### Table 3. Bias and Monte Carlo variation in conventional PEA Parameterizations

1997

"... In PAGE 39: ....2.1. Conventional PEA Table3 provides information on the performance of conventional PEA in approximating the conditional expectation, exp [e(k, O)] , that is the focus of Marcet (1988) apos;s analysis. The results in Marcet and Marshall (1994) indicate that conventional PEA is arbitrarily accurate for sufficiently large M and N.... In PAGE 39: ... The question that interests us here is how well the algorithm works for the values of M and N used in practice. For the results in Table3 , we set M = 10,000. By way of comparison, to solve the growth model, den Haan and Marcet (1990) use M = 2,500, den Haan and Marcet (1994) use M = 29,000, and den Haan (1995) uses M = 25,000.... In PAGE 39: ...aphson method to solve a* - S(a*; N. M) = 0. When this method is successful at finding a solution, we found it does so more quickly than does the successive approximation method. The first three terms in each cluster of four numbers in Table3 provide information about bias. The unbracketed term is the value of the statistic, s, indicated in the first column implied by the dynamic programming solution.... In PAGE 40: ...The results in Panel A of Table3 pertain to various second moment properties of consump tion, investment, and output. Here, aj , j = y, c, i denote the standard deviation of gross output, consumption and gross investment, respectively, and p(y, j), j = c, i denote the correlation of gross output with consumption and gross investment, respectively.... In PAGE 40: ... Here, aj , j = y, c, i denote the standard deviation of gross output, consumption and gross investment, respectively, and p(y, j), j = c, i denote the correlation of gross output with consumption and gross investment, respectively. The results in Panel B of Table3 pertain to first and second moment properties of Tobin apos;s q and asset returns. The results in Panel A indicate that, at least for parameterizations (1)-(6), the conventional PEA performs reasonably well.... In PAGE 41: ...results are based on I = 50). These are reported in column 2 of Table 4 (column 1 simply reproduces the results from Table3 for convenience.) TO diagnose the reasons for the poor performance of conventional PEA for model (7), consider the results in Figure 4.... In PAGE 43: ...Approximating Marcet apos;s Conditional Expectat ion Function by PEA Collocation We applied PEA collocation to approximate e in all seven models, and obtained acceptable accuracy with N = M = 3 for models (1) to (6). By apos;acceptable apos;, we mean that all statistics analyzed in Table3 and 4 are within 10 percent of their exact values. We only study bias for this method, since Monte Carlo uncertainty is not applicable.... In PAGE 58: ...ith a value for a) to construct mz, ..., mso,ool in the manner described in step #l. These data were then used in the nonlinear regression specified in step #2. collocation N=3 N=5 (ii) The entries in the first column are reproduced from the last column in Table3 . There, I = 500, though 48 of these had to be discarded because capital converges to zero in simulation.... ..."

### Table 2: Parameterization of k

1995

"... In PAGE 10: ... It was heartening to see that any possible size increase due to loss of precision was almost always overtaken by the reduction due to many-to-one object name mapping. Table2 summarizes our observations. Note that for deriv2, the size of alias solution and precision of function resolution su ered for a lower k.... ..."

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### Table 1: Parameterized vertices

"... In PAGE 9: ... How- ever, each vertex is given as the intersection of hyperplanes defined by the constraints with faces of A1BG. Figure 2 shows the general position of these intersection points, and Table1 presents them as a list. The third column of Table 1 states the conditions on the parameters for when the intersection point is an actual vertex of the polytope.... In PAGE 9: ... Figure 2 shows the general position of these intersection points, and Table 1 presents them as a list. The third column of Table1 states the conditions on the parameters for when the intersection point is an actual vertex of the polytope. Such conditions we will subsequently encounter in great numbers; they are formally introduced by the following definition.... In PAGE 10: ...(r)=0.4, I(s)=0.2 (0,0,1,0) (0,0,1,0) I(r)=0, I(s)=0 (0,0,1,0) Figure 1: Polytopes for different parameter values 3 5 2 4 8 6 7 1 Figure 2: General vertex positions problem statement for generating a complete parameterized vertex list can now be refined as follows: given input constraints BV, we have to find a list DACY BM APCY B4BD AK CY AK C5B5 (13) where each DACY is a parameterized vertex as in (7), and the APCY are lists of p-constraints, such that for every parameter instantiation C1 the set of vertices of A1B4C1B4BV B5B5 is just CUC1B4DACYB5 CY C1 satisfies APCYCV (where, naturally, C1 satisfies AP iff for every APCX AH D4CX AO BC BE AP: C1B4D4CXB5 AO BC). Table1 provides this list for the input constraints (9) and (10). To obtain a systematic method for generating such a list it is convenient to consider one by one the different faces of A1BEC3, in... In PAGE 12: ...icularly suitable method is fraction free Gaussian elimination (see e.g. [7]). This is a variant of Gaussian elimination that avoids divisions, which is useful for us, as otherwise we would have to divide by symbolic expressions that might be zero for some parameter values and nonzero for others, thereby requiring us to make a number of case distinctions. As an illustration for the working of the algorithm we retrace how vertex 8 in Table1 was generated. This vertex is the solution of the system (14)-(16) defined by CS BP BE, C0 BP CUBDBN BEBN BFCV and the (then mandatory) selection of both constraints CRBDBN CRBE for (16).... In PAGE 14: ... To illustrate the general method, we continue with our example, taking C8 B4BMBT CY BUB5 to be the target probability of the inference rule to be derived. The probability of BMBT given BU at the vertices listed in Table1 is evaluated by computing DABFBPB4DABDB7DABFB5, which leads to the values listed in Table 4. Note that the possible values of C8 B4BMBT CY BUB5 are still annotated with the parameter constraints on the vertices at which they are attained, and that for vertices 5 and 8 the new p-constraint D7 BO BD has been added.... In PAGE 21: ... Minimal irredundant sets of values for minimization and maximization of C8 B4BT CY BU CM BWB5 are indicated by the +-marks in the columns 8 and 9, respectively. The final bound functions we obtain now are C4B4D6BN D8BN D9BN DAB5 BP minCJD6BPDA BM AQ BN DA BQ BCCL (40) CDB4D6BN D8BN D9BN DAB5 BP maxCJBC BM D6 BP BCBN DA BP BDBN BD BM AQ BN D9 AK D6BN D6 BQ BCBN BD BM AQ BN D9 AK D8BN D8 AK DABN D8 BQ BCBN BD BM AQ BN D6 AK D9BN D9 AK D8BN D9 BQ BCBN BD BM AQ BN DA AK D8BN DA BQ BCBN D8BPD9 BM AQ BN D8 AK D9BN D9 BQ BCCLBM (41) where the p-constraints AQ suppressed in Table1 have been reinstated. Remembering the con- ventions min BN BP BDBN max BN BP BC, and taking into account that the conditions D6 AK D8BN D9 AK DA are taken for granted in (29), these functions can be seen to be the same as (30) and (31).... ..."

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### Table 1. Parameters of the parameterized luminosity function.

"... In PAGE 5: ... (1992). The optimal values of the parameters for p = 1; 2; 3 are tabulated in Table1 ; the case p = 3 will be included in some of the tests we present for methodological perspective. For graphical purposes, it is often convenient to work with the distribution per volume per log luminosity, given by ^ (L) ? d d(log10L) = 1 log10eL (L) : (12) We shall use this form in making plots of the luminosity function.... In PAGE 6: ... We plot the parameterized luminosity function (Eq. 11, using the parameters of Table1 ) as a solid curve, and superimpose the luminosity functions derived using the nonparametric method for several di erent binnings: the bin size used by SNWZ (in log L), nll = 10 as recommended by SNWZ, and nll = 253. We plot error bars only for the non-parametric luminosity function that most closely approaches the parametric one.... In PAGE 9: ... We make two points here: rst, we have found qualitatively very similar results to those presented here when we use the parameterized luminosity function of Eq. (11), which uses only three parameters ( Table1 ). Second, note that the t to the luminosity function is done for galaxies with redshifts between 500 and 12,000 km s?1; for p = 1, this t remains good for galaxies between 12,000 and 20,000 km s?1, while for p = 2 and p = 3, the t is unacceptable in this range.... ..."

### Table 1 Loss functions for the physically parameterized models.

2001

"... In PAGE 9: ... This means that in the rst case ten physical parameters are identi ed and in the second case ve additional parameters for the noise description are estimated. In Table1 the value of the loss function of the estimated models are shown for the three data sets and the two model structures. As expected the loss function is much smaller when the noise model is included.... ..."

### Table 2: Link functions and parameterizations of the linear predictor.

2004

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### Table 21: Non-approximability of various levels of the Function Bounded NP Query Hierarchy. y Applies only to complete problems.

"... In PAGE 89: ...P query hierarchy. Though the correspondence is not exact ([Kre88, p. 492]; [CP91, p. 243]), there is a pattern of approximability and non-approximability (see Table21 ). This pattern may assume greater signi cance in the light of future discoveries of lower limits on approximability.... ..."

### Table 21: Non-approximability of various levels of the Function Bounded NP Query Hierarchy. y Applies only to complete problems.

"... In PAGE 82: ...P query hierarchy. Though the correspondence is not exact ([Kre88, p. 492]; [CP91, p. 243]), there is a pattern of approximability and non-approximability (see Table21 ). This pattern may assume greater signi cance in the light of future discoveries of lower limits on approximability.... ..."

### Table 2: Surface Parameterization for business jet problem

1997

"... In PAGE 12: ... Six design variables were used whose associated mode shapes were combinations of 3 chordwise functions and 2 spanwise functions. The selected design variables result in a wing parameterization given by equation (36) and the functions fi and gj for this case are listed in Table2 . Note that the chordwise functions are given by a shear function (which is similar to a twist variable for small geometry perturbations), and two Hicks-Henne functions.... ..."

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