Characterizations of Pareto, Weibull and Power Function Distributions Based On Generalized Order Statistics

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1 Marquette University Mathematics, Statistics and Computer Science Faculty Research and Publications Mathematics, Statistics and Computer Science, Department of Characterizations of Pareto, Weibull and Power Function Distributions Based On Generalized Order Statistics M. Ahsanullah Rider University-Lawrenceville Gholamhossein Hamedani Marquette University, Published version. Journal of the Chungcheong Mathematical Society, Vol. 29, No. 3 August 26): DOI. 26 Chungcheong Mathematical Society. Used with permission.

2 JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 29, No. 3, August 26 CHARACTERIZATIONS OF PARETO, WEIBULL AND POWER FUNCTION DISTRIBUTIONS BASED ON GENERALIZED ORDER STATISTICS Mohammad Ahsanullah* and G.G. Hamedani** Abstract. Characterizations of probability distributions by different regression conditions on generalized order statistics has attracted the attention of many researchers. We present here, characterization of Pareto and Weibull distributions based on the conditional epectation of generalized order statistics etending the characterization results reported by Jin and Lee 24). We also present a characterization of the power function distribution based on the conditional epectation of lower generalized order statistics.. Introduction The concept of generalized order statistics gos) was introduced by Kamps [8] in terms of their joint pdf probability density function). The order statistics, record values, k-record values, Pfifer records and progressive type II order statistics are special cases of the gos. The rv s random variables) X, n, m, k), X 2, n, m, k),..., X n, n, m, k), k >, m R, are n gos from an absolutely continuous cdf cumulative distribution function) F with corresponding pdf f if their joint pdf, f,2,...,n, 2,..., n ), can be written as.) ) [ f,2,...,n, 2,..., n ) k Πj n γ j Πj n F j ) ) ] m f j ) F n ) ) k f n ), F +) < < 2 <... < n < F ), Received September 2, 24; Accepted July 5, Mathematics Subject Classification: Primary 62E, 6E5. Key words and phrases: conditional epectation, continuous univariate distributions, generalized order statistics, characterizations. Correspondence should be addressed to G.G. Hamedani, g.hamedani@mu.edu.

3 386 Mohammad Ahsanullah and G.G. Hamedani where F ) F ) and γ j k + n j) m + ) for all j, j n, k is a positive integer and m. If k and m, then X r, n, m, k) reduces to the ordinary rth order statistic and.) will be the joint pdf of order statistics X,n X 2,n... X n,n from F. If k and m, then.) will be the joint pdf of the first n upper record values of the i.i.d. independent and identically distributed) rv s with cdf F and pdf f. Integrating out, 2,..., r, r+,..., n from.) we obtain the pdf f r,n,m,k of X r, n, m, k).2) f r,n,m,k ) c r r )! where c r and Π r j γ j and F ) ) γr f ) g r m F )), [ ) m+], m, g m ) h m ) h m ) m + ln ), m,, ) h m ) m + )m+, m, ln ), m., ) Note that, since lim m m+ write g m ) m + g ) [ ) m+] ln ), we will [ ) m+], for all, ) and all m with lim g m ). m The joint pdf of X s, n, m, k) and by see Kamps [8], p.68) f s, r,n,m,k, y) c s r )!s r )! X r, n, m, k), r < s, is given F ) ) m f ) g r m F )) [h m F y)) h m F ))] s r F y) ) γ s f y), y. Consequently, the conditional pdf of X s, n, m, k) given X r, n, m, k), for s s + l, s > 2 and m, is.3) f s r,n,m,k y ) c s c r s r )! [h mf y)) h m F ))] s r F y))γs fy), y >. F )) γ r+

4 Characterizations of Pareto, Weibull and power function distributions 387 Burkschat et al. [5] introduced lower generalized order statistics lgos) as follows: The rv s X, n, m, k), X 2, n, m, k),..., X n, n, m, k), k >, m R, are n lgos from an absolutely continuous cdf F with corresponding pdf f if their joint pdf f, 2,..., n ), can be written as ) [ ] f, 2,..., n ) k Πj n γ j Πj n F j)) m f j ).4) F n )) k f n ), F ) > > 2 >... > n > F +). The marginal pdf fr,n,m,k ) is.5) f r,n,m,k ) c r r )! F ))γr f ) q r m F )), where The joint pdf by.6) f s,r,n,m,k, y) where q m m + )m+, for m ln ), for m. of X s, n, m, k) and X r, n, m, k), r < s, is given c s r )! s r )! F ))m f ) v r m F )) [h m F y)) h m F ))] s r F y) ) γ s f y), y, h m) m + m+, for m ln ), for m. Consequently, the conditional pdf of X s, n, m, k) given X r, n, m, k, is.7) c s fs r,n,m,k y ) c r s r )! [h mf y)) h mf ))] s r F y)) γ s F )) γ fy), y <. r+

5 388 Mohammad Ahsanullah and G.G. Hamedani 2. Characterization results Characterizations of probability distributions by different regression conditions on generalized order statistics has attracted the attention of many researchers e.g., Bieniek and Szynal [4], Cramer et al. [6] and Bieniek [3]), to name a few. In [4], the authors consider all cdf s F for which the following linearity of regression holds: E [ X r + l, n, m, k)) X r, n, m, k)] a X r, n, m, k) + b. They conclude that only eponential, Pareto and power function distributions satisfy this equation. Using this result they obtain characterizations of these distributions based on sequential order statistics, records and progressive type II censored order statistics. Cramer et al. [6], point out that characterizations of distributions based on linear regressions 2.) E [ψ X r + l, n, m, k)) X r, n, m, k) ] ϕ ) have been studied etensively for order statistics and record values r N, l. Since gos provide a unifying approach to these models, they set up a comprehensive solution related to characterization problems. They started with the case of adjacent gos l ) and then pointed out that for larger l the calculations become more difficult. In order to obtain an eplicit result, they restricted themselves to a linear function ϕ. They showed that the linearity of the conditional epectation provides a characterization of generalized Pareto distributions. Ahsanullah and Hamedani [] presented characterizations of continuous distributions based on 2.) for l but without the assumptions of monotonicity of ψ ) and linearity of ϕ ). Jin and Lee [7] presented the characterizations of Pareto and Weibull distributions based on conditional epectations of the upper record values. Following [7] we present similar characterization based on conditional epectations of gos etending the characterization results of Jin and Lee [7]. We also present a characterization of the power function distribution based on the conditional epectation of lower generalized order statistics. Theorem 2.. Let {X n, n } be a sequence of i.i.d. random variables with an absolutely continuous cdf F ), corresponding pdf f ) and E [X p ] <, p N. For l N and p < θ R + 2.2) θγ s+ p)θγ s+2 p)...θγ s+l p) EX p s+l, n, m, k) Xr, n, m, k) ) γ s+ γ s+2...γ s+l

6 Characterizations of Pareto, Weibull and power function distributions 389 θ l EX p s + l, n, m, k) Xr, n, m, k) ), if and only if 2.3) F ) θ,. Proof. Suppose 2.3) holds and as before, let s s + l. c s f s r,n,m,k y ) c r s r )! m + )s r and y θm+) + θm+) )s r y θ )γ s θ ) γ r+ Then θ y θ+, EX p s, n, m, k) Xr, n, m, k) ) c s θ θ ) γ r+ c r s r )! m + )s r y θγs p) y θm+) + θm+) )s r dy. Since θ p >, γ s >, then θγ s p > and using integration by parts on the right hand side of the above equality, we arrive at EX p s, n, m, k) Xr, n, m, k) ) c s θ 2 θ ) γ r+ c r s r 2)! θγ s p) m + )s r 2 y θγ s p) θm+) y θm+) + θm+) )s r 2 dy. Observing that θγ s p) + θ m + ) θγ s p, we have EX p s, n, m, k) Xr, n, m, k) ) c s θ 2 θ ) γ r+ c r s r 2)! θγ s p) m + )s r 2 y θγ s p) y θm+) + θm+) )s r 2 dy. Employing integration by parts on the right hand side of the above equality, results in EX p s, n, m, k) Xr, n, m, k) ) c s θ 3 θ ) γ r+ c r s r 3)! θγ s p) θγ s p) m + )s r 3 y θγ s p) θm+) y θm+) + θm+) )s r 3 dy. Continuing in this manner, we arrive at

7 39 Mohammad Ahsanullah and G.G. Hamedani EX p s, n, m, k) Xr, n, m, k) ) c s θ l θ ) γ r+ c r s r l)! Π l i θγ s i p) m + )s r l y θγ s l p) y θm+) + θm+) )s r l dy. θ l Π s+l is+ γ i Π l i θγ s i p) EXp s, n, m, k) Xr, n, m, k) ), which is now 2.2). Now assume 2.2) holds. Then 2.4) θγ s+ p)θγ s+2 p)...θγ s+l p) γ s+ γ s+2...γ s+l s r )! y p [h m F y)) h m F ))] s r F y)) γ s fy)dy θ l s r )! y p [h m F y)) h m F ))] s r F y)) γs fy)dy. Differentiating both sides of 2.4) s r) times with respect to, we obtain 2.5) y p [h m F y)) h m F ))] l F y)) γ s fy)dy θ l γ s+ γ s+2...γ s+l θγ s+ p)θγ s+2 p)...θγ s p) p F )) γs+. Letting u F y) F )) γs in 2.5), we arrive at )) [ ) m+ ) F p u /γ s F ) F ) u /γ s m+ ] l F ) m + ) ) u /γ γs s F ) u γs γs F ) du γ s or θ l γ s+ γ s+2...γ s+l θγ s+ p)θγ s+2 p)...θγ s p) p F )) γ s+,

8 Characterizations of Pareto, Weibull and power function distributions 39 θγ s+ p)θγ s+2 p)...θγ s p) γ s+ γ s+2...γ s+l )) F p u m+ u /γs γs F ) m + ) l du θ l p, and letting v u /γ s and F ) t, we have 2.6) θγ s+ p)θγ s+2 p)...θγ s p) γ s+ γ s+2...γ s+l ) ) F p v m+ l ) vt) v γs θ l F p t). m + and t e w in 2.6), we obtain, upon simplifi- Putting v e u cation θγ s+ p)θγ s+2 p)...θγ s p) γ s+ γ s+2...γ s+l F ) l e u+w))) p e m+)u e γsu du θ l F e w )) p, m + for < w <. Now, the rest follows from the proof of Theorem 2. of Jin and Lee [7], page 245). We need the following definition for our characterization of the Weibull distribution. Definition 2.2. The random variable X with cdf F belongs to the class C if for some δ >, either F + y)) /δ F )) /δ F y)) /δ, for all, y, or F + y)) /δ F )) /δ F y)) /δ, for all, y. Theorem 2.3. Let {X n, n } be a sequence of i.i.d. random variables with an absolutely continuous cdf F ), corresponding pdf f ) and E [ X δ] <, δ >. We assume that F ) and F ) > for all >. For l N, the following statements are equivalent: i) F ) e δ,.

9 392 Mohammad Ahsanullah and G.G. Hamedani ii) The random variable X belongs to the class C and 2.7) [ ] E Xs + l, m.n.k)) δ Xr, m.n.k) γ s+l + γ s+l γ s+ + E [ Xs, m.n.k)) δ Xr, m.n.k) Proof. Suppose i) holds and let Y X δ, then Y has an eponential distribution with parameter. We know that see, [2], page 7) Y r, n, m, k) d Y s, n, m, k) d r j s j Y j γ j, Y j γ j and Y s + l, n, m, k) d s+l j Y j γ j, where Y j, j, 2,..., s + l are i.i.d with cdf F Y y) e y, y, and EY s, n, m, k)) Thus, s j EY s + l, n, m, k) Er, n, m, k) t) s+l js+ s+l and EY s + l, n, m, k)). γ j γ j j γ j + EY s, n, m, k) Y r, n, m, k) t), s > r. Writing the above equation in terms of X, we have 2.7). Now, assume ii) holds. Then, the left hand side of 2.7) can be epressed as E [ Xs + l, m.n.k)) δ Xr, m.n.k) ] c s+l c r s + l r )! Thus, ]. y δ h m y) h m )) s+l r F y))γ s+l fy)dy. F )) γ r+

10 Characterizations of Pareto, Weibull and power function distributions ) c s+l y δ h m y) h m )) s+l r F y)) γs+l fy)dy c r s + l r )! ) F )) γ r+ γ s+l γ s+l + c s c r s r )! γ s+ y δ h m y) h m )) s r F y)) γs fy)dy. Now, differentiating both sides of 2.8) with respect to, s r) times, we arrive at c s+l c r l )! y δ h m y) h m )) l F y)) γ s+l fy)dy γ r+ γ r+2...γ s γ s+l + γ s+l γ s+ )F )) γ r+ + c s c r δ F )) γ r+. We can write the above equation as 2.9) l )! y δ hm y) h m ) F ) ) l F y) F ) c s ) + c s δ. c s+l γ s+l γ s+l γ s+ c s+l ) γs+l fy) F ) dy 2.) Putting u F y) F ) l )! in 2.9), we obtain F uf ))) δ u m+ ) l u γ s+l du c s ) + c s δ. c s+l γ s+l γ s+l γ s+ c s+l Substituting u e v and F ) e w in 2.), we have 2.) l )! F e w+v) )) δ e m+)v ) l e v ) γ s+l c s F e w ) c s+l ) δ c s c s+l γ s+l + γ s+l γ s+ ).

11 394 Mohammad Ahsanullah and G.G. Hamedani We observe that the right hand side of 2.) is independent of w, so letting w in 2.) and noting that F ), we arrive at 2.2) 2.3) l )! F e v )) δ e m+)v ) l e v ) γ s+l c s c s+l γ s+l + γ s+l γ s+ ). Now, in view of 2.) and 2.2), we have l )! l )! F e w+v) )) δ e m+)v ) l e v ) γ s+l c s c s+l F e w )) δ. F e v )) δ e m+)v ) l e v ) γ s+l Let H e m+)v ) l e v ) γ s+l. Putting z m + )v, we have 2.4) H l )! m + e m+)v ) l e v ) γ s+l e z ) l e γ s+l m+ z dz. Upon integration by parts on the right hand side of 2.4), l and substituting v m+) z, we have F δ e )) v e m+)v ) l e v ) γ s+l 2.5) c s c s+l F e w )) δ. In view of 2.3) and 2.5), we obtain F δ e )) w+v) e m+)v ) l e v ) γ s+l F e v )) δ e m+)v ) l e v ) γ s+l F e w )) δ e m+)v ) l e v ) γ s+l. time

12 Characterizations of Pareto, Weibull and power function distributions 395 Upon simplification of the above equality, we arrive at [ F δ ) e )) w+v) F δ ) ] e v ) F δ e w ) 2.6) e m+)v ) l e v ) γ s+l. In view of the fact that X belongs to the class C, we must have 2.7) ) F δ ) e w+v) ) F δ e v ) + F δ e )) w, for all v and w. Putting G u) F e u )) δ, we can write 2.7) as 2.8) G + y) G ) + G y), for all and y. Equation 2.8) is the well-known Cauchy functional equation whose solution is G ) θ, for all where θ is a constant. Thus F δ e )) θ from which we have F ) e θ. In view of the boundary conditions F ) and lim F ), we must have 2.9) F ) e θδ,, θ >, δ >. Note that we can assume without loss of generality θ. Remark 2.4. For δ, Theorem 2.2 gives a characterization of the eponential distribution based on the generalized order statistics. The following theorem gives characterization of power function distribution using lgos. Theorem 2.5. Let {X n, n } be a sequence of i.i.d. random variables with an absolutely continuous cdf F ), corresponding pdf f ) and E [X p ] <. For θ > and l N 2.2) if and only if θγ s+ + p)θγ s+2 + p)...θγ s+l + p) γ s+ γ s+2...γ s+l EX s + l, n, m, k)) p Xr, n, m, k) ) θ l EX s, n, m, k)) p Xr, n, m, k) ), 2.2) F ) θ,.

13 396 Mohammad Ahsanullah and G.G. Hamedani Proof. Is similar to that of Theorem 2.. References [] M. Ahsanullah and G. G. Hamedani, Characterizations of continuous distributions based on conditional epectation of generalized order statistics, Comm. Statist. Theory-Methods 42 23), [2] M. Ahsanullah and G. G. Hamedani, Eponential Distribution: Theory and Methods, NOVA Publishers, 2). [3] M. Bieniek, A note on characterizations of distributions by regressions of nonadjacent generalized order statistics, Aust. N. Z. J. Stat. 5 29), [4] M. Bieniek and D. Szynal, Characterizations of distributions via linearity of regression of generalized order statistics, Metrika 58 23), [5] M. Burkschat, H. Cramer, and U. Kamps, Dual Generalized Order Statistics, Metron, LXI, 23), [6] E. Cramer, U. Kamps, and C. Keseling, Characterizations via linear regression of ordered random variables: a unifying approach, Comm. Statist. Theory- Methods 33 24), [7] H-W. Jin and M-Y. Lee, On characterizations of Pareto and Weibull distribution by considering conditional epectations of upper record values, J. of Chungcheong Math. Sci ), no. 2, [8] U. Kamps, A concept of generalized order statistics, Teubner Skripten zur Mathematischen Stochastik. [Teubner Tets on Mathematical Stochastics]. B.G. Teubner, Stuttgart, 995). * Department of Management Sciences Rider University Lawrenceville, NJ ahsan@rider.edu ** Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI g.hamedani@mu.edu