Identifying the generating distribution of business and economics data: an empirical method.

ABSTRACT

A general framework for empirically identifying familial membership within the Pearson system of distributions is proposed. Specifically, the research addresses the problem of constructing a point and a non-parametric confidence interval estimate of familial membership. Unlike most identification procedures that require the data to support or negate membership in a specific hypothesized family, e.g., Chi-Square and Empirical Distribution Function goodness of fit, the presented approach uses bootstrap re-sampling techniques to help identify the likely generating family of the data.

1. INTRODUCTION

The study of systems of probability distributions first appeared in the literature in the late 1800's when Karl Pearson (1895) began to question the basic assumption of normal theory. His empirical studies revealed non-normal characteristics to be inherent features of many populations. Initially, many of Pearson's contemporaries doubted the need for curves (systems of distributions) other than the normal density. However, by the turn of the century, theoreticians and empiricists had accepted the possibility of non-normality, and began to explore various typologies for alternative distributions. Of these typologies, the Pearson (1902a; 1902b) system of probability distributions became the most cited.

The Pearson, as well as complementary systems of distributions, such as Johnson's [S.sub.U] and [S.sub.B] system (Johnson 1949) are particularly relevant to research in economic and business disciplines when the phenomenon under inquiry is assumed to be governed by a probability law. Often, the specific law in question is unknown, but its realizations (data) are available. In such instances these systems provide a general familial space of distributions in which one member, i.e., a point in the space, can be identified as the "most likely" candidate generating the observed phenomenon. Thus, knowledge of general systems of distributions (families) can aid the researcher in addressing the basic problem: Given a sample set of independently, identically distributed (iid) observations, how does one empirically determine the implied generating family?

In much economics and business research however, the problem of familial identification is circumvented by imposing an explicit a' priori functional form for the generating family, e.g., X is distributed normal. Mathematical tractability often determines the choice of the assumed family, e.g., closure of normally distributed variables under addition, rather than empirical or theoretical arguments. Simplifying the problem in this manner reduces the general research issue of familial identification to parameter estimation within a given family; in the case of the normal density family, the estimation of [mu] and [sigma]. Imposing a' priori distribution can be costly in terms of the congruity of the research model with the phenomena being studied, and hence the usefulness of the research. While there are many procedures to reject a specific distribution, such as normality, a more useful approach would be to determine the family or families of distributions that are consistent with the data.

We can formally state the simplified problem as follows: Let the random component of the phenomenon of interest be represented by an iid random variable X. Further, let the distribution of X belong to a family of distributions {f(x;[omega]): [omega] [member of] [OMEGA]}. Each member f (x;[omega]) of {f(x;[omega]): [omega] [member of] [OMEGA]} is uniquely defined (identified) by the values of the parameter set [omega] [member of] [OMEGA]. For example, assume that the distribution of a random variable X follows the normal probability law [psi](x;[mu],[sigma]). The law [psi](x;[mu],[sigma]), however, can be viewed as a general family of distributions {[psi](x;[omega]): [omega][member of] [OMEGA]}, where [omega] = ([mu], [sigma]), and [OMEGA] = {([mu],[sigma])| -[infinity], < [mu] < +[infinity], 0 < [sigma] < +[infinity]}. Since a unique member of this family exists for each pair ([mu], [sigma]) [member of] [OMEGA], the simplified research problem is to decide on the basis of data which member, or members of the assumed family, "best" represents the distribution of X. Thus, the problem of within-family identification is one of statistical estimation, estimation of the underlying parameter(s) that uniquely identify a member of the assumed family implied by the data. It begs the question however, of the empirical or conceptual validity of the a'priori familial specification. This question is important when parameter estimation, and hence, membership inference depends on the general family in question.

This paper proposes a statistical procedure to identify the likely candidate(s) of probability distribution(s) that generated a set of observed data. The procedure is useful when data realizations are assumed to be governed by a single, unknown, continuous distribution having membership in the Pearson system of probability densities. Using this procedure a researcher can construct a point and, more informatively, a joint confidence interval estimate that respectively identify a single Pearson class or classes (families) of distributions that could have propagated the observed data. The procedure employs a computationally intensive technique referred to as "bootstrap" (Efron, 1985; Efron, 1982; Efron & Tibshirani, 1993; Efron & Tibshirani, 1986).

First, the basic properties of the Pearson system (1895) are discussed along with an exposition of Craig's (1936) parsimonious representation. Second, having defined the familial space of interest, the theory of bootstrap estimation and its application to familial identification is presented. Here, bootstrap statistics for two parameters that jointly define a unique generating family is developed along with their bias corrected joint confidence interval.

2. THE PEARSON SYSTEM AND CRAIG'S REPRESENTATION

For every member of the Pearson system the probability density function f(x) must satisfy the differential equation

(1) (1/f)(df/dx) = -{(a+x)/([b.sub.0]+[b.sub.1]x+[b.sub.2][x.sup.2])},

subject to the conditions f(x) [greater than or equal to] 0, and [integral]f(x)dx = 1 (Elderton & Johnson 1969; Fisher 1922; Johnson & Kotz 1970; Ord 1972). The form of the solution to the differential equation above, and hence, the form of a particular family, depends on the nature of the roots of [b.sub.0]+[b.sub.1]x+[b.sub.2][x.sup.2] = 0. The shape of the distribution is in turn governed by the values of a, [b.sub.0], [b.sub.1] and [b.sub.2]. For example, if [b.sub.1] = [b.sub.2] = 0, then (1) can be written as dlogf(x)/dx = -{(x+a)/[b.sub.0]), giving the solution f(x) = [Constant][exp{-(x+a)2/2[b.sub.0])}], i.e., the normal family with mean -a and variance [b.sub.0].

Altogether, there are three "main" Pearson types I, IV and VI, (these are conventional, albeit, not intuitively logical labels), and ten "transition" types (special cases of I, IV and VI), the normal, II, III, V, and VII to XII types, with the normal being the limiting case of I, IV, and VI. For the empiricist, the attractiveness of the Pearson system is that it includes, among others, the beta (Type I), uniform (Type II), gamma (Type III), inverse gaussian (Type V), central f (Type VI), central t (Type VII), exponential (Type X), and the normal density families (Johnson & Kotz, 1970).

If we define [square root of [[beta].sub.1]] = [[mu].sub.3]/ [[[mu].sub.2]).sup.3/2] and [[beta].sub.2] = [[mu].sub.4]/([mu.sub.2)]).sup.2] where [[mu].sub.i] = ith moment about the mean [[mu].sub.1], ([square root of [[beta].sub.1] is the moment coefficient of skewness, and [[beta].sub.2] the moment coefficient of kurtosis), then every pair ([[beta].sub.1], [[beta].sub.2]) defines unique familial membership within the Pearson system (Elderton & Johnson, 1969; Johnson et al, 1963; Ord, 1972). For example, ([[beta].sub.1], [[beta].sub.2]) = (0,3) and ([[beta].sub.1], [[beta].sub.2]) = (4,9) respectively identify the normal and exponential density families. Given the solutions to the differential equation in (1), and all theoretically possible pairs ([[beta].sub.1], [[beta].sub.2]), the complete Pearson system can be defined on the {[[beta].sub.1]}x{[[beta].sub.2]} plane.

Craig (1932) however suggests an elegant and, for our purpose, more tractable alternative to the standard [[beta].sub.1]-[[beta].sub.2] representation of the Pearson family of curves. Craig maps every Pearson family onto the {[[beta].sub.1]}x{[delta]} plane, where [delta] is defined by [delta] = (2[[beta].sub.2]-3[[beta].sub.1] - 6)/([[beta].sub.2]+3). The corresponding divisions of Pearson types in Craig's [[beta].sub.1]-[delta] plane are given by the five lines:

(2) [delta] = -1 ([delta] > -1)

(3) [delta] = -.5

(4) [delta] = 0

(5) [delta] = .4 (5 < .4)

(6) [[beta].sub.1] = 0

and the two curves,

(7) [[beta].sub.1] = 4[delta]([delta]+2), and

(8) (2+3[delta])[[beta].sub.1] = 4(1+2[delta]).sup.2]([delta]+2).

The points of Pearson's Type V lie on the graph of (7) and (8), agreeing with the functions of Type VIII, IX, X and XI. Of particular importance are the two linear boundaries given by (2) and (5) above. Values of [delta] < -1 denote the "Impossible Area" of all frequency curves where [integral]f(x)dx [not equal to] 1. Values of [delta] > .4 designate the "Heterotypic Area" where [[mu].sub.8] fails to exist, or equivalently, the standard deviation of the fourth moment in samples would be infinite (Craig, 1932).

Since Craig's mapping (and the standard [[beta].sub.1]-[[beta].sub.2] depiction of Pearson's system) is expressed in terms of [[beta].sub.1] ([[beta].sub.1] [greater than or equal to] 0), information about the direction of skewness is lost. Reflecting (7) and (8) about [[square root of [[beta].sub.1] ]= 0, a modified version of Craig's map in [[square root of][[beta].sub.1]]-[delta] space is possible. This modified representation is presented in Figure 1. Starting at the top half of the graph we have the three Pearson "main" types, I, IV, VI, and their corresponding shapes, U, J, J', B(ell) depicted in boxes. For completeness, the remaining ten "transition" classes, and several families of special interest have also been included. The bottom portion of the graph presents information about the tail characteristics of their respective families (symmetric about [square root of [[beta].sub.1] = 0).

[FIGURE 1 OMITTED]

All symmetric families of distributions reside along the abscissa ([[square root of [[beta].sub.1]] = 0) in Figure 1. Beginning at the left with [delta] > -1, one encounters the Pearson Type II "U" shaped curves. As [delta] is increased, the "U" becomes less pronounced, and then completely vanishes at [delta] = -.5, the uniform family. Moving just to the right of the uniform family the "U" inverts to form a dome with vertical sides of height [approximately equal to] 1.0. Eventually, the vertical sides of the dome uniformly diminish to form the shape of a bell. As the normal family ([delta] = 0) is approached, the mass within the tails gradually shifts to the center of the bell causing the curve to drop very quickly to zero as one moves in either direction from the mode. At the point [delta] = 0, the ordinate [square root of [[beta].sub.1] is the boundary between finite and infinite tailed distributions. The skewed bell shaped gamma family (Type III) is bounded above by the J shaped exponential (Type X) and below by the normal distribution. For values of [delta] > 0, the Pearson Type IV bell shaped families have two "fat", albeit infinite, tails, e.g., central t. Finally, the uniform is the limiting case of all finite tail "U", "J", and bell shaped distributions; the normal of all Pearson main types I, IV and VI.

The general shape of Pearson Type I distributions for [[square root of [beta].sub.1]] > 0 is governed by values of [delta]. Beginning with -1 < [delta] < -.667), are the "U" shaped families I(U). Then, as VIII(J) is further approached from the left, there is a continual flattening of the right (upturned) portion of the "U" until a "J" is formed at VIII(J). At VIII(J) the right end point ordinate while still positive is now finite. Progressing from VIII(J) to XII(J') we encounter the "twisted-J" shaped distributions, where the right side of the "J" bends downwards and approaches zero at XII(J'). Increasing 5 beyond -.5 the "twisted-J" forms a "standard-J." Here, the curve continuously decreases at decreasing rates, with infinite and zero ordinates at respectively the left and right end points. The "standard-J" shape is maintained at the IX(J) curve where both end points have finite ordinates. Finally, crossing over IX(J), we enter into the region of finite tailed, bell shaped densities. With the exception of the tail(s), the basic "J" and bell shapes are maintained as we vertically traverse [[square root of [[beta].sub.1] at [delta] = 0.

Within the context of [[square root of [beta].sub.1]] and [delta] the general problem of familial identification can be reduced to knowledge of the central moments [[mu].sub.2], [[mu].sub.3], and [[mu].sub.4]; [[mu].sub.i] = (x-[[mu].sub.1]).sup.i]f(x)dx of a random variable X having density function f(x). By constructing the moment coefficient of skewness [[square root of [[beta].sub.1], and kurtosis [[beta].sub.2], we can identify the membership of f(x) on the {[[square root of [beta].sub.1]1}[]{[[beta].sub.2]} (Johnson et al, 1963) or {[[beta].sub.1]} []{[[beta].sub.2]} plane (Elderton & Johnson 1969; Ord 1972; Pearson 1902a; Pearson 1902b; Solomon & Stephens, 1978). In terms of Craig's mapping the empirical issue is straight forward. First, take a random sample [X.sub.1], ..., [X.sub.n] of size n from unknown density f(x), and obtain the point estimates [[beta].sub.1] and [[beta].sub.2] (bold-underscored terms denote estimates) subject to the moment bias corrections given by Deutsch (1965). Having ([[beta].sub.1], [[beta].sub.2]), and knowing the sign of [[mu].sub.3], we may then obtain the estimate [bar.[delta]], and "plot" the realized coordinate ([[square root of [bar.[beta].sub.1]], [bar.[delta]]) on the {[[square root of [[beta].sub.1]}x{[delta]} plane. The reader should be cautioned that while the mapping [delta] = ([[beta].sub.2] - [[beta].sub.1]-6)/([[beta].sub.2]+3) is one-to-one onto from the [[beta].sub.1] - [[beta].sub.2] to the [[beta].sub.1]-[delta] space, the mapping defined by [delta] = (2[[beta].sub.2]-3[[beta].sub.1]-6)/([[beta].sub.2]+ 3) is not one-to-one from the [][[beta].sub.1]-[[beta].sub.2] to the [[beta].sub.1][delta] space; and [delta] = (2[[beta].sub.2]-3[[beta].sub.1]-6)/([[beta].sub.2] + 3) from the [[beta].sub.1]-[[beta].sub.2] to the [[beta].sub.1]-[delta] space does not preserve convexity (see Conjectures 1-3, Appendix A for proofs).

Just how accurate are the estimates [square root of [bar.[beta].sub.1], and [bar.[delta]]? How confident can we be in realized coordinate ([square root [bar.[beta].sub.1]], [[bar.[delta]), and hence, in the implied generating family? Statistically, the answer to these questions requires knowledge of the standard errors of [[square root of [beta].sub.1]], and [bar.[delta], and more complex, knowledge of the values of [[zeta].sub.L], [[zeta].sub.U], [v.sub.L], and [v.sub.u] such that Prob[[[zeta].sub.L] [less than or equal to] [[square root of [beta].sub.1] [less than or equal to] [[zeta].sub.U], [v.sub.L] [less than or equal to] [delta] [less than or equal to] [v.sub.U]] = 1-[alpha]. Although the marginal and joint distribution of [square root of [bar.[beta].sub.1]] and [bar.[beta].sub.2]] has been approximated for samples drawn from the normal and Type I family (Bowman and Shenton, 1973; D'Agostino & Pearson, 1973; Shenton & Bowman, 1977), virtually nothing is known about the frequency behavior of [bar.[delta]]. More importantly, employing traditional statistical analysis to address these two questions requires a' priori knowledge of the parent distribution f, i.e., [X.sub.i] is iid f for i = 1, ..., n. Clearly, possession of such knowledge is logically circular relative to the identification problem at hand.

The bootstrap, is a computationally based methodology designed to appraise statistical problems that are not mathematically tractable (Efron & Tishirani, 1993; Efron & Tishirani, 1986). In essence, bootstrap methodology substitutes numerical for theoretical analysis. The following section discusses the general theory of the bootstrap and its application to the problem of constructing a joint confidence interval for [[square root of [bar.[beta].sub.1], and [bar.[delta]].

3. BOOTSTRAP ESTIMATIONS OF [check][[beta].sub.1] AND [delta]

Suppose that [X.sub.1], ... ,[X.sub.n] constitutes a random sample of size n from a population with unknown density function f. For our purpose, f is taken to be a member of the Pearson system. Further, assume that we wish to make some inference about a population parameter [theta], e.g., [check][[beta].sub.1] or [delta]. Let [theta]([X.sub.1], ... ,[X.sub.n]) be a statistic for [theta], and let f represent the sample distribution of f, i.e., f is the relative frequency distribution of the observed data. Here, f can be taken as the non-parametric maximum likelihood estimate of f, assigning mass 1/n to each [X.sub.i]. The bootstrap seeks to approximate the sampling distribution of [theta] using f and Monte Carlo simulation (Beran, 1982; Efron, 1982; Effon & Tishirani, 1993; Efron & Tibshirani, 1993; Efron & Tishirani, 1986). First, f is constructed using sample data [X.sub.1] = [x.sub.1], ... ,[X.sub.n] = [x.sub.n]. Then, an lid bootstrap sample [X'.sub.1], ... ,[X'.sub.n] is drawn with replacement from f, and the statistic [theta]' = [theta]([X'.sub.1], ... ,[X'.sub.n]) is calculated. This second step is repeated B times. In effect, we draw B independent samples of size n from f, and for each such sample, obtain an estimate [theta]'. Given the B estimates of [theta]', denoted by [[theta]'.sub.1], ... , construct a bootstrap distribution of [theta], denoted by [THETA]', having standard error

(9) [sigma] = [{[(B-1).sup.-1] [B.summation over (j = 1)] [([[theta]'.sub.j] - [theta]").sup.2]}.sup.1/2], where [theta]" = [(B).sup.-1] [B.summation over (j = 1)] [[theta]'.sub.j].

Thus, the bootstrap procedure outlined above provides a technique to empirically estimate the sampling distribution of a statistic [theta] without explicit knowledge of f. If the bootstrap sample size is equal to n, then as B [right arrow] [infinity], [sigma] approaches the true standard error of [theta]([X.sub.1], ... ,[X.sub.n]) under f (Efron 1982). Moreover, as n [right arrow] [infinity], we have [THETA]' [right arrow] [THETA]. It is important to note, however, that the "accuracy" of the bootstrap results clearly depends on how well f images f, i.e., the sufficiency of the original sample [X.sub.1], ... , [X.sub.n] to image its parent distribution.

For our purpose, the advantage of the bootstrap lies in its ability to approximate a confidence interval for a parameter [theta]. While some have expressed concerns about the coverage probability of bootstrap confidence intervals, e.g., Martin (1990), Schenker (1985), their efficacy in situations where the standard maximum likelihood (1-a)100% estimate [theta] [member of] {[theta] [+ or -] [sigma][z.sub.a/2]} is neither appropriate nor possible has been well documented (Bickel & Krieger, 1989; Efron, 1985; Efron & Tibshirani, 1986; Tibshirani, 1988). The bootstrap method of assigning a confidence interval to any real-valued parameter [theta] is based on the distribution [THETA]. Let F(t) = Prob'[theta]' [less than or equal to] t] denote the distribution function of [theta]' [~.sub.iid] [THETA]. Here, F(t) may be approximated by the number of elements in the set {[theta]' < t} normalized by B, i.e., F(t) = #{[theta]' < t}/B. For any a on the open interval (0,1), define [[theta].sub.Low](a/2) = [F.sup.-1](a/2), and [[theta].sub.Up](a/2) = [F.sup.-1](1-a/2). Efron's (Efron & Tibshirani, 1986; Efron 1982) percentile method for a 1-a central confidence interval for [theta] is given by

(10) [[[theta].sub.Low](a/2),[[theta].sub.Up](a/2)].

Since, a/2 = F([[theta].sub.Low]) and 1-a/2 = F([[theta].sub.Up]), the interval consists of the central 1-a proportion of the bootstrap distribution [THETA]. Operationally, the construction of a (1-a)100% bootstrap interval is quite simple. First, sequence the B estimates [[theta]'.sub.1], ... ,[[theta]'.sub.B]. Now, trim the bottom and top (a/2)B realizations. The end points of the remaining (1-a)B estimates will be respectively, [[theta].sub.Low](a/2), and [[theta].sub.Up](a/2). When Prob'[[theta]' < [theta]"] [not equal to] .5, where _[theta]" is the estimate of [theta] based on the original sample [X.sub.1] = [x.sub.1], ... ,[X.sub.n] = [X.sub.n] (not a bootstrap), then Efron's bias-corrected percentile method should be used (Efron, 1982). The 1-a bias-corrected interval is defined by

(11) [[F.sup.-1] ([[theta](2[z.sub.0]-[z.aub.a])), [F.sup.-1] ([theta](2[z.sub.0] + [z.sub.a]))],

where [PHI] is the distribution function for the standard normal variate, [z.suub.0] = [[PHI].sup.-1]F([theta]")), and [z.sub.a/2] the upper a/2 point [PHI]([z.sub.a/2]) = 1 - a/2. Note that (11) reduces to (10) if Prob'[[theta]' < [theta]"] = .5, or equivalently [z.sub.0] = 0. In effect, the bias-corrected percentile method attempts to incorporate the skewness of f by redistributing the probability unequally in the two tails of [THETA].

While the estimation of the standard errors of [cheak][[beta].sub.1], and [delta] can be accomplished by (9) above, how one should construct their joint confidence interval is not clear. Using (11), we can construct separate (1-a)100% confidence intervals for [check][[beta].sub.1] and [delta]. Further, it is well known (Wilks, 1963), if Prob[([[zeta.sub.L] < [check][[beta].sub.1] < [[zeta].sub.U]] = 1-a and Prob[[v.sub.L] < [delta] < [v.sub.u]] = 1- [epsilon], then in general Prob[[[zeta].sub.L] < [check][[beta].sub.1] < [[zeta].sub.U], [v.sub.L] < [delta] < [v.sub.U] VU] [not equal to] (1-a)(1-[epsilon]). For the rectangular confidence region on the {[check][[beta].sub.1]} x {[delta]} plane, i.e., {([check][[beta].sub.1], [delta])|[[zeta].sub.L] < [check][[beta].sub.1] < [[zeta].sub.U], [v.sub.L] < [delta] < [V.sub.U]}, we have

(12) Prob[[[zeta].sub.L] < [check][[beta].sub.1] < [[zeta].sub.U], [v.sub.L] < [delta] < [v.sub.U]] > 1-(a+[epsilon]).

Thus, for [epsilon] = a, and using the results in (11) and (12), we can construct the joint rectangular confidence interval for [check][[beta].sub.1] and [delta] having confidence coefficient of at least 1-2a. Given the one-to-one correspondence between a coordinate ([check][[beta].sub.1],[delta]), and a specific family within the Pearson system, a (1-2a)100% joint confidence region for [check][[beta].sub.1] and [delta], represents a (1-2a)100% confidence region of familial membership (see proof of Conjecture 4 in Appendix A).

3.1 Operational Summary

The operational steps of the proposed [check][[beta].sub.1]-[delta] bootstrap approach to familial identification can be summarized as follows:

Step 1: Draw a random sample [X.sub.1], ... ,[X.sub.n] of size n from unknown parent distribution f (an assumed member of the Pearson system).

Step 2: Estimate the central moments [[mu].sub.r], r = 2,3,4 using their unbiased estimators (Deutsch, 1965)

(13) [m.sub.2] = {n/(n-1)}[[mu].sub.2],

(14) [m.sub.3] = {[n.sup.2]/[(n-1)(n-2)]}[[mu].sub.3] [m.sub.4] = {[n([n.sup.2] - 2n + 3]/[(n - 1)(n - 2)(n - 3)]}[[mu].sub.4]

(15) - {[3n(2n - 3)]/[n - 1)(n - 2)(n - 3)]}([[mu].sub.2]),

where,

(16) [[mu].sub.r] = (1/n) [n.summation over (i = 1)] [([X.sub.i] - X).sup.r], and X = (1/n) [n.summation over (i = 1)] [X.sub.j].

Step 3: Using the results obtained in Step 2, calculate [[beta].sub.1] ([check][[beta].sub.1]), [[beta].sub.2], and [delta].

Step 4: Draw a simple random sample [X'.sub.1], ... ,[X'.sub.n] of size n from f [equivalent to] {[X.sub.1]/n, ... ,[X.sub.n]/n}, and obtain the bootstrap estimates following Steps 1 and 2.

Step 5: Repeat Step 4 B-times.

Step 6: Calculate the separate and joint confidence intervals for [check][[beta].sub.1] and [delta] using (11) and (12) above.

In steps 1 through 3, the point estimates [check][[beta].sub.1], [[beta].sub.2], and [delta] are calculated using the sample observations [X.sub.1] = [x.sub.1], ... ,[X.sub.n] = [x.sub.n], [X.sub.i] [~.sub.iid] f. The bootstrap resampling algorithm is then evoked in steps 4 and 6, yielding [check][[beta]'.sub.1i], [[beta]'.sub.2i], and [[delta]'.sub.i], i = 1, ... ,B. If only the bootstrap standard errors are of interest, then B [congruent to] 200 is appropriate. However, if confidence intervals of the form (10), (11) or (12) are required, then B > 1000 is essential (Efron, 1990; Efron & Tibshirani, 1986). This rather large difference in B should not be surprising, since interval estimates are both theoretically and computationally more ambitious measures of statistical accuracy than standard errors.

4. DISCUSSION

By employing nonparametric bootstrap re-sampling techniques to construct confidence intervals for the Pearson-Craig parameters [check][[beta].sub.1] and [delta] we have in effect substituted computational power for theoretical analysis. This substitution however, is not without its problems. First, it was seen that the percentile interval (10) is a special case of the bias-corrected form given in (11). There is, however, a more general and robust bootstrap confidence interval, the accelerated bias-corrected interval, currently being discussed in the literature (Andrews & Buchinsky, 1999; DiCiccio & Efron, 1996; Diciccio & Romano, 1988; Efron, 1987). While superior in its level of specificity, i.e., the bias-corrected interval is a special case of the accelerated interval, and having confidence limits that are second-order correct, the accelerated method requires knowledge of an additional parameter. Also, the bias-corrected interval, the pivot needs to be only normalizing and variance stabilizing. The most general and least restrictive case is the accelerated interval, which requires merely that g is normalizing (Efron, 1988; Efron & Tibshirani, 1987). Given the difficulties associated with estimating this parameter (Diciccio & Romano, 1988); the uncertainty of its gain when constructing two-sided intervals (Martin (1990); and the favorable results of Lei and Smith (2003) when using the bias-corrected interval estimates for skewness and kurtosis, the bias-corrected approach is proposed at this time.

Second, under ideal conditions B should be infinite, but we, at least for now, must work within the technological scope of our computational and memory limits. Efron (1990) presents improved computational methods that may, under certain conditions, reduce B by a factor of ten, and Andrews and Buchinsky (1999) propose a three-step procedure to choose B that ensures that the lower and upper lengths of a accelerated bias corrected interval deviate from those obtained when B [right arrow] [infinity] by at most a small percentage with high probability. The employment of these methods may prove to be fruitful when an empirical test of the proposed procedure is conducted.

Third, is the possible inapplicability of the assumptions underlying the construction of bootstrap confidence intervals (Schenker, 1985). For the three bootstrap confidence intervals, percentile, bias-corrected, and accelerated bias-corrected, we assume the existence of a monotone transformation g such that g([theta]) - g([theta]) is a normal pivotal quantity. We need not know the exact functional form of g, only that g exists. When constructing a percentile interval, g([theta]) - g([theta]) is assumed to be normalizing, variance stabilizing and median unbiased in the sense that Prob'[[theta]' < [theta]"] = .5, or equivalently, [Z.sub.o] = 0 in (11) above.

Finally, the legitimacy of the variance stabilizing assumption, and hence, use of the bias-corrected interval for [check][[beta].sub.1] and [delta] is unknown. When discussing the coverage performance of the bootstrap intervals for [check][[beta].sub.1] and [delta], we implicitly assumed that [a.sub.n] [right arrow] a as n [right arrow [infinity]. While this assumption may have strong intuitive appeal, it is not in general valid (Loh, 1987). That is, while [a.sub.n] [right arrow] [[tau].sub.n] in large samples, we may not have [[tau].sub.n] [right arrow] a as n [right arrow] [infinity]. With the exception of a few special cases, e.g., estimating the mean, the conditions under which [[tau].sub.n] [right arrow] a for bootstrap confidence intervals are not fully understood at this time.

5. APPENDIX A: PROOFS

Conjecture 1: The mapping defined by [delta] = (2[[beta].sub.2]-3[[beta].sub.1]-6)/([[beta].sub.2]+3) is one-to-one onto from the [[beta].sub.1-] [[beta].sub.2] to the [delta]- [[beta].sub.1] space.

Proof 1: In general a function [??] is 1:1 whenever [??] (a) = [??] (b) implies a = b. A function [??] is onto B, if for all b [epsilon] B, there exists an a [epsilon] A such that [??](a) = b. For our purpose we would like to prove g([[beta].sub.1], [[beta].sub.2]) = ([[beta].sub.1], [delta]([[beta].sub.1], [[beta].sub.2])) is 1-1 onto, i.e., Craig's mapping from [[beta].sub.1]-[[beta].sub.2] space to [delta]-[[beta].sub.1] space is 1:1 onto. We have, [delta]([[beta].sub.1], [[beta].sub.2]) = (2[[beta].sub.2]-3[[beta].sub.1]-6)/([[beta].sub.2]+3) for [[beta].sub.1], [[beta].sub.2] [greater than or equal to] 0. Fix the second coordinate of [delta] to say a constant [omega]. Then, for fixed [??], [delta]([[beta].sub.1], [??]) = (2[??]-3[beta] [sub.1]-6)/([??] + 3), or equivalently, [delta]([[beta].sub.1], [??]) = ((22[??]-6)/([??]+3))-(3/([??]+3))[[beta].sub.1]. Clearly, [delta] is linear in the first coordinate ([[beta].sub.1]). Thus, the function [delta] takes horizontal segments in [[beta].sub.1]-[[beta].sub.2] space linearly to line segments in [delta]- [[beta].sub.1] space. Since linear functions a+[??]x are 1-1 onto, Craig's mapping is 1:1 onto.

Conjecture 2: The mapping defined by [delta] = (2[beta]2-3[beta]1-6)/([beta]2+3) is not one-to-one from the [check][beta]1-[beta]2 to the [check][[beta].sub.1]-[delta] space.

Proof 2: From above we have, [delta]([??][[beta].sub.1], [[beta].sub.2]) = (2[[beta].sub.2]-3 [([??][[beta].sub.1]).sup.2]-6)/([[beta].sub.2]+3) for [([??][[beta].sub.1]).sup.2], [[beta].sub.2] [greater than or equal to] 0. Fix the second coordinate of [delta] to say, a constant [omega]. Then, for fixed [omega], [delta]([??][[beta].sub.1], [omega]) = (2[omega]-3[([??][[beta].sub.1].sup.2]-6)/([omega]+3), or equivalently, [delta]([??][[beta].sub.1], [omega]) = ((2[omega]-6)/([omega]+3)) - (3/ ([omega]+3))[([??][[beta].sub.1]).sup.2]. Let a = [??][[beta].sub.1], and b = + [??][[beta].sub.1]. Clearly, [delta](a,[omega]) = [delta](b,w) but a, b except for a [not equal to] b = 0. Thus, [delta]([??][[beta].sub.1], [[beta].sub.2]) is not 1:1.

Conjecture 3: The mapping defined by [delta] = (2[[beta].sub.2]-3[[beta].sub.1]-6)/([[beta].sub.2]+3) from the [[beta].sub.1]-[[beta].sub.2] to the [delta]-[[beta].sub.1] space does not preserve convexity.

Proof 3: For our purpose we would like to prove g([[beta].sub.1], [[beta].sub.2])=([[beta].sub.1],[delta]([[beta].sub.1],[[beta].sub.2])) does not preserve convexity from [[beta].sub.1]-[[beta].sub.2] to [[beta].sub.1]-[delta] space. We have, [delta]([[beta].sub.1], [[beta].sub.2]) = (2[[beta].sub.2]-3[[beta].sub.1]-6)/([[beta].sub.2]+3) for [[beta].sub.1], [[beta].sub.2] [greater than or equal to] 0. Consider the of [DELTA] = {([beta], [beta])|[beta] [member of] {[R.sup.+]}[intersection]{0}}, and define [GAMMA] ([beta],[beta])= ([beta],(2[beta],3[beta]-6)/([beta]+3) or, [GAMMA]([beta],[beta]) = ([beta],-(([beta]+6)/([beta]+3)). The second coordinate of [GAMMA] is clearly the set of points on the graph of the function given by g([beta]) = - (([beta]+6)/([beta]+3)), 13 [greater than or equal to] 0. Taking the first and second derivative of g we have g'([beta]) = {3/[([beta]+3).sup.2]} > 0 for [beta] [greater than or equal to] 0, and g"([beta]) = - {6/ [([beta]+3).sup.3]} < 0 for [beta] [greater than or equal to] 0. As such we see that g is concave down for all [beta] [greater than or equal to] 0. However, since the image of [DELTA] in [[beta].sub.1]-[[beta].sub.2] space is convex, all linear functions are convex functions, and an identity mapping [beta] = [beta] is a linear function, and [GAMMA] is non-convex by above, it is then clear that Craig's mapping does not preserve convexity.

Conjecture 4 Let I1 denote the confidence interval for parameter [[beta]1] and I2 the confidence for [beta]2, each having confidence coefficient 1-a. The joint rectangular confidence interval for [beta]1 and [beta]2 defined by {I1}x{I2} has confidence coefficient at least equal to 1-2a.

Proof 4: Let Prob[[[beta].sub.1] [member of] [I.sub.1]] = 1-a and Prob[[[beta].sub.2] [member of] [I.sub.2]] = 1-a. If we write [not member of] for "not included in," we have Prob[[[beta].sub.1] [member of] [I.sub.1], [[beta].sub.2] [member of] [I.sub.2]] = 1 - Prob[[[beta].sub.1] [not member of] [I.sub.1], [[beta].sub.2] [member of] [I.sub.2]] - Prob[[[beta].sub.1] [member of] [I.sub.1], [[beta].sub.2] [not member of] [I.sub.2]] - Prob[[[beta].sub.1] [not member of] [I.sub.1], [[beta].sub.2] [not member of] [I.sub.2]] [greater than or equal to] 1 - {Prob[[[beta].sub.1] [not member of] [I.sub.1], [[beta].sub.2] [member of] [I.sub.2]] + Prob[[[beta].sub.1] [not member of] [I.sub.1], [[beta].sub.2] [not member of] [I.sub.2]]} - {Prob[[[beta].sub.1] [member of] [I.sub.1], [[beta].sub.2] [not member of] [I.sub.2]] + Prob[[[beta].sub.1] [not member of] [I.sub.1], [[beta].sub.2] [member of] [I.sub.2]]} = 1 - Prob[[[beta].sub.1] [not member of] [I.sub.1]] - P[[[beta].sub.2] [not member of] [I.sub.2]] = 1 - a - a = 1 - 2a. Thus, Prob[[[beta].sub.1] [member of] [I.sub.1], [[beta].sub.2] [member of] [I.sub.2] [greater than or equal to] 1-2a.

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Author Profiles:

Dr. Laurence R. Takeuchi earned his Ph.D. at the Claremont Graduate School, Claremont California. Currently he is a professor of marketing at California State University, Sacramento.

Dr. Joseph Richards earned his Ph.D. at Syracuse University, New York. Currently he is an assistant professor of marketing at California State University, Sacramento.