On the distribution of the inverted linear compound of dependent F-variates and its application to the combination of forecasts
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(3) Journal of Applied Statistics Vol. 33, No. 9, 961 –973, November 2006. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. On the Distribution of the Inverted Linear Compound of Dependent F-Variates and its Application to the Combination of Forecasts KUO-YUAN LIANG , JACK C. LEE & KURT S.H. SHAO† . Polaris Research Institute and Department of Economics, National Taiwan University, Taiwan, National Chiao Tung University, Taiwan, †Polaris Research Institute, Taiwan. . ABSTRACT This paper establishes a sampling theory for an inverted linear combination of two dependent F-variates. It is found that the random variable is approximately expressible in terms of a mixture of weighted beta distributions. Operational results, including rth-order raw moments and critical values of the density are subsequently obtained by using the Pearson Type I approximation technique. As a contribution to the probability theory, our findings extend Lee & Hu’s (1996) recent investigation on the distribution of the linear compound of two independent F-variates. In terms of relevant applied works, our results refine Dickinson’s (1973) inquiry on the distribution of the optimal combining weights estimates based on combining two independent rival forecasts, and provide a further advancement to the general case of combining three independent competing forecasts. Accordingly, our conclusions give a new perception of constructing the confidence intervals for the optimal combining weights estimates studied in the literature of the linear combination of forecasts. KEY WORDS : Combining weights, critical values, error-variance minimizing criterion, inverted F-variates, Pearson Type I approximation. Introduction In this paper, we study the distribution of an inverted linear compound of dependent F-variates in the form: 1 1 þ a1 F1 (T, T) þ a2 F2 (T, T). (1). with degrees of freedom as indicated. Here, the two constants a1 and a2 lie in the interval (0,1]. This distribution is useful in constructing confidence intervals of the Correspondence Address: Jack C. Lee, Graduate Institute of Finance, National Chio Tung University, Hsinchu, Taiwan. Email: jclee@stat.nctu.edu.tw. 0266-4763 Print=1360-0532 Online=06=090961–13 # 2006 Taylor & Francis DOI: 10.1080=02664760600744330.
(4) 962 K.-Y. Liang et al. minimum variance weights that can be attached to the components of the linear composite forecasts. From Reid (1969), Dickinson (1973) or Newbold & Granger (1974), it is well known that given a history of unbiased forecast errors for k (k 2) models, under the errorvariance minimizing criterion, the optimal weighting vector (W) of the combined forecasts becomes P1. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. W¼. u0. u P1 u. (2). P where u is a (k 1) vector of ones and a (k k) positive definite covariance matrix of forecasting errors between the k models. It is worth noting that despite the popularity of equation (2), very little is known about the sampling properties of its estimator. A notable exception to this issue is the work P of Dickinson (1973). When k ¼ 2, based on the maximum likelihood estimator (S) of with zero off-diagonal elements and normally distributed forecasting errors, Dickinson ˆ is expressible (1973) demonstrated that each component of the estimated weight vector W as a weighted beta or beta (if homoscedasticity is further imposed) distribution. Although the combining procedure may involve more P than two competing forecasts, we will restrict our attention to the k ¼ 3 set-up with having zero off-diagonal elements and normally distributed forecasting errors only. This proves necessary as use of the general k k set-up is technically difficult. Our restricted set-up hence extends the initial work of Dickinson (1973). From the statistical viewpoint, a corollary of Dickinson’s (1973) result is that an 1 inverted F-variate of the form: 1þaF(T, T) where a is an arbitrary constant, is expressible as a weighted beta (if a , 1) or as a beta (if a ¼ 1) distribution, i.e., 1 weighted beta 1 þ aF(T, T). (3). Another relevant theoretical contribution to our investigation is the work of Lee & Hu (1996). According to them, an arbitrary linear combination of two independent F-variates can be expressed approximately as a suitable constant (c) times an F density function, i.e., a1 F(u1 , u2 ) þ a2 F(u1 , u2 ) _ cF(m1 , m2 ). (4). where m1 and m2 are two positive constants. Using the restricted set-up (detailed above), this paper derives the sampling distribution of the estimated combining weights. To achieve P this goal, we begin in the next section with a reformulation of equation (2) by replacing with S, and show that each estimated weight is an inverted linear compound of dependent F-variates. We then relax the independence assumption on equation (4), and demonstrate in the third section that an expression of the right-hand side of equation (4) approximately still holds. Using this result and random variables transformation techniques, we also show in the third section that the distribution of an inverted linear compound of dependent F-variates of equation (1) is approximately expressible in terms of a mixture of weighted beta distributions. Additionally, we conduct extensive simulations to assess the accuracy of these approximations. Our results thus generalize those of Lee & Hu (1996) as well as Dickinson (1973). A notable implication of our theoretical results to the equal weighting scheme is elaborated as well..
(5) Distribution of the Inverted Linear Compound of Dependent F-Variates 963 Owing to the complexity of the derived distribution, the fourth section presents several operational results, including rth-order raw moments and critical values of the density based on the Pearson Type I approximation technique (Johnson et al., 1963). The fifth section summarizes our findings and indicates future research directions.. Model and Related Results P Consider equation (2) in the restrictedPcase where ¼ diag(s11 , s22 , s33 ). In practice, the parameters sii are unknown, and is estimated by: PT. 2 t¼1 e1t , T. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. S ¼ diag. PT. 2 t¼1 e2t , T. PT. 2 t¼1 e3t T. !. where eit is the error in the tth forecast value, using the ith forecasting method. Assuming eit to be normally P distributed with zero mean and variance sii , the maximum likelihood estimator of T is given by: TS ¼ diag. T X. e21t ,. t¼1. T X. e22t ,. t¼1. T X. ! e23t :. (5). t¼1. It follows that: T X. e2it sii x2 (T). (6). t¼1. From above, the ith weight in equation (2) is estimated by: P 1= Tt¼1 e2it w^ i ¼ PT 2 PT 2 PT 2 , 1= t¼1 e1t þ 1= t¼1 e2t þ 1= t¼1 e3t. i ¼ 1, 2, 3. (7). Based on equation (7), each estimated weight can thus be written as an inverted linear compound of dependent F-variates of the form stated in equation (1). For example, equation (7) implies that another expression for the first estimated weight is: w^ 1 ¼. 1þ. PT. 2 t¼1 e1t =. 1 PT 2 PT 2 2 þ e t¼1 2t t¼1 e1t = t¼1 e3t. PT. (8). Using equation (6) in equation (8), we can conclude that: w^ 1 . 1 1 þ a1 F1 (T, T) þ a2 F2 (T, T). (9). with degrees of freedom as indicated, and a1 ¼ s11 =s22 , a2 ¼ s11 =s33 are two positive constants. Comparing this set-up with equation (1), we see that s11 s22 and s11 s33 are assumed here for illustrational P convenience. Similar expressions for w^ 2 and w^ 3 can also be readily derived. Since Tt¼1 e21t appears in the second and third denominator terms of the right-hand expression of equation (8), it can be shown, that if T . 4 these.
(6) 964 K.-Y. Liang et al. two F-variates are indeed dependent and their correlation is given by: corr(F1 , F2 ) ¼. T 4 2(T 1). (10). See Appendix A.1 for the proof.. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. Theorems, Simulations and Implications Having verified the dependency between F1 and F2 in equation (10), we now turn our attention to the problem of finding the probability density of its linear compound of a1 F1 (T, T) and a2 F2 (T, T). As the following result indicates, this linear compound can be approximated by a constant (h) times an F(m1 , m2 ) variate, with the degrees of freedom as indicated.. Theorem 1 a1 F1 (T, T) þ a2 F2 (T, T) _ hF(m1 , m2 ). (11). where the right-hand expression of equation (11) comes from the denominator of equation (9), and the parameters h, m1 and m2 can be expressed in explicit forms in terms of a1, a2 and T. Specifically (Lee & Hu, 1996),. h¼. 2A2 C 2AB2 A2 B þ 3AC 4B2. m1 ¼. 4A2 C 4AB2 AB2 2A2 C þ BC. m2 ¼. 2A2 B þ 6AC 8B2 A2 B þ AC 2B2. and (12). where (a1 þ a2 )T T 2 T(T þ 2) a21 þ a22 2a1 a2 B¼ þ T 2 T 4 T 2 3 T(T þ 2)(T þ 4) a1 þ a32 3a21 a2 þ 3a1 a22 þ C¼ (T 2)(T 4) T 6 T 2 A¼. (13). and T . 6. Similar to Lee & Hu (1996), we conduct an extensive simulation study to assess the accuracy of this approximation. The results of our study are summarized in Table 1. In the simulation, we conduct 15,000 runs for each linear compound of the form a1 F1 (T, T) þ a2 F2 (T, T) and compute the probabilities of exceeding the 1%, 5% and 10% points. From Table 1, we see that the approximation to the assigned probability.
(7) Distribution of the Inverted Linear Compound of Dependent F-Variates 965 Table 1. Simulated results for Theorem 1. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. Tail probability Linear compound. a ¼ 0.01. a ¼ 0.05. a ¼ 0.1. F1(10,10) þ F2(10,10) F1(10,10) þ 0.5F2(10,10) 0.3F1(10,10) þ 0.01F2(10,10) 0.7F1(10,10) þ 0.01F2(10,10) 0.9F1(10,10) þ 0.01F2(10,10) F1(10,10) þ 0.01F2(10,10) 0.9F1(10,10) þ 0.7F2(10,10) 0.9F1(30,30) þ 0.01F2(30,30) F1(30,30) þ 0.01F2(30,30). 0.0101 0.0101 0.0106 0.0103 0.0097 0.0098 0.0106 0.0098 0.0098. 0.0510 0.0510 0.0491 0.0493 0.0491 0.0500 0.0507 0.0493 0.0493. 0.1019 0.1024 0.0994 0.0993 0.1000 0.0984 0.1039 0.0978 0.0972. Table entries are the simulated probabilities in the right-hand tail of the listed linear compound of dependent F-variates.. (a) in the right-hand tail of the listed linear compound of dependent F-variates is generally quite accurate. By virtue of Theorem 1, w^ 1 in equation (9) can thus be reasonably approximated by: w^ 1 _. 1 1 þ hF(m1 , m2 ). (14). Likewise, similar expressions for w^ 2 and w^ 3 can be obtained. Using the approximations derived in equation (14), we are ready to apply the variable transformation techniques to derive the probability density of f (w^ i ) (i ¼ 1, 2, 3). Theorem 2 Let eit (i ¼ 1, 2, 3; t ¼ 1, 2, . . . , T) be the error in the tth forecast value using theP ith fore0 casting model. Assume at a particular point of time, e ¼ (e , e , e ) N(0, ) with 1t 2t 3t P ¼ diag(s11 , s22 , s33 ), then under the error-variance minimizing criterion, the distribution of the ith optimal combining weight estimator w^ i is approximately a mixture of beta random variables with the probability density function of the form: f (w^ i ) ¼. 1 h m m (1 b)m2 =2 X m1 m1 i 2 2 þ j, þ j, Cjmþj1 bj B Beta B(m2 =2, m1 =2) j¼0 2 2 2 2. (15). hm2 where Cjmþj1 ¼ (mþj1)! j!(m1)! , b ¼ 1 m1 , B(p, q) is a beta function and Beta(p, q) is a beta density function with parameters p, q, respectively.. Proof For the proof see Appendix A.2. The following two theorems show that the Ðcondition jbj , 1 is sufficient for the integr1 ability of f (w^ i ) over (0,1], the satisfaction of 0 f (w^ i )dw^ i ¼ 1, and the existence of the rth order raw moment..
(8) 966 K.-Y. Liang et al. Theorem 3 If jbj , 1, then. Ð1 0. f (w^ i )dw^ i ¼ 1.. Proof For the proof see Appendix A.3. Theorem 4 If jbj , 1, then the rth-order raw moment of f (w^ i ) exists.. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. Proof For the proof see Appendix A.4. A notable P implication of the condition jbj , 1 is elaborated P as follows. Suppose the matrix in equation (2) is further restricted to ¼ diag(s11 , s22 , s33 ) and s11 ¼ s22 ¼ s33 . It would immediately appear that this case would generate w1 ¼ w2 ¼ w3 ¼ 1=3. The case in which each forecast receives equal weight is of particular interest, because it may be reasonable in many realistic applications. More specifically, the usual rationale for the equal weighting scheme is as follows. First, ‘if (a) there is only a small data base and/or (b) P the error covariance structure is not stationary’ (Bunn, 1986, p. 152), then specifying as an unrestricted real symmetric positive definite matrix tends to cause the robustness problems due to poor estimation P of its elements. A resolution is therefore suggested to specify the matrix in our restricted setup as diag(s11 , s22 , s33 ). Second, if no information is known or no reason to believe a priori on the relative accuracy of the competing forecasts, an even more extreme response is to further impose the constraint s11 ¼ s22 ¼ s33 into the above diagonal setting and utilize the equal weighting scheme (Bunn, 1986). This extreme case means that a1 ¼ a2 ¼ 1 in equation (9). Substituting a1 ¼ a2 ¼ 1 for b in equation (15) and using the condition jbj , 1 produces: 0,. 4(T 6)(3T 2 10T 10) ,2 (T 2)(3T 2 16T þ 28). Significantly, the above inequality holds only when T ¼ 7,8,9. Therefore, as a practical matter, the existing conditions of f (w^ i ) and its rth-order moment in this particular equal weighting scheme are extremely hard to satisfy. A cautious approach is suggested when applying this method, where other sources also share this view (Bunn, 1986; Winkler & Clemen, 1992) Theorem 4 gives the following corollary. Corollary 1 Each rth-order moment of f (w^ i ) is expressible as a monotonically decreasing sequence. Proof For the proof see Appendix A.5 To check the validity of the properties expressed in Theorems 2, 3, 4 and Corollary 1, the raw moments of w^ i up to the fourth-order with sample sizes 10, 30 and 100 are studied.
(9) Distribution of the Inverted Linear Compound of Dependent F-Variates 967. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. separately. Because each of the three weights studied leads to similar conclusions, only the numerical results of the first weight (w1 ) are reported in Table 2. Three important points are noted as follows. First, for each reported inverted linear compound with three different sample sizes, the condition jbj , 1 is satisfied. For example, for the case, 1/(1 þ 0.7 F1 þ 0.01 F2) with sample sizes 10, 30, and 100, jbj ¼ 0:5804, 0:4999, and 0.4971, respectively. Second, the numerical results are consistent with Corollary 1, displaying a monotonically decreasing sequence pattern. Third, all E(w^ 1 ) entries have a downward bias, i.e. E(w^ 1 ) ¼ E(1=1 þ a1 F1 þ a2 F2 ) , (1=1 þ a1 þ a2 ) ¼ w1 . However, the magnitude of this downward bias shrinks as the sample size increases, implying that w^ 1 tends to be asymptotically unbiased for w1 .. Pearson Type I Approximation Although the density of w^ i has been derived in the previous section, the critical values for interval estimation and hypothesis testing purposes are still extremely hard to obtain. However, since the moments are available, we can approximate the distribution of w^ i by the Pearson type I distribution which is defined as (Lee & Hu, 1996) f (x) ¼ ½b(a þ 1, b þ 1)(s1 s0 )aþbþ1 1 (x s0 )a (s1 x)b where s0 x s1 , a, b [ R.. Table 2. w1 and the raw moments of w^ 1 up to the fourth-order w1. Ew^ 1. Ew^ 21. Ew^ 31. Ew^ 41. 1/(1 þ 0.02F1 þ 0.01F2) 1/(1 þ 0.002F1 þ 0.001F2) 1/(1 þ 0.3F1 þ 0.01F2) 1/(1 þ 0.7F1 þ 0.01F2) 1/(1 þ 0.9F1 þ 0.01F2) 1/(1 þ F1 þ 0.01F2) 1/(1 þ 0.9F1 þ 0.7F2) 1/(1 þ F1 þ 0.5F2). 0.970874 0.997009 0.763359 0.584795 0.523561 0.497512 0.384618 0.400000. 0.964700 0.996307 0.744231 0.576654 0.520868 0.497211 0.382487 0.397275. Sample size 10 0.931100 0.992634 0.567644 0.354011 0.293674 0.269679 0.162398 0.174562. 0.899077 0.988980 0.441612 0.228156 0.156437 0.156437 0.075405 0.083274. 0.868543 0.985345 0.349256 0.152900 0.095475 0.095745 0.037481 0.042524. 1/(1 þ 0.02F1 þ 0.01F2) 1/(1 þ 0.002F1 þ 0.001F2) 1/(1 þ 0.3F1 þ 0.01F2) 1/(1 þ 0.7F1 þ 0.01F2) 1/(1 þ 0.9F1 þ 0.01F2) 1/(1 þ F1 þ 0.5F2). 0.970874 0.997009 0.763359 0.584795 0.523561 0.497512. 0.968968 0.996798 0.756860 0.584420 0.522624 0.497424. Sample size 30 0.939000 0.993607 0.577213 0.346758 0.281083 0.255416. 0.910054 0.990427 0.443337 0.210638 0.155174 0.134998. 0.882093 0.987259 0.342773 0.130330 0.087736 0.073263. 1/(1 þ 0.02F1 þ 0.01F2) 1/(1 þ 0.3F1 þ 0.01F2) 1/(1 þ 0.7F1 þ 0.01F2) 1/(1 þ 0.9F1 þ 0.01F2) 1/(1 þ F1 þ 0.5F2). 0.970874 0.763359 0.584795 0.523561 0.497512. 0.970324 0.761404 0.583922 0.523275 0.497487. Sample size 100 0.941556 0.581016 0.343273 0.276255 0.249945. 0.913665 0.444323 0.203130 0.147106 0.126783. 0.886626 0.340508 0.120970 0.078992 0.064911. w^ 1.
(10) Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. 968 K.-Y. Liang et al.. Table 3. Critical values for w^ 1 based on the Pearson Type I approximation Tail Probability a w^ 1. w1. a ¼ 0.005. a ¼ 0.01. a ¼ 0.025. a ¼ 0.05. a ¼ 0.95. a ¼ 0.975. a ¼ 0.99. a ¼ 0.995. 1/(1 þ 0.3F1 þ 0.01F2) 1/(1 þ 0.7F1 þ 0.01F2) 1/(1 þ F1 þ 0.5F2). 0.763359 0.584795 0.497512. 0.361607 0.185902 0.125202. 0.406223 0.223393 0.142536. 0.470608 0.279248 0.171396. Sample size 10 0.524328 0.902542 0.327863 0.810466 0.199804 0.625318. 0.918216 0.849467 0.668873. 0.932646 0.892380 0.717642. 0.940401 0.920052 0.749386. 1/(1 þ 0.02F1 þ 0.01F2) 1/(1 þ 0.3F1 þ 0.01F2) 1/(1 þ 0.7F1 þ 0.01F2). 0.970874 0.763359 0.584795. 0.934439 0.584444 0.348087. 0.939205 0.598797 0.371800. 0.945599 0.614514 0.405815. Sample size 30 0.950575 0.982675 0.642355 0.860101 0.435148 0.721016. 0.984428 0.874330 0.744503. 0.986272 0.888617 0.770518. 0.987420 0.896978 0.787430. 1/(1 þ 0.02F1 þ 0.01F2) 1/(1 þ 0.3F1 þ 0.01F2). 0.970874 0.763359. 0.954803 0.651932. 0.956636 0.664615. 0.959200 0.682600. Sample size 100 0.961292 0.978035 0.697790 0.814163. 0.979237 0.821643. 0.980566 0.829568. 0.981427 0.834512. Table entries are the critical values with the probability a lying beneath..
(11) Distribution of the Inverted Linear Compound of Dependent F-Variates 969. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. In order to utilize the Pearson Type I approximation, we need the first four moments of w^ i , which can be obtained as demonstrated numerically in Table 2. Let m ¼ E(w^ i ), mh ¼ E(w^ i m)h , h ¼ 2, 3, 4 and b1 ¼ m23 =m32 , b2 ¼ m4 =m22 . Then the Pearson Type I distribution requires that 6 þ 3b1 2b2 . 0, b2 b1 1 . 0. Instead of computing from the density directly, we will make use of the tables produced by Johnson et al. (1963). For this purpose, we need the following double entry interpolations. p ffiffiffiffiffi Linear interpolation is often possible for b2 , while second differences are needed for This procedure allows us to interpolate first for b2 at each of the nearest four values bp 1 .ffiffiffiffiffi of b pffiffiffiffiffi 1 . Furthermore, it also tabulates first x1 , x0 , x1 , x2 , and finally to interpolate for b1 , using the formula 1 x(u) ¼ (1 u)x0 þ ux1 u(1 u)½D2 x0 þ D2 x1 4 where u is the appropriate fraction in the tabular interval. Based on the Pearson Type I approximation, as briefed above, Table 3 gives critical values of w^ 1 with a ¼ 0:005, 0:01, 0:025, 0:05, 0:95, 0:975, 0:99, 0:995 for several cases considered in Table 2. In reference to Table 3, two major results emerged. First, by picking up a ¼ 0:025 and a ¼ 0:975, it can be seen that with sample sizes 10, 30 and 100 the 95% interval estimates of w1 for w^ 1 ¼ 1=(1 þ 0:3F1 þ 0:01F2 ) lie in the interval [0.470608, 0.918216], [0.614514, 0.874330] and [0.6826, 0.821643], respectively. Most importantly by using the data in Table 3, the same method also applies to the construction of interval estimates of w1 based on particular w^ 1 and distinctive width considerations. Second, as expected, we note that, under the preassigned percentage, the larger are the sample sizes the narrower are the interval weight estimates. Conclusions Among methods of combining forecasts (Liang, 1992), the formula (2) proposed by Reid (1969), Dickinson (1973) or Newbold & Granger (1974) is perhaps the single most extensively used measure of the optimal weights. Despite the popularity of this formula, very little is known about the sampling properties of its estimator. Although Dickinson (1973) has studied this issue, it only dealt with the combination of two forecasts exhibiting no covariance between their errors. Dickinson (1973, p. 259) also mentioned that ‘[t]he exact derivation of confidence intervals for the weights . . . of the combined forecasts is extremely complex when more forecasts, or covariance between errors, are introduced’. In this paper, attention has been directed mainly to the combination of three forecasts exhibiting no covariance between their errors. With normally distributed forecasting errors, we show that each estimated weight is expressible as an inverted linear compound of dependent F-variates and has approximately a mixture of weighted beta distributions. Operational results, including rth-order raw moments and critical values of the density are subsequently obtained by using the Pearson Type I approximation technique. As a contribution to the probability theory, our findings extend Lee & Hu’s (1996) recent investigation on the distribution of the linear compound of two independent F-variates. In terms of relevant applied works, our results refine Dickinson’s (1973) inquiry on the distribution of the optimal combining weights estimates based on combining two independent rival forecasts, and provide a further advancement to the general case of combining three independent competing forecasts. Accordingly,.
(12) 970 K.-Y. Liang et al. this paper gives a new perception of constructing the confidence intervals for the optimal combining weights estimates studied in the literature of the linear combination of forecasts. A cautious approach is also suggested when applying the popular equal weighting combining method, because the existing conditions of f (w^ i ) and its rth-order moment are practically very hard to satisfy. In this paper, we have enlarged the forecasting error covariance matrix from Dickinson’s (1973) 2 2 diagonal setting to 3 3. We strongly hope that this can serve as a stepping stone into the studies based on the more general formulation of the matrix.. Appendix: Proofs. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. Appendix A.1. Proof of equation (10) Suppose X1 , X2 and X3 are three independent chi-square distributed random variables with T degrees of freedom. By independence, we have fX1 X2 X3 (x1 ,x2 ,x3 ) ¼ fX1 (x1 )fX2 (x2 )fX3 (x3 ) Using the following transformations of variables F1 ¼. X1 X1 , F2 ¼ , F 3 ¼ X 1 X2 X3. we obtain the joint probability density function of F1 , F2 and F3 fF1 F2 F3 (f1 , f2 , f3 ) ¼. 1 f1ðT=2Þ1 f2ðT=2Þ1 f3ð3T=2Þ1 e(1þð1=f1 Þþð1=f2 Þ)f3 =2 Þ, ½G(T=2)3 2ð3T=2Þ. and the joint probability density function of F1 and F2 fF1 F2 ¼. G(3T=2) f1ðT=2Þ1 f2ðT=2Þ1 3 ½G(T=2) (1 þ ð1=f1 Þ þ ð1=f2 Þ)ð3T=2Þ. It is easy to verify that r. E(Fj j ) ¼. G(T=2 þ rj )G(T=2 rj ) , 8rj [ N ½G(T=2)2. and E(F1r1 F2r2 ) ¼. G(T=2 þ r)G(T=2 r1 )G(T=2 r2 ) , r ¼ r1 þ r2 ½G(T=2)3. Hence, E(Fj ) ¼. T T 2.
(13) Distribution of the Inverted Linear Compound of Dependent F-Variates 971 E(Fj2 ) ¼ E(F1 F2 ) ¼. T(T þ 2) (T 2)(T 4) T(T þ 2) (T 2)2. and cov(F1 , F2 ) ¼. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. var(Fj ) ¼ corr(F1 , F2 ) ¼. 2T (T 2)2 4T(T 1) (T 2)2 (T 4) T 4 2(T 1). Appendix A.2. Proof of Theorem 2 Since w^ i . 1 1 _ 1 þ a1 F1 þ a2 F2 1 þ hF(m1 , m2 ). we have the following approximate probability density function of w^ i f (w^ i ) ¼. (1 b)m2 =2 w^ iðm2 =2Þ1 (1 w^ i )ðm1 =2Þ1 B(m2 =2, m1 =2) (1 bw^ i )m. Using a negative binomial expansion we can express f (w^ i ) as f (w^ i ) ¼. 1 (1 b)m2 =2 X CðmþjÞ1 bj w^ iðm2 =2Þþðj1Þ (1 w^ i )ðm1 =2Þ1 B(m2 =2, m1 =2) j¼0 j. or alternatively as f (w^ i ) ¼. 1 h m m (1 b)m2 =2 X m1 m1 i 2 2 þ j, þ j, CðmþjÞ1 bj B Beta j B(m2 =2, m1 =2) j¼0 2 2 2 2. Appendix A.3. Proof of Theorem 3 # ð1 ð1" 1 m m (1 b)ðm2 =2Þ X m1 m1 2 2 mþj1 j þ j, þ j, f (w^ i )dw^ i ¼ Cj bB Beta dw^ i 2 2 2 2 0 0 B(m2 =2, m1 =2) j¼0.
(14) 972 K.-Y. Liang et al.. ¼ (1 b)m2 =2. 1 X B(ðm2 =2Þ þ j, m1 =2) CjðmþjÞ1 bj B(m2 =2, m1 =2) j¼0. Consider a non-negative infinite series fan g, and let. anþ1. R ¼ lim sup. an. n!1. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. If R , 1, then fan g is absolutely convergent, by the ratio test (Apostol, 1974, p.173). By the previous theorem, with. m2 þ 2n. b ¼ jbj R ¼ lim sup. 2 þ 2n. n!1 if R ¼ jbj , 1, then ð1 f (w^ i )dw^ i ¼ 1 0. Appendix A.4. Proof of Theorem 4 ð1 r E(w^ i ) ¼ w^ ri f (w^ i )dw^ i 0. ¼ (1 b). m2 =2. 1 X ðmþjÞ1 j B(ðm2 =2Þ þ r þ j, m1 =2) Cj b B(m2 =2, m1 =2) j¼0. Again, by the previous theorem (Apostol, 1974, p. 193), with. (m þ n)(m2 þ 2r þ 2n). b. ¼ jbj Rr ¼ lim sup. n!1 (1 þ n)(2m þ 2r þ 2n) Rr ¼ jbj , 1, then the rth-order raw moment of w^ i exists.. Appendix A.5. Proof of Corollary 1 Let Brj ¼ ¼ Brþ1 j. B(ðm2 =2Þ þ r þ j, m2 =2) B(m2 =2, m1 =2) m2 þ 2r þ 2j Br m1 þ m2 þ 2r þ 2j j. Then Brþ1 , Brj , 8r [ N j.
(15) Distribution of the Inverted Linear Compound of Dependent F-Variates 973 and therefore m2 =2 E(w^ rþ1 i ) ¼ (1 b). 1 X. CjðmþjÞ1 bj Brþ1 j. j¼0. , (1 b)m2 =2. 1 X. CjðmþjÞ1 bj Brj ¼ E(w^ r ), 8r [ N. j¼0. Appendix A.6. Pearson Type I Approximation. Downloaded by [National Chiao Tung University ] at 01:29 26 April 2014. We must compute some important coefficients such as. b1 ¼. m23 m ,b ¼ 4 m32 2 m22. then check the following conditions 6 þ 3b1 2b2 . 0, b2 b1 1 . 0 and the interpolation between x0 and x1 is 1 x(u) ¼ (1 u)x0 þ ux1 u(1 u)½D2 x0 þ D2 x1 4 References Apostol, T.M. (1974) Mathematical Analysis (New York: Wiley). Bunn, D.W. (1986) Statistical efficiency in the linear combination of forecasts, International Journal of Forecasting, 1, pp. 151–163. Dickinson, J.P. (1973) Some statistical results in the combination of forecasts, Operational Research Quarterly, 24, pp. 253–260. Johnson, N.L., pffiffiffiffiffiNixon, E., Amos, D.E. & Pearson, E.S. (1963) Table of percentage points of Pearson curves, for given b1 and b2 expressed in standard measure, Biometrika, 50, pp. 459–498. Lee, J.C. & Hu, L. (1996) On the distribution of linear functions of independent F and U variates, Statistics & Probability Letters, 26, pp. 339– 346. Liang, K.Y. (1992) On the sign of the optimal combining weights under the error-variance minimizing criterion, Journal of Forecasting, 11, pp. 719–723. Newbold, P. & Granger, C.W.J. (1974) Experience with forecasting univariate time series and combination of forecasts, Journal of Royal Statistics Society, Series A, pp. 131–146. Reid, D.J. (1969) A comparative study of time-series prediction techniques on economic data. PhD thesis, Department of Mathematics, University of Nottingham. Winkler, R.L. & Clemen, R.T. (1992) Sensitivity of weights in combining forecasts, Operations Research, 40, pp. 609 –614..
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