ORDER RESTRICTED STATISTICAL INFERENCE ON LORENZ CURVES OF PARETO DISTRIBUTIONS. Myongsik Oh. 1. Introduction

J. Appl. Math & Computing Vol. 13(2003), No. 1-2, pp. 457-470 ORDER RESTRICTED STATISTICAL INFERENCE ON LORENZ CURVES OF PARETO DISTRIBUTIONS Myongsik Oh Abstract. The comparison of two or more Lorenz curves of Pareto distributions of first kind under arbitrary order restriction is studied. The problem is turned out to be a statistical inference problem concerning scale parameters under order restriction. We assume that the location parameters of Parato distributions are completely unknown. In this paper the maximum likelihood estimation and likelihood ratio tests for and against order restriction are proposed. AMS Mathematics Subject Classification: 62H12, 62F03 Key words and Phrases : Isotonic regression, Lorenz curve, Lorenz order, maximum likelihood estimation, Pareto distribution. 1. Introduction A distribution of a positive random variable can be characterized by so-called Lorenz curve. The Lorenz curve of a distribution function of a positive random variable is defined as, for u [0, 1], u 0 L(u) = F 1 X (t)dt 1 0 F 1 X (t)dt, where F 1 X (t) = inf x{x : F X (x) t} and F X is the distribution function of random variable X. The typical Lorenz curve is concave down. The area between the line L(u) =u and the Lorenz curve may be regarded as a measure of inequality of income, or more generally, of variability in the distribution of X. This area is also called the area of concentration. The Gini s concentration index is twice this area. A concise account of properties of Lorenz curve is found in Dagum (1985). Received January 14, 2003. Revised July 14, 2003. c 2003 Korean Society for Computational & Applied Mathematics. 457

458 Myongsik Oh Let X and Y be positive random variables and the corresponding Lorenz curves be L X (u) and L Y (u), respectively. The Lorenz ordering is defined according to L X (u) ( )L Y (u) for all u (0, 1). If two curves cross at least one point of u, then X and Y are not comparable. The Lorenz ordering is widely used for comparing the amounts of inequality in two or more distributions. Arnold (1987) wrote an excellent research book about Lorenz curve and majorization. See also Shaked and Shanthikumar (1994) for brief review of Lorenz Order. The comparison of Lorenz curve may be quite complicated. This is true for general distributions but becomes quite simple for some well-known distributions, such as Pareto distribution of first kind. The probability density function of Pareto distribution of first kind is written as, f(x) =θδ θ x θ 1,θ>0,x>δ>0, and the Lorenz curve is L(u) =1 (1 u) (θ 1)/θ. Since the Lorenz curve is monotonic with respect to parameter θ, Lorenz ordering for Pareto distribution is equivalent to the ordering among parameters θ s. The Pareto distribution is widely used for many areas of science, such as astronomy, economics, and so on. Donnison (1990) and Donnison and Peers (1992) used Pareto distributions to model the brightness (same as the magnitude) of comets. They assumed a trend in the brightness of comets as a function of some factors such as perihelion distance or orbital period. The statistical method they employed is so-called order restricted statistical inference concerning Pareto distributions under trend, which is usually called the simple ordering. Pettit (1993) considered Bayesian inferential method for ordered parameter using the same data. They, however, did not considered the unknown nuisance parameter, which is lower bound for random variable X. In this paper we consider the statistical inference concerning scale parameters of Pareto distributions under arbitrary ordering when location parameters are assumed to be unknown. The ordering generated by Lorenz ordering becomes a partial order among k scale parameters of Pareto distributions. In Section 2, we discuss maximum likelihood (ML) estimation of Pareto parameters. We consider both common location parameter case and completely unknown case. In Section 3, likelihood ratio test statistics for and against partial order restriction are given and their null distributions are studied. Two types of approximation for deriving null distributions are discussed when underlying order is simple ordering. Critical values for both tests are given for simple order case. In Section 4, other types of testing procedures, such as ad hoc test, are briefly discussed and the further research topics are also discussed. 2. Maximum Likelihood Estimation Suppose we have k populations which follow Pareto distributions with probability distribution functions f(x; θ i,δ i )=θ i δ θi i x θi 1, θ i > 0,x > δ i > 0 for

Inference on Lorenz curves of Pareto distributions 459 i =1,...,k. The Lorenz curve for ith population is L i (u) =1 (1 u) (θi 1)/θi. As we mentioned in Sectio, the Lorenz ordering can be represented by a certain ordering among scale parameters θ i s. Let I = {1, 2,...,k} be an index set. We consider a partial order,, on I. For i, j I, i j whenever L i (u) L j (u) for all u (0, 1). In terms of parameters θ s, we have, for i, j I, i j whenever θ i θ j. Let C = {y = (y 1,...,y k ) R k : y i y j whenever i j} and A = {y R k : y C }.We note that C is a closed convex cone associated with partial order. Let Θ = {θ R k : θ i θ j whenever i j, θ i > 0,i =1,...,k} with θ =(θ 1,...,θ k ). The most frequently used orderings among others are simple ordering, simple tree ordering. For example, the simple order is 1 2 k and the simple tree order is 1 j for j =2,...,k. Now we discuss the ML estimation of θ i s under an arbitrary partial ordering. For each i, let X ij,j =1,...,n i be random sample of size n i from ith population. Then the likelihood function is k i=1 θ ni i δ niθi i ni j=1 xθi+1 ij. (1) It is straightforward to show that the unrestricted maximum likelihood estimates (MLEs, hereafter), ˆθ i, ˆδ i,ofθ i and δ i are n i / n i j=1 ln(x ij/ mi j ni {x ij }) and mi j ni {x ij }, for each i. Donnison(1990) and Donnison and Peers(1992) studied the ML estimation when δ i s are known with application to astronomical data. Transforming the data, which divided by δ i first and taken natural logarithm, enable us to use generalized isotonic regression for exponential family. They, however, assumed that the values of δ i. Here in this paper we do not assume that δ i s are known to us. We need to consider the two cases: (1) δ i s are the same but the common value is unknown, and (2) δ i s are completely unknown. First we consider the case that δ 1 = = δ k = δ. We assume that the common value of δ is known temporally. Focusing on the restriction, which does not relate the two types of parameters θ i s and δ s, it is not difficult to expect that the estimation for parameter θ is quite similar to those of Donnison and Peers(1992). Note that δ<x ij for j =1,...,n i, i =1,...,k. Now the log of the likelihood function is given by n i n i n i ln θ i θ i ln (x ij /δ) ln x ij. (2) i=1 j=1 i=1 j=1 The leading term is of form of exponential likelihood function. Hence it follows from Theorem 1.5.1 of Robertson, Wright and Dykstra (1988) that the MLE of θ s under restriction can be obtained by an isotonic regression. Note that ˆθ i = n i / n i j=1 ln(x ij/δ) for i =1,...,k., i.e., ˆθ i is the reciprocal of the mean of log transformed data divided by δ. We need to be careful to use

460 Myongsik Oh the isotonic regression notation. We use the reciprocal of θ rather than θ. Let θ be the restricted MLE of θ. Now θ 1 = E n (ˆθ 1 A ), where n =(,...,n k ), x 1 =(x 1 1,...,x 1 k ). The projection operator E w( ) is defined as follows. For any collection of positive weights, w =(w 1,w 2,...,w k ), let < x, y > w be the inner product on R k defined by < x, y > w = k i=1 x iy i w i ; let w denote the induced norm, i.e., x 2 w = k i=1 x2 i w i; and for any subset A of R k let E w (x A) denote the projection (i.e., closest point under w )ofx onto A provided it exists and is unique. For details of projection theory, see Robertson et al. (1988). The Pool-Adjacent Violators Algorithm (PAVA) is a widely used computational method for finding isotonic regression. If violation occurs, the algorithm pools those estimators into a common estimator using appropriate weights. Assume, for example, i j but ˆθ i > ˆθ j. These two estimators do not satisfy the order restriction. The PAVA combines these two estimators into a common estimator which is n i + n j ni l=1 ln(x il/δ)+ n j l=1 ln(x jl/δ). (3) The PAVA is basically the repetition of this pooling process until all the violations are cleared up. Now by plugging θ i s into (2) we have [ ni ln θ ] n i i n i ln x ij. (4) i=1 Now let δ vary. Note that each of θ i is monotonic increasing function of δ s as we see in (3). To maximize the log-likelihood function (2) we need to choose maximum value as the MLE of δ. Hence the MLE, δ, ofδ becomes min { min {x ij}}. 1 i k 1 j n i, The equality assumption of δ i does not affect on estimation procedure. It, however, causes some difficulties in finding the null distributions of test statistics. We will discuss it later in Section 3. The above discussion is summarized in the following Theorem. i=1 j Theorem 2.1. Let X ij,j =1,...,n i,i =1,...,k be random sample from the populations with probability density functions f(x) =θ i δ θi x θi 1, θ i > 0,x > δ>0. The MLEs, θ and δ, ofθ and δ under the restriction θ Θ are given by θ = E n (ˆθ 1 A ) 1 and δ = min 1 i k { min 1 j n i {x ij }}.

Inference on Lorenz curves of Pareto distributions 461 Now we consider the case that δ i s are completely unknown. Note that δ i < mi j ni {x ij }. The log likelihood function is n i n i n i ln θ i θ i ln (x ij /δ i ) ln x ij. (5) i=1 j=1 i=1 j=1 Since it suffices to consider the case of k = 2, we assume k = 2. And we also assume θ 1 θ 2. Let ˆθ i ( θ i ) be the unrestricted MLEs of θ i when δ i are fixed to be ˆδ i ( δ i ). Assume δ i < ˆδ i. Suppose ˆθ 1 ˆθ 2, i.e. ˆθi do not violate the order restriction and may serve as the MLEs of θ i under the order restriction. For this case, the value of log likelihood function becomes ln n1 j=1 ln(x 1j/ˆδ 1 ) +n 2 ln n2 j=1 ln(x 2j/ˆδ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ). j=1 j=1 Suppose this is not true but θ i is the true unrestricted MLE of θ i. Then the corresponding MLEs are given by θ 1 = n1 j=1 ln(x 1j/ δ 1 ), θ2 = n2 j=1 ln(x 2j/ δ 2 ), if n1 j=1 ln(x 1j/ δ 1 ) n2 j=1 ln(x 2j/ δ 2 ), (6) θ 1 = θ + 2 = n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) if n1 j=1 ln(x 1j/ δ 1 ) > n2 j=1 ln(x 2j/ δ 2 ). (7) Then the log likelihood functions for each of (6) and (7) become ln n1 j=1 ln(x 1j/ δ 1 ) + ln n2 j=1 ln(x 2j/ δ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ), and + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ), j=1 j=1 respectively. First assume case (6). Since the log likelihood function is a monotonic increasing function with respect to each of δ i, we have ln n1 j=1 ln(x 1j/ˆδ 1 ) + ln ln n2 j=1 ln(x 2j/ˆδ 2 ) n1 j=1 ln(x 1j/ δ 1 ) + ln j=1 j=1 n2 j=1 ln(x 2j/ δ 2 ).

462 Myongsik Oh Hence for this case, θ i and δ i can not be a candidate for the MLE. Next assume (7) is the case. Since we assume θ 1 > θ 2, we have + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ n2 n1 j=1 ln(x 1j/ δ 1 ) = ( + )ln n1 j=1 ln(x 1j/ δ 1 ). Noting that ˆθ 1 ˆθ 2, we also have ln n1 j=1 ln(x 1j/ˆδ 1 ) + n 2 ln n2 j=1 ln(x 2j/ˆδ 2 ) (n 1 + )ln n1 j=1 ln(x 1j/ˆδ 1 ). From the above two inequalities, we have ln n1 j=1 ln(x 1j/ˆδ 1 ) + ln n2 j=1 ln(x 2j/ˆδ 2 ) + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ). This means that θ i and δ i can not be a candidate for the MLE for this case. Next suppose ˆθ 1 > ˆθ 2, i.e., ˆθ i violate the order restriction. If ˆθ i are the true unrestricted MLEs, then the value of log likelihood function is given by + ( + )ln n1 j=1 ln(x 1j/ˆδ 1 )+ j=1 ln(x 2j/ˆδ 2 ) (n 1+ ) ( ln x 1j + ln x 2j ). j=1 j=1 Suppose this is not true but θ i are the true unrestricted MLEs of θ i. First suppose that no violations occur i.e., θ 1 < θ 2. Then we have ln n1 j=1 ln(x 1j/ δ 1 ) + ln n2 j=1 ln(x 2j/ δ 2 ) ( + )ln n2 j=1 ln(x 2j/ δ 2 ) ( + )ln n2 j=1 ln(x 2j/ˆδ 2 ) + ( + )ln n1 j=1 ln(x 1j/ˆδ 1 )+ j=1 ln(x 2j/ˆδ 2 ) The last inequality is due to ˆθ 1 > ˆθ 2. Hence θ i and δ i can not be the MLEs for this case. Next suppose that violation occurs and hence θ 1 and θ 2 need to be pooled. The common value is + ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ).

Inference on Lorenz curves of Pareto distributions 463 Since the likelihood function is monotonically increasing with respect to δ i s, we have + ( + )ln n1 j=1 ln(x 1j/ δ 1 )+ j=1 ln(x 2j/ δ 2 ) + ( + )ln n1 j=1 ln(x 1j/ˆδ 1 )+ j=1 ln(x 2j/ˆδ 2 ). This shows that the MLE for δ i s should be ˆδ i s. Now the following theorem summarizes the above arguments. Theorem 2.2. Let X ij,j =1,...,n i,i =1,...,k be random sample from the populations with probability density functions f(x) =θ i δ θi i x θi 1, θ i > 0,x > δ i > 0. The MLEs, θ and δ i,ofθ and δ i s under the restriction θ Θ are given by θ = E n (ˆθ 1 A ) 1 and δ i = min {x ij} for i =1,...,k. 1 j n i, Next we need to find the MLEs of θ i under the equality assumption of θ i s. We briefly state the MLEs, θ i s, when θ 1 = = θ k. Theorem 2.3. Let X ij,j =1,...,n i,i =1,...,k be random sample from the populations with probability density functions f(x) =θ i δ θi x θi 1, θ i > 0,x > δ>0. The MLEs, θ and δ,ofθ and δ s under the restriction θ 1 = = θ k are given by θ 1 = = θ k = + + n k k i=1 ni j=1 ln(x ij/δ ) and δ = ˆδ = min i i k { min 1 j n i, {x ij}}. Theorem 2.4. Let X, j =1,...,n i,i =1,...,k be random sample from the population with probability density functions f(x) =θ i δ θi i x θi 1, θ i > 0,x > δ i > 0. The MLEs, θ and δi,ofθ and δ i s under the restriction θ 1 = = θ k are given by θ1 = = θk + + n k = k ni i=1 j=1 ln(x ij/δi ) and δi = ˆδ i = min {x ij} for i =1,...,k. 1 j n i, To avoid notational confusion we use ˆδ i and ˆδ rather than δ i and δ. Finally we briefly mention the strong consistency of the estimators given above. It follows from the fact that the projection operators are continuous with

464 Myongsik Oh respect to weights and the arguments given polyhedral cone and the unrestricted estimators are strongly consistency. 3. Likelihood Ratio Tests Consider the following three hypotheses: H 0 : θ 1 = = θ k, H 1 : θ Θ, H 2 : No restriction on θ i s except θ i > 0. In this section we consider the likelihood ratio tests for H 0 versus H 1 H 0 and for H 1 versus H 2 H 1. As we discussed in previous section we need to consider two cases according to the assumption on δ s. First we consider the case that all the δ i s are unknown completely. For the case of equal δ i, we need some modifications to derive the null distribution of the test statistic. First consider the test for H 0 versus H 1 H 0. It is not difficult to show that the test reject H 0 for large value of T 01 = 2 ln(λ) = 2 n i ln θ i θ, (8) i=1 i where Λ is the likelihood ratio. Malik (1970) shows that mi j ni {x ij } and n i / n i j=1 ln(x ij/ mi j ni {x ij }) are jointly sufficient for (δ i,θ i ) and independent to each other, for each i = 1,...,k. It also has been shown that 2n i θ ˆθ 1 i i is distributed as chi-square with 2(n i 1) degrees of freedom. Hence n i j=1 ln(x ij/ˆδ i ) is distributed as gamma with shape parameter n i 1 and scale parameter θ 1 i. Noting that the shape parameter is known, this is just a testing problem for order restriction in the scale parameters of gamma distributions with known shape parameters. Appealing to Theorem 4.1.1 of Robertson et al. (1988) or Robertson and Wegman (1978), we obtain the following theorem. Theorem 3.1. If H 0 is true, then, for all real t, lim P [T 01 t] = P (l, k; w, )P [χ 2 l 1 t], (9) N l=1 where N = k i=1 n i, w =(,...,n k ), and P (l, k; w, ) is the level probabilities. The level probability P (l, k; w, ) is the probability that E w (X A ) has exactly l distinct levels, where X =(X 1,...,X l ) and X 1,...,X l are independent normal variables with zero means and variances w1 1,...,w 1, respectively. k

Inference on Lorenz curves of Pareto distributions 465 Table 1. Critical values for T 01 and T 12 with k populations, simple order, equal sample sizes, and approximation based on Robertson and Wegman (1978) significance k test level 3 4 5 6 7 8 T 01 0.01 6.8228 7.7089 8.3563 8.8653 9.2838 9.6383 0.05 3.8201 4.5283 5.0491 5.4604 5.7998 6.0882 T 12 0.01 7.6717 9.5646 11.3053 12.9578 14.5503 16.0983 0.05 4.5773 6.1749 7.6647 9.0946 10.4848 11.8461 9 10 11 12 13 14 T 01 0.01 9.9456 10.2162 10.4579 10.6760 10.8747 11.0570 0.05 6.3388 6.5601 6.7581 6.9371 7.1004 7.2505 T 12 0.01 17.6114 19.0962 20.5571 21.9977 23.4205 24.8278 0.05 13.1849 14.5054 15.8108 17.1031 18.3842 19.6554 15 16 17 18 19 20 T 01 0.01 11.2252 11.3815 11.5273 11.6638 11.7922 11.9134 0.05 7.3892 7.5181 7.6386 7.7515 7.8579 7.9583 T 12 0.01 26.2208 27.6017 28.9714 30.3307 31.6807 33.0219 0.05 20.9177 22.1721 23.4192 24.6598 25.8943 27.1232 Usually the exact values of level probabilities are intractable. But the level probabilities are known to be robust with respect to weights. So we may use critical values for T 01 with equal weights. Table 1 display critical values for T 01 for equal weights i.e., equal sample sizes. This approximation is generally acceptable because gamma distribution is pretty much same as normal distribution when shape parameters get large, for instance, n i > 10 for i =1,...,k. However, if is the usual simple order and the sample sizes are the same, then other type of approximation is also available. The details for this type of approximation can be found in Guffey and Wright (1986) and Robertson et al. (1988). Let τ = 1= = n k 1. Under H 0 we have P [T 01 t] P S (l, k)p [χ 2 b(τk/l,l)(l 1) c(τk/l, l)t], (10) l=2 where P S (l, k) is the level probability for simple ordering with equal weights, and c(τ,l) = lφ 1(τ) φ 1 (lτ) lφ 2 (τ) φ 2 (lτ), φ 1 (τ) φ 2 (τ) b(τ,l)= [lφ 1(τ) φ 1 (lτ)]c(τ,l), l 1. = 3 1 1+τ + τ 6(1 + τ) 2 2τ ln(1 + 1 τ. = 1+ 2+τ 3(1 + τ) 2 + 1 3(1 + τ) 3. ), and

466 Myongsik Oh Table 2. Critical values for T 01 with k populations, simple order, equal sample sizes, and approximation based on Bain and Engelhardt (1975) Sample Size k 3 5 10 15 20 25 50 100 Significance level = 0.01 3 7.2502 7.0843 6.9550 6.9112 6.8893 6.8760 6.8495 6.8362 4 8.1157 7.9565 7.8337 7.7923 7.7716 7.7591 7.7341 7.7215 5 8.7398 8.5891 8.4735 8.4346 8.4151 8.4034 8.3799 8.3681 6 9.2267 9.0842 8.9754 8.9388 8.9205 8.9095 8.8874 8.8763 7 9.6252 9.4902 9.3875 9.3530 9.3357 9.3253 9.3046 9.2941 8 9.9619 9.8338 9.7365 9.7038 9.6875 9.6776 9.6580 9.6481 9 10.2532 10.1313 10.0388 10.0078 9.9922 9.9829 9.9642 9.9549 10 10.5096 10.3932 10.3050 10.2755 10.2607 10.2518 10.2340 10.2251 11 10.7384 10.6271 10.5428 10.5145 10.5004 10.4919 10.4749 10.4664 12 10.9449 10.8381 10.7573 10.7302 10.7167 10.7086 10.6923 10.6841 13 11.1331 11.0303 10.9527 10.9268 10.9138 10.9059 10.8903 10.8825 14 11.3057 11.2067 11.1320 11.1071 11.0946 11.0871 11.0721 11.0645 15 11.4651 11.3696 11.2976 11.2735 11.2614 11.2542 11.2397 11.2325 16 11.6133 11.5210 11.4514 11.4281 11.4164 11.4095 11.3955 11.3885 17 11.7516 11.6622 11.5949 11.5724 11.5611 11.5544 11.5408 11.5341 18 11.8811 11.7945 11.7292 11.7075 11.6965 11.6899 11.6768 11.6703 19 12.0031 11.9191 11.8557 11.8346 11.8240 11.8176 11.8049 11.7985 20 12.1183 12.0366 11.9751 11.9545 11.9442 11.9381 11.9257 11.9196 Significance level = 0.05 3 4.0585 3.9648 3.8929 3.8687 3.8566 3.8493 3.8347 3.8274 4 4.7627 4.6702 4.5996 4.5759 4.5641 4.5569 4.5426 4.5355 5 5.2741 5.1851 5.1174 5.0947 5.0833 5.0765 5.0628 5.0559 6 5.6752 5.5900 5.5254 5.5038 5.4930 5.4865 5.4734 5.4669 7 6.0046 5.9233 5.8617 5.8411 5.8308 5.8246 5.8122 5.8060 8 6.2838 6.2060 6.1473 6.1276 6.1178 6.1119 6.1000 6.0941 9 6.5258 6.4514 6.3953 6.3765 6.3671 6.3614 6.3501 6.3445 10 6.7393 6.6680 6.6142 6.5962 6.5871 6.5817 6.5709 6.5655 11 6.9303 6.8617 6.8100 6.7927 6.7841 6.7789 6.7685 6.7633 12 7.1028 7.0368 6.9870 6.9704 6.9621 6.9571 6.9471 6.9421 13 7.2601 7.1965 7.1485 7.1325 7.1245 7.1197 7.1100 7.1052 14 7.4047 7.3432 7.2969 7.2814 7.2737 7.2690 7.2598 7.2551 15 7.5384 7.4789 7.4341 7.4191 7.4116 7.4072 7.3982 7.3937 16 7.6627 7.6050 7.5616 7.5471 7.5399 7.5356 7.5269 7.5225 17 7.7788 7.7229 7.6808 7.6667 7.6597 7.6555 7.6470 7.6428 18 7.8877 7.8334 7.7925 7.7789 7.7720 7.7679 7.7598 7.7556 19 7.9902 7.9374 7.8977 7.8844 7.8778 7.8738 7.8659 7.8619 20 8.0871 8.0357 7.9971 7.9842 7.9777 7.9738 7.9661 7.9622

Inference on Lorenz curves of Pareto distributions 467 For the details of this approximation, see also Bain and Engelhardt(1975). For other ordering, such as simple tree or unimodal ordering, some modifications are required. The approximation for these types of ordering appears elsewhere. Table 2 shows the critical values for T 01 for k =3,...,20 and various sample sizes based on (10), i.e., approximation of Bain and Engelhardt (1975). For each k, the critical values converge to the corresponding value. The rate of convergence is rather slow. Even for moderate sample size, however, the critical value is quite close the limiting value which is given in Table 1. Therefore we recommend the reader to use Table 1 for moderate or large sample size cases and to use Table 2 for small sample sizes cases. Now consider the likelihood ratio test for H 1 against H 2 H 1. The test rejects H 1 for the large value of T 12 = 2 ln(λ) = 2 n i ln ˆθ i. (11) θ i Consider the partial order, θ induced by and θ on I which requires that, for i, j I, i θ j only when i j and θ i = θ j. Suppose i j and θ i <θ j. By the strong law of large numbers, for sufficiently large sample sizes, we have ˆθ i < ˆθ j with probability one. This means that there will be no violation and hence no amalgamation between these two θ s. Since this happens with probability one, no order needs to be defined between these two. For this reason, the partial ordering associated with the limiting distribution of T 12 depends upon and θ through θ. Now we have Theorem 3.2. If H 1 is true and θ is true value, then, for all real t, lim P [T 12 t] = P (l, k; w, N θ )P [χ 2 k l t]. (12) l=1 Moreover, lim P [T 12 t] P (l, k; w, )P [χ 2 k l t]. (13) N l=1 Since the true value of θ is unknown, (12) can not be used for finding critical value for the test. The equation (13) provides the critical value for conservative test. These critical values may be found in Table 1. The distribution associated to the right side of (13) is so called least favorable distribution which is the stochastically largest distribution with respect to parameter θ. The least favorable configuration is obtained when θ 1 = = θ k. For some values of θ, however, the test based on (13) will give substantially lower power. i=1

468 Myongsik Oh Table 3. Critical values for T 12 with k populations, simple order, equal sample sizes, and approximation based on Bain and Engelhardt (1975) Sample Size k 3 5 10 15 20 25 50 100 Significance level = 0.01 3 8.2493 8.0280 7.8527 7.7929 7.7628 7.7447 7.7083 7.6900 4 10.2600 9.9927 9.7818 9.7100 9.6739 9.6521 9.6084 9.5865 5 12.1059 11.7972 11.5546 11.4721 11.4306 11.4056 11.3556 11.3304 6 13.8564 13.5092 13.2370 13.1446 13.0981 13.0701 13.0141 12.9859 7 15.5423 15.1583 14.8579 14.7560 14.7048 14.6740 14.6122 14.5813 8 17.1804 16.7609 16.4333 16.3223 16.2665 16.2330 16.1658 16.1320 9 18.7809 18.3269 17.9730 17.8532 17.7930 17.7568 17.6842 17.6478 10 20.3511 19.8634 19.4838 19.3553 19.2908 19.2520 19.1742 19.1353 11 21.8957 21.3749 20.9701 20.8331 20.7644 20.7230 20.6401 20.5986 12 23.4185 22.8654 22.4357 22.2905 22.2175 22.1737 22.0858 22.0417 13 24.9224 24.3372 23.8831 23.7297 23.6526 23.6063 23.5135 23.4670 14 26.4095 25.7928 25.3148 25.1532 25.0721 25.0234 24.9257 24.8768 15 27.8815 27.2337 26.7318 26.5623 26.4772 26.4260 26.3235 26.2722 16 29.3405 28.6619 28.1364 27.9590 27.8700 27.8165 27.7092 27.6555 17 30.7875 30.0784 29.5295 29.3444 29.2514 29.1955 29.0836 29.0275 18 32.2235 31.4841 30.9122 30.7192 30.6224 30.5642 30.4476 30.3892 19 33.6494 32.8800 32.2852 32.0845 31.9839 31.9233 31.8021 31.7415 20 35.0661 34.2669 33.6494 33.4411 33.3365 33.2738 33.1479 33.0850 Significance level = 0.05 3 4.9321 4.7946 4.6872 4.6508 4.6325 4.6215 4.5995 4.5884 4 6.6363 6.4574 6.3177 6.2704 6.2466 6.2323 6.2036 6.1893 5 8.2215 8.0052 7.8368 7.7798 7.7511 7.7339 7.6994 7.6821 6 9.7404 9.4892 9.2938 9.2278 9.1946 9.1747 9.1347 9.1147 7 11.2157 10.9309 10.7100 10.6353 10.5978 10.5753 10.5301 10.5075 8 12.6591 12.3420 12.0962 12.0133 11.9716 11.9466 11.8964 11.8713 9 14.0778 13.7291 13.4593 13.3682 13.3225 13.2951 13.2400 13.2124 10 15.4765 15.0970 14.8037 14.7047 14.6550 14.6252 14.5654 14.5354 11 16.8586 16.4489 16.1323 16.0256 15.9720 15.9398 15.8754 15.8431 12 18.2265 17.7870 17.4476 17.3333 17.2759 17.2414 17.1723 17.1378 13 19.5822 19.1132 18.7514 18.6295 18.5683 18.5316 18.4580 18.4211 14 20.9270 20.4289 20.0449 19.9156 19.8507 19.8117 19.7336 19.6945 15 22.2622 21.7353 21.3294 21.1927 21.1241 21.0829 21.0004 20.9590 16 23.5888 23.0334 22.6057 22.4616 22.3894 22.3460 22.2591 22.2156 17 24.9075 24.3238 23.8746 23.7233 23.6475 23.6019 23.5106 23.4650 18 26.2191 25.6074 25.1367 24.9783 24.8988 24.8511 24.7555 24.7077 19 27.5242 26.8846 26.3927 26.2271 26.1441 26.0942 25.9944 25.9444 20 28.8232 28.1559 27.6428 27.4702 27.3836 27.3316 27.2275 27.1754

Inference on Lorenz curves of Pareto distributions 469 Oh(1994) studied the approximate test for resolving this difficulties. One might approximate θ using an estimate of θ. This will not be discussed here. Another approximation similar to (10) is also available. Under H 1 we have P [T 12 t] P S (l, k)p [χ 2 b(τ,k/l)(k l) c(τ,k/l)t]. (14) l=2 The critical values based on (14) are displayed in Table 3. As we have seen in T 01 the rate of convergence to the vales in Table 1 is rather slow. Even for moderate sample size, however, the critical value is quite close the limiting value which is given in Table 1. Therefore we also recommend the reader to use Table 1 for moderate or large sample size cases and to use Table 3 for small sample sizes cases. Next we consider the case that the location parameters are the same i.e., δ i are assumed to be the same. For this case we need a slight modification but use the same critical values as in the case of different location parameters. Let δ 1 = = δ k = δ. Note that n i ˆθ i = ni j=1 ln(x ij/ mi i k {mi j ni {x ij }}). That means 2n i θ ˆθ 1 i i is no longer distributed as chi-square with 2(n i 1) degrees of freedom. We also note that θ i s are not independent to each other. This fact makes us unable to find the null distributions of test statistics. However, we can find null distribution for the case of sufficiently large sample size. It is straightforward to show that mi i k {mi j ni {x ij }} converges to δ with probability one. And, for each i =1,...,k, mi j ni {x ij } also converges to δ with probability one. Then we have, for sufficiently large sample sizes, n i ˆθ i ni j=1 ln(x ij/ mi j ni {x ij }). Hence Theorem 3.1 and 3.2 are still valid for the case that equal value of δ. We, however, are not aware of whether (10) and (14) are still valid. We conjecture that those approximations are valid since the derivation of the fact does not depend heavily upon independence assumption. 4. Concluding Remarks It is generally known that a likelihood ratio test is a powerful test but under order restriction it has major drawback. That is, it is not easy to find level probabilities even for moderate value of k. Several approximation methods are suggested but it is still unsatisfactory. So some considered ad hoc tests and other type of tests. We will not pursue these tests in this paper. Interested reader refers to Robertson et al. (1988). Pettit (1993) studied Bayesian test for ordered

470 Myongsik Oh scale parameter of Pareto distributions when shape parameters are completely known. As we have seen in Donnison (1990) and Donnison and Peers (1992) it is not easy to expect that we have sufficiently large sample sizes. We need to find near-exact null distribution or some other approximation method for finding critical value. Bootstrapping method is likely to be a candidate for resolving this problem. References 1. Arnold. B. C. (1987). Majorization and Lorenz Order, Lecture Notes in Statistics No. 43, Springer-Verlag., New York and Berlin. 2. Bain, L. J., and Engelhardt, M. (1975). A Two-Moment Chi-Square Approximation for the Statistic log( x/ x), Journal of the American Statistical Association, 70, 948-950. 3. Dagum, C. (1985). Lorenz Curve, Encyclopedia of Statistical Sciences, 5, S. Kotz, N. L. Johnson, and C. B. Read (editors), 156-161, Wiley, New York. 4. Donnison, J. R. (1990). The Distribution of Cometary Magnitudes, Monthly Notice of Royal Astronomical Society, 245, 658-664. 5. Donnison, J. R., and Peers, H. W. (1992). An Approach to trend Analysis in Data with Special Reference to Cometary Magnitudes, Monthly Notice of Royal Astronomical Society, 256, 647-654 6. Guffey, J. M., and Wright, F. T. (1986). Testig for trends in a nonhomogeneous Poisson process Technical report, Department of Mathematics and Statistics, University of Missori- Rolla. 7. Malik, H. J. (1970). Estimation of the Parameter of the Pareto Distribution, Metrika, 15, 126-132. 8. Oh, M. (1994). Statistical Tests Concerning a Set of Multinomial Parameters under Order Restrictions: Approximations to Null Hypotheses Distributions, Ph. D. Thesis, The University of Iowa. 9. Pettit, L. I. (1993). Inferences about Ordered Parameters - An Astronomical Problem. Journal of the Royal Statistical Society, Series D, The Statistician, Vol. 42, 491-496. 10. Robertson, T., and Wegman, E. J. (1978). Likelihood ratio tests for order restrictions in exponential families, The Annals of Statistics, 6, 485-505. 11. Robertson, T., F. T. Wright, and R. L. Dykstra (1988). Order Restricted Statistical Inference, Wiley, Chichester. 12. Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications, Academic Press, New York. Myongsik Oh received his Ph.D at the University of Iowa. He is an associate professor in Department of statistics, Pusan University of Foreign Studies. His research interest focuses on order restricted statistical inference and mathematical statistics. Department of Statistics, Pusan University of Foreign Studies, Pusan 608-738, Korea. e-mail: moh@stat.pufs.ac.kr