JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No., March 2007 CHARACTERIZATIONS OF THE PARETO DISTRIBUTION BY THE INDEPENDENCE OF RECORD VALUES Se-Kyung Chang* Abstract. In this paper, we establish characterizations of the Pareto distribution by the independence of record alues. We proe that X P AR(, β) for β > 0, if and only if and are independent for n. And we show that X P AR(, β) for β > 0, if and X U(n+) only if X U(n+) and are independent for n.. Introduction Let {X n, n } be a sequence of independent and identically distributed (i.i.d.) random ariables with cumulatie distribution function (cdf) F (x) and probability density function (pdf) f(x). Suppose Y n = max{x, X 2,, X n } for n. We say X j is an upper record alue of this sequence, if Y j > Y j for j >. By definition, X is an upper as well as a lower record alue. The indices at which the upper record alues occur are gien by the record times {U(n), n }, where U(n) = min{j j > U(n ), X j > X U(n ), n 2} with U() =. A continuous random ariable X is said to hae the Pareto distribution with two parameters α > 0 and β > 0 if it has a cdf F (x) of the form ( x ) β, x >, α > 0, β > 0 () F (x) = α 0, otherwise. Receied January 30, 2007. 2000 Mathematics Subject Classifications: Primary 62E0; Secondary 60E05, 62G30, 62H05, 62H0. Key words and phrases: record alue, characterization, independence, Pareto distribution.
52 SE KYUNG CHANG A notation that designates that X has the cdf () is X P AR(α, β). Some characterizations by the independence of the record alues are known. In [] and [2], Ahsanullah studied, if X P AR(α, β) for α > 0 and β > 0, then X U(m) and X U(m) are independent for 0 < m < n. And Ahsanullah proed, if X W EI(θ, α) for θ > 0 and α > 0, then X U(m) are independent for 0 < m < n. In [4], Lee and Chang characterized that X P AR(, β) for β > 0, if and only if X U(n+) and are independent for n and X EXP (σ) for σ > 0, if and only if X U(n+) are independent for n. and X U(n+) and In this paper, we will gie characterizations of the Pareto distribution with the parameter α = by the independence of record alues. 2. Main results Theorem. Let {X n, n } be a sequence of i.i.d. random ariables with cdf F (x) which is an absolutely continuous with pdf f(x) and F () = 0 and F (x) < for all x >. Then F (x) = x β for all x > and β > 0, if and only if and are independent for n. X U(n+) Proof. If F (x) = x β for all x > and β > 0, then the joint pdf f n,n+ (x, y) of and X U(n+) is for all < x < y, β > 0 and n. f n,n+ (x, y) = βn+ (ln x) n x y β+ Consider the functions V = and W =. It follows X U(n+) ( )w that x U(n) = w, x U(n+) = and J = w. Thus we can write the 2 joint pdf f V,W (, w) of V and W as (2) f V,W (, w) = βn+ β (ln w) n ( ) β+ w β+
CHARACTERIZATIONS OF THE PARETO DISTRIBUTION 53 for all > 0, w >, β > 0 and n. (3) The marginal pdf f V () of V is gien by f V () = = f V,W (, w) dw β n+ β ( ) β+ = β β ( ) β+ for all > 0, β > 0 and n. Also, the pdf f W (w) of W is gien by (ln w) n w β+ dw (4) for all w >, β > 0 and n. f W (w) = (R(w))n f(w) = βn (ln w) n w β+ From (2), (3) and (4), we obtain f V ()f W (w) = f V,W (, w). Hence V = and W = are independent for n. X U(n+) Now we will proe the sufficient condition. The joint pdf f n,n+ (x, y) of and X U(n+) is f n,n+ (x, y) = (R(x))n r(x) f(y) for all < x < y, β > 0 and n, where R(x) = ln( F (x)) and r(x) = d ( ) f(x) R(x) = dx F (x). Let us use the transformations V = and W =. X U(n+) The Jacobian of the transformations is J = w. Thus we can write the 2 joint pdf f V,W (, w) of V and W as f (5) f V,W (, w) = for all > 0, w > and n. ( ( ) w ) (R(w)) n r(w) w 2
54 SE KYUNG CHANG The pdf f W (w) of W is gien by (6) f W (w) = (R(w))n f(w) for all w > and n. From (5) and (6), we obtain the pdf f V () of V f f V () = ( ) ( ) w r(w) w 2 f(w) for all > 0, w > and n, where r(x) = That is, f V () = F F (w) f(x) F (x). ( ( ) w ) Since V and W are independent, we must hae ( ) ( ( )) ( )w (7) F = F ( F (w)). for all > 0 and w >. ( ) Upon substituting = in (7), then we get (8) F ( w) = ( F ( ))( F (w)) for all > and w >. By the monotonic transformations of the exponential distribution (see, [3]), the most general solution of (8) with the boundary condition F () = 0 is F (x) = x β for all x > and β > 0. This completes the proof.
CHARACTERIZATIONS OF THE PARETO DISTRIBUTION 55 Theorem 2. Let {X n, n } be a sequence of i.i.d. random ariables with cdf F (x) which is an absolutely continuous with pdf f(x) and F () = 0 and F (x) < for all x >. Then F (x) = x β for all x > and β > 0, if and only if X U(n+) and are independent for n. Proof. In the same manner as Theorem, we consider the functions V = X U(n+) and W =. It follows that x U(n) = w, x U(n+) = ( )w and J = w. Thus we can write the joint pdf f V,W (, w) of V and W as (9) f V,W (, w) = for all < 0, w >, β > 0 and n. The marginal pdf f V () of V is gien by β n+ (ln w) n ( ) β+ w β+ (0) f V () = = = f V,W (, w) dw β n+ ( ) β+ β ( ) β+ (ln w) n w β+ dw for all < 0, β > 0 and n. Also, the pdf f W (w) of W is gien by () f W (w) = (R(w))n f(w) = βn (ln w) n w β+ for all w >, β > 0 and n. From (9), (0) and (), we obtain f V ()f W (w) = f V,W (, w). Hence V = X U(n+) and W = are independent for n. Now we will proe the sufficient condition. By the same manner as Theorem, let us use the transformations V = X U(n+) and
56 SE KYUNG CHANG W =. The Jacobian of the transformations is J = w. can write the joint pdf f V,W (, w) of V and W as Thus we (2) f V,W (, w) = f(( )w) (R(w))n r(w) w for all < 0, w > and n. The pdf f W (w) of W is gien by (3) f W (w) = (R(w))n f(w) for all w > and n. From (2) and (3), we obtain the pdf f V () of V f V () = f(( )w) r(w) w f(w) for all < 0, w > and n, where r(x) = That is, f V () = f(x) F (x). ( F (( )w) F (w) Since V and W are independent, we must hae ). (4) F (( )w) = ( F ( ))( F (w)) for all < 0 and w >. Upon substituting ( ) = 2 in (4), then we get (5) F ( 2 w) = ( F ( 2 ))( F (w)) for all 2 > and w >. By the monotonic transformations of the exponential distribution (see, [3]), the most general solution of (5) with the boundary condition F () = 0 is F (x) = x β for all x > and β > 0. This completes the proof.
CHARACTERIZATIONS OF THE PARETO DISTRIBUTION 57 References. M. Ahsanuallah, Record Statistics, Noa Science Publishers, Inc., NY, 995. 2. M. Ahsanuallah, Record Values-Theory and Applications, Uniersity Press of America, Inc., NY, 2004. 3. J. Galambos and S. Kotz, Characterization of Probability Distributions. Lecture Notes in Mathematics. No. 675, Springer-Verlag, NY, 978. 4. M. Y. Lee and S. K. Chang, Characterizations based on the independence of the exponential and Pareto distributions by record alues, J. Appl. Math. & Computing, Vol. 8, No. -2, (2005), 497-503. * Department of Mathematics Education Cheongju Uniersity Cheongju, Chungbuk 360-764, Republic of Korea E-mail: skchang@cju.ac.kr