltjes transform of a distribution H by ()()()1,d. SHzH xzx=−For every z in the set of complex numbers which does not belong to the support of H, i.e., . ()suppHcz∈The following lemma indicates how the Stieljes trans-form of Z1 and X1, X2 are related. Lemma 2.2. Let Z1 be a random variables that satisfies (1.1). Suppose that the random variables X1 and X2 are independent and continuous with distribution functions 1XF and 2XF respectively. Then ()()()()()()1111,,suppH .nnZXXcSFz SFzSFznz−=−∈2,, Proof. It follows from (2.6) that ()()()()111 222,1,indZzXZxx iSFzFgF x==∏i. And ()()()111 222,11d ,d!d inZzXZx xniSF zFgFxnz ==∏i for ()1zgx zx=−. Now, it follows that ()()()()11 2,11221 2111d1.d!zZxxninniiFgzxzzxxx nixx−−=−=− −−−−ni And by using Lemma 2.1, we have () ()()()11 2,121.nzZxx nFgzxzx−=−− Therefore, ()()()()()12211211d ,d!dinnZXnniSF zFxnzzx zx=−=−−∏,i and ()()()()()()1111,,suppH .nnZXXcSFz SFzSFznz−=−∈2,,)2, (2.7) This finishes the proof. Note that Van Asch’s lemma is the case of n = 1: ()()(11,,ZXXSF zSFzSFz′−= . We also note that the Stieltjes transform of Cauchy distribution, i.e., ()1,SFz zc=+ satisfies (2.7). 3. Characterizations Now, we apply Lemma 2.2 for some characterizations, when X1 and X2 are not identically distributed. Theorem 3.1. Let X1 and X2 be independent random variables and Z be a randomly weighted average given in (1.1). For n = 2 we have, a) if X1 has uniform distribution on [−1, 1], then Z1 has semicircle distribution on [−1, 1] if and only if X2 has Arcsin distribution on [−1, 1]; b) if X1 has uniform distribution on [−1, 1], then Z1 has power semicircle distribution on [−1, 1] if and only if X2 has power semicircle distribution i.e., ()()231 , 114zfzz−=−≤≤. c) if X1 has Beta (1,1) distribution on [0, 1], then Z1 has Beta 33,22 distribution if and only if X2 has Beta Copyright © 2013 SciRes. AM H. HAJIR ET AL. 134311,22 distribution; d) if X1 has uniform distribution on [0, 1], then Z1 has Beta (2, 2) distribution if and only if X2 has Beta (2, 2) distribution. Proof. 1) For the “if” part we note that the random variable X1 has uniform distribution and X2 has Arcsin distribution on [−1, 1]; then ()()11,ln1ln2XSF zzz=+−−1. And ()221,.1XSF zz=− From Lemma 2.2 and substituting the corresponding Stieltjes transforms of distributions, we get ()()13222,.1ZSFzz′′ =− The solution ()()12,2 1.ZSF zzz=−− Which is the Stieltjes transform of the semicircle dis- tribution on [−1, 1]. For the “only if” part we assume that the random variable Z1 has semicircle distribution. Then it follows from Lemma 2.2 that ()()2232211,11XSF zzz−=−−. The proof is completed. 2) By an argument similar to that given in 1) and solving the following differential equations, ()()()()()12,26lnln161ZSFzzz zzzzz′′−=−−−−−3.+ (for the “if” part), and ()()()22S,6 lnln16X3Fzzzzzz=− −−+−. (for the “only if” part). The proof can be completed. 3) By Lemma (2.2), we have ()()()11,,211ZSFz zz zz−′′−=−− (for the“if” part), and ()()()()211,111XSF zzz zz zz−−=−−−, (for the “only if ” part). The proof can be completed by solving the above dif-ferential equations. 4) By Lemma (2.2), we have ()()()()()12,26lnln161ZSFzzz zzzzz′′−=−−−−−3+ (for the “if” part), and ()()()22S,6 lnln16X3Fzzzzzz=− −−+− (for the “only if ” part). Solving the differential equations, can complete the proof. 4. TSP Random Variables In Section 3, we used a powerful method, based on the use of Stieltjes transforms, to obtain the distribution of z1 given in (1.1). It seems that one can not use that method to find distribution of z2 given in (1.2). So we employ a direct method to find the distribution of z2. Let us follow Lemma 4.1 to find a simple method to get the distribu- tion of z2 following [2] and the work of them leads us to the following lemma. Lemma 4.1. Suppose W has a power distribution with parameter n, n ≥ 1, n is an integer, and let 212=1,,n()11min ,yXX=, (),yX2max X where XX random variables are. Let independent()121XYWYY=+−. Then 1) X is a TSP random variable. 2) X can be equivalently defined by ()12 121122XXX WXX=++−− . Proof. 1) ()( )()()()12 121 112,112121,.Xx xn2FzPYWYYzXxXxzyPy Wyyzyy=+ −≤==−=+ −≤=− Proof. 2) () ()11221212, min,,max,,XXx XxUxxxx== ()() ()[]1212 12min ,0,1 ,max ,min ,XxxWUxx xx−=− and also ()12 1212min ,,2xxxxxx +−−= ()12 1212max ,.2xxxxxx ++−= Copyright © 2013 SciRes. AM H. HAJIR ET AL. 1344 then 12121222,xxxxXWxx−+−−=− so ()12 1211.22Xxx Wxx=++−− 4.1. Moments of TSP Random Variables The following theorem provides equivalent conditions For . 2Theorem 4.1.1. Suppose that z2 is a TSP random vari- ablesatisfying (1.2). If X1 and X2 are random variables and kXEzμ′=kiEx =∞, for all integers k then 1, 2i=()1) ()()()()20111kiEz nEy ynk ki−==Γ++ Γ−+12ikikkkin−Γ+ Γ+. 2) ()211122ki iikikkEzEWEX XX Xi−− =−+   212+. 3) ()()21ikkiknEzE yyyini−=−+ 21. Proof. 1) By using Lemma 2.1, we obtain that ()()()()()()()()()()()210210120111.11kikiki iikikiki iikikiikEWWEY YikEWWEY Yikkinnnk ki−−=−−=−==−=−Γ+ Γ−+=Γ++ Γ−+Eyy Proof. 2) This can be easily proved by Lemma 4.1 2). Proof. 3) ()()()()()( )()21 2112101210kkkiikiikiikiiEzYW YYkEWyyyikEWE yyyi−=−==+ −=−=− ()()21ikkiknEzE yyyini−=+ 21−. Let us consider expectation and variance of z2. First, we suppose that 11EYμ=, 22EYμ=12σ, , , and . Then 211VarYσ=2Y=Var22σ()12,YY =Cov1221nEZ nμμ+=+, and also, if then 120EXEX==() ()21 21nE ZEYEYEYn=+ −+1. By 1212XXYY+=+. We have () ()()21 11211nnEZ EYEYEYnn−=+−=++1. (4.2) It can easily follow from (4.2) that the Arcsin result of Van Asch [1] is only true for n = 1, about the variance, we have ( )()()()()( )222221221 122Var121.12Znnnnnnnμμσσ σ−++++ +=++ Following the computation of expectation and vari- ance, we evaluate them for some well-known distribu- tions. If X1 and X2 have standard normal distributions, then from Theorem 3.1.1 2) and the fact that 12XX− and 12XX+ are independent, it follows that their first, second and third order moments are equal, respectively, to 2111πnEZ n−=+, ()( )222212nnEZ nn++=++, and ()()()3232151213 303122πnnnEZ nnn++−=+++. Also, in case X1 and X2 have uniform distributions, Theorem 4.1.1 2) implies that, ()()()()()()2012112kkikkinEznnkkiki=Γ+Γ−+=Γ++Γ−++ +1. ()22131nEZ n+=+, and ()()3222136Var 18 11nnnZnn+++=++2. Theorem 4.1.2. Suppose that z2 is a TSP random variable satisfying (4.1), then 1) z2 is location invariant; 2) If X1 and X2 have symmetric distribution around μ, then z2 has symmetric distribution around μ, only when n = 1. Proof. 1) Is immediate. 2) We can assume without loss of generality that 0μ= If Z2 has a symmetric distribution around zero, then Copyright © 2013 SciRes. AM H. HAJIR ET AL. 1345() (121121dYWYY YWYY+−=−+−). We note that () ()()1211 21dYWYYYWY Y+−=−+−−−. Since ()(1212min ,max,)XXX−=−X11d−, XX=− , And 22dXX=− , we have () (12121dYWYYYWYY+−=+−)2. (4.3) By equating the conditional distributions given at 11Xx= and 22Xx= in (3.3), we conclude that n = 1 It can also easily follow from Theorem (4.1.1) that the Cauchy result of Van Asch [1] is true only for n = 1. 4.2. Distributions of TSP Random Variables In this subsection, we investigate computing distributions by the direct method. We will give two examples of derivation based on (4.1). This method may be compli- cated in some cases, but we have chosen some easy to find examples. We use randomly weighted average on order statistics to find the distribution of z2. Gauss hyper geometric function (,,;)Fabcz which is a well-known special function that we used in this way. Example 4.2.1. Let X1, X2 and W be independent ran- dom variables such that X1 and X2 are uniformly distrib- uted over [0, 1], and W has a power function distribution with parameter. We find the value (2;)Zfzn by means of ()2z;ZWwf therefore ()()220,21 1.1ZWzwzzwwfzwzw<<=−<<−) (4.4) By using the distribution of W, the density function 2(;Zfzn , can be expressed in terms of the Gauss hyper geometric function (,,;)Fabcz which is a well-known special function. Indeed according to Euler’s formula, the Gauss hyper geometric function assumes the integral representation ()()() ()()( )1110,,;11cb abF abczctt tzbcb−− −−Γ=−ΓΓ−d,t− where a, b, c are parameters subject to , , whenever they are real and z is the variable. a−∞ <<∞0cb>>()()()( )21;2121 1,,1,1Znnfznnz zzzFnnn−=−+−+−where n > 0 and . There are some important func-tions as a Gauss hyper geometric function. 1n≠,z) (4.5) () (log11,1; 2;zzFZ+= −. elim ,;;. zbzFabb→∞=a() ()1, 1; 1;.azFaz−−= When n = 1 similar calculations lead to the following distribution ()( )()2log12log, 01Z( )21fzz−zzz=−−−<1,1d, ,,xbaxIabtttabBab−−=−. 5. Conclusion And by using partial fractional rule, we have We have described how directed methods could be used for obtaining the distributions, Characterizations and properties of the random mixture of variables defined in (1.1). The TSP random variable when X1 and X2 have uniform distributions, led us to a new family of distribu- tion which can be regarded as some generalization of “uniformly randomly modified tine”. The proposed model in the direct method can easily lead to distribution generalizations, though this is not possible for the Stielt- jes method, but here the characteristics can be easily computed. ()()()()()()21 2,22 2121221121.zZxxFgzx zxzxzxx x=+−−−− − Therefore, ()()()()()()()222212221122111,221 d.iZXiiSFz zxzxFxzx zxxx =′′′−=−−+−−−∏ 6. Acknowledgements And ()()()(12 121,,,2,2ZXX XXSFz SFzSFzSFFz′′′′ ′−=+),. The author is deeply grateful to the anonymous referee for reading the original manuscript very carefully and for making valuable suggestions. This finishes the proof. It is worth mentioning that the present method yields other extensions too; the following is such an example. REFERENCES Example 4.2.3. Suppose that X1, X2 W are independent random variables. If X1 and X2 have Uniform distributions on [0, 1] and W has Beta (2, 2) distribution, then z2 has the same distribution as W. [1] W. Van Asch, “A Random Variable Uniformly Distrib- uted between Two Independent Random Variables,” Sankhaya, Vol. 49, No. 2, 1987, pp. 207-211. doi:10.1080/00031305.1990.10475730 If the product moments of order statistics are known, those of W can be derived from that of z2. By using Theorem 4.1.1 1). Then the distribution of W is charac- terized by that of z2. [2] A. R. Soltani and H. Homei, “Weighted Averages with Random Proportions That Are Jointly Uniformly Distrib- uted over the Unit Simplex,” Statistics & Probability Letters, Vol. 79, No. 9, 2009, pp. 1215-1218. doi:10.1016/j.spl.2009.01.009 By an argument similar to the one given in example 4.2.1, when W has a Beta distribution with Parameters n and m, we find the distribution ()2;,Zfznm as [3] N. L. Johnson and S. Kotz, “Randomly Weighted Aver- ages,” The American Statistician, Vol. 44, No. 3, 1990, pp. 245-249. doi:10.2307/2685351 ()() ()()()()()( )1, 211,,,121,1,,01zzBnm zInmBnmBnm zI nmBnmz−−−−+−<<[4] A. I. Zayed, “Handbook of Function and Generalized Function Transformations,” CRC Press, London, 1996. [5] A. R. Soltani and H. Homei, “A Generalization for Two- Sided Power Distributions and Adjusted Method of Mo- ments,” Statistics, Vol. 43, No. 6, 2009, pp. 611-620. doi:10.1080/02331880802689506 − where (,z)Iab is incomplete Beta function: