Abstract
The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate of an unknown vector w in , as a standard estimate, if as , and for the rth-order cumulants of have magnitude and can be expanded in Here we present a significant extension. We give the expansion of the distribution of any smooth function of , say in giving its distribution to . We do this by showing that , is a standard estimate of . This provides far more accurate approximations for the distribution of than its asymptotic normality.
Keywords:
Edgeworth expansions; parametric inference; standard estimates; chain rules for cumulant coefficients MSC:
60B12; 60B20; 60E05; 62E20; 62F12; 62G86; 62H10
1. Introduction and Summary
Suppose that is a standard or Type A estimate of an unknown w in with respect to a given parameter n. That is, as and for its rth-order cumulants have magnitude and can be expanded as
where the cumulant coefficients do not depend on n, or at least are bounded as . So For example, (1) holds for a function of a sample mean. We show that if is a smooth function of a standard estimate , then it is a standard estimate of . We establish this for unbiased in Theorem 2, and for biased in Theorem 3. More generally, we define as a Type B estimate if as and for ,
For example, this type arises when considering one-sided confidence regions. If is a smooth function of a Type B estimate, then it is a Type B estimate of . So for a Type A estimate, is for and 0 for d odd. n is typically the sample size or the minimum sample size if there is more than one sample.
Section 3 and Section 4 show that a smooth function of , say , is a standard estimate of . These sections provide the cumulant coefficients of in terms of those of and the derivatives of . Section 3 does this for unbiased and Section 4 for biased. So they can be thought of as chain rules for obtaining the cumulant coefficients for from those of . We use the notation to mean that is bounded as We provide the cumulant coefficients required for Edgeworth expansions of to . Cumulant coefficients up to were given in [1]. Cumulant coefficients up to use the rth derivatives of . Section 5 specialises to univariate with examples. Theorem 3 and Corollary 4 rectify and on pages 67 and 59 of [2]. Section 2 extends the shorthand bar notation above and gives the foundation theorem.
We now summarise the expressions for Edgeworth expansions of for standard and Type B estimates in terms of the cumulant coefficients and given in [3,4,5]:
is the multivariate normal distribution with zero mean and covariance , is the complete ordinary Bell polynomial of [6]:
This equation provides the 5th-order Edgeworth expansion for the distribution of , extending it up to . It is important to note that (5) utilises the tensor summation convention of implicitly summing over their range . For example,
for a standard estimate. For a standard estimate, in (3) and the cumulant coefficients needed for of (2) are ,
Therefore, to derive the 5th-order Edgeworth expansion for the distribution of for a standard estimate, we simply substitute the coefficients in (6) and (7) in the expression for , with those corresponding to as provided in Section 3, Section 4 and Section 5.
Equation (9) of [3] provides for the more general case where is the distribution function of Y in which depends on n but is asymptotic to and has a Type B expansion. One can choose so that the number of terms in each greatly reduces: see Withers and Nadarajah (2012d) [7,8]. When is lattice, further terms need to be added: see for example Chapter 5 of [9], [10], and for the density of , p211 of [11], Section 5 of [12], and Section 6 of [13]. Corollary 1 of [3] gives the tilted Edgeworth expansion for , sometimes called the saddlepoint approximation, or the small sample expansion as it is a series in not just . It is very useful for the tails of the distribution where Edgeworth expansions perform poorly. Cumulant coefficients are also needed for bias reduction, Bayesian inference, confidence regions and power. See [7,8,14,15,16,17,18]. For examples. For a historical overview of Edgeworth expansions, refer to Section 7.
In summary, this paper gives high-order expansions for the distribution of a wide range of estimates, by determining the cumulant coefficients required for any smooth function of a standard estimate. This approach offers unprecedented accuracy for these distributions and eliminates the necessity for simulation methods.
2. Foundations
Considering in and an estimate assume that as and that for its rth-order cumulants have magnitude . Given in , we write these cumulants in shorthand as
For example, if is the mean of a random sample of size n, then (8) holds since where is the ith component of X. According to Theorem 1, Equation (8) is valid if is a smooth function of one or more sample means. Let be a smooth function in a neighbourhood of w with jth component and finite partial derivatives
where Superscripts i are reserved for the cumulants of and subscripts for partial derivatives of . Superscripts j are reserved for the components of and for the joint cumulants of . This bar shorthand allows us to shorten expressions by suppressing the is and js. We write the cumulants of as
For example, and imply that the covariance of is represented by , and the covariance of is represented by , both of which scale as . Next, we demonstrate that
In other words, employing the tensor sum convention. The rest of this section and all proofs can be skipped on a first reading. Theorem 1 provides the cumulants of when is unbiased.
We use the notation to denote summing over all N permutations of resulting distinct terms.
Theorem 1.
Note 1.
For reference regarding N in , refer to page 48 of [19]. It is important to note that the notation in terms like only applies for in the context where they are used. For example, writing and recalling that only permutes superscripts but leaves subscripts alone, we have
with not since
say, when multiplied by , as in , gives say, where for . For example, in above is shorthand for . For,
Proof
This result can be derived by substituting by according to [19]. □
Likewise, one can readily derive from pages 51–53 of [19]. The tensor form can be conceptualised as a molecule or molecular structure of 2 atoms, and , connected by the double bond 1, 2, represented as . is a linear combination of , 2 atoms linked by the triple bond 1,2,3, and secondly . The last expression has the structure of CO2, with 2 identical atoms each linked by a double bond to a central atom. Just as such bonds are depicted in chemistry to illustrate the structure of a molecule, they can be very useful here to illustrate the difference in structure of similar mathematical expressions. of Note 1 is a linear molecular form with the 4 single bonds 1,2,3,4 and 4 distinct atoms, and Other expressions have more complex structures. Doubling the last term in yields where exhibits a linear structure with a double bond between 1 and 2, followed by two single bonds, 3 and 4. Additionally, forms a square or rectangle with four single bonds 1,2,4,3 arranged along successive edges of the square. These pictorial forms are a very useful way to distinguish similar expressions in .
Section 6 provides the ’more complicated’ terms referred to (but not given) on p49 of [19] when is biased. It can be used for an alternative Proof of Theorem 3 below. From Theorem 1, Edgeworth expansions can be obtained for the distribution and density of the standardised form of ,
of the form
where are The of Theorem 1 needed for are as follows.
3. Cumulant Coefficients for when
We now show that for and , the cumulant coefficient from Equation (10) can be expanded as
Substituting with on the right-hand side of (4), denoted as RHS (4), provides the Edgeworth expansion for as in Equation (12). If is a product of cumulants as in Equation (1), let denote the coefficient of in the expansion of . For example,
Now, let us provide the elements of the expansion (14) when .
Theorem 2.
Assume that is an unbiased estimate of w satisfying Equation (1) and has finite derivatives. In this case, Equation (14) holds with bounded cumulant coefficients
and so forth. The leading coefficients needed for of (4) for the distribution of of (12) are given in the notation of Theorem 1 as follows.
Note 2.
(11) made explicit the 3 terms needed in for of Theorem 2. Similarly needs the 12 terms
where It also needs the terms where
4. Cumulant Coefficients for when
We proceed by removing the assumption of being unbiased. We utilise from Theorem 2, and the shorthand where again A significant distinction arises compared to Theorem 2: in that case, was treated as an algebraic expression. However, now we must consider each of them as a function of w. Thus, we assume that the distribution of is determined by w. This assumption is necessary to derive higher order confidence intervals for when : see [20]. It is demonstrated that for from Equation (12), require the first derivatives where , need the 1st derivatives , and so on. The derivatives of are computed using Leibniz’s rule for the derivatives of a product. For example,
Theorem 3.
Proof
and are functions of w. By (14)
where by (1), has th component Consider the Taylor series expansion
Substituting into (14) gives (17) with
Also so that (18) holds with
□
An alternative proof can be obtained using Section 6. This corrects given in Appendix B of [21]. Ref. [2] uses for but the expression for on p67, lines 2–3 omitted the term . That is, the last term in of Theorem 3 was omitted. Similarly the results on p67 for are only true when the is unbiased or the cumulant coefficients of do not depend on w, as they omit the derivatives of . The examples given there are not affected as is unbiased. Nor are the nonparametric examples of [22] and [23] affected, as the empirical distribution is an unbiased estimate of a distribution. Likewise is unbiased for the examples of [20]. M-estimates are biased but the results of [16] are not affected as only are given. No changes are needed for [3,4,17,24]. Applications to non-parametric and parametric confidence intervals were given in [22] and [20,23] and to ellipsoidal confidence regions and power in [4] and [25]. For nonparametric problems, and its empirical distribution play the role of w and ; since it is unbiased, no corrections are needed. For were given for parametric and non-parametric problems in [22] and [2,23] and expressions for the classic Edgeworth expansion of in terms of were given in [14]. For , for parametric problems were given in [2], and can be obtained easily from given when for 1-sample and multi-sample non-parametric problems in [22] and [23] and for semi-parametric problems in [16,24]. All these results can be extended to samples with independent non-identically distributed residuals, as done in [26] Section 6 and [17]. The extension to matrix just needs a slight change in notation. For example, in [17], can be viewed as a function of the mean of n independent complex random matrices, although n is actually the number of transmitters or receivers. Extensions to dependent random variables are also possible: see [27].
5. Cumulant Coefficients for Univariate
Now suppose that . Let be the coefficient of in . We write as . For , (14), (16) and (20) become
For , (17)–(19) become
Here, we give the cumulant coefficients needed for the Edgeworth expansion of of (12) for . We do this when in Corollary 1 and when in Corollaries 3 and 4. To show more clearly the expressions we need in molecular form, we introduce the following ions, (expressions with unpaired suffixes),
where a suffix does not have a match then summation does not occur. For example, the RHS of sums over but not . Let be the 27 functions of given on p4234–4235 of [20], labelled there as . By Corollaries 1 and 3 below, those needed for , of (4), that is, for the Edgeworth expansion of of (12) to , are the following molecules.
Each molecule can be written as a shape. For example, is a rectangle. We now give the molecules needed for the Edgeworth expansion to , that is, for for . Note that needs the derivatives of up to order .
These and do not use derivatives of , the cumulant coefficients of .
Corollary 1.
Proof
Since becomes N. We write as . By Theorem 2 we need the following.
□
Example 1.
Suppose that and is linear in w. Then for . For , the needed for of (4) for the distribution of of (12) are as follows.
For, are 0, as are most and
So for , for we only need to calculate these 3 and 5 .
Let be a gamma random variable with known mean Its rth cumulant is For a standard exponential random variable
Example 2.
Linear combinations of scale parameters. Suppose that and is linear, the components of are independent, and for has a distribution with known rth cumulant . Then, for and
For example, if is a gamma random variable with mean then .
For and any function , set summed over their range. In Example 3 their range is ; for example in In Example 4 their range is ; for example
Example 3.
Suppose that and are independent, where has magnitude n. Set . Then for , and cross-cumulants of are zero. Take . Then by Corollary 1, are given in terms of
as follows.
Similarly, one can write down the Ls needed for .
Example 4.
Suppose that we have the summary statistics from k samples of size from normal populations with means and variances . Take . So we have p independent statistics, and where has magnitude n, the total sample size. Set
Then for , and cross-cumulants of are zero. Suppose that only depends on , as in Example 3.3 of [2]. (The notation there is slightly different). Then
and by Corollary 1, the coefficients needed are as follows.
Corollary 2.
Set Then
where is of Corollary 1 with replaced by
Proof
This is straightforward. □
Looking at as functions of w, we denote their partial derivatives with respect to , say. by and similarly for higher derivatives. We shall give the ones we need in Lemma 1. When constructing confidence regions, one needs to assume that the distribution of is determined by w. So far, we have not assumed this. For biased, we need
Corollary 3.
Proof
This follows from Theorem 3. where is the coefficient of in the expansion of about . □
For , and any , let sums over all N permutations of giving distinct terms. For example,
The derivatives of and needed for Corollary 3 are given by
Lemma 1.
Proof
For example, substitute into □
So now we can write needed for Corollary 3 in molecular form:
Corollary 4.
Assume that the conditions of Corollary 3 hold. Then and given there satisfy
Proof
were given for by Theorem 3. Corollaries 5.3, 5.4 agree with given for on p59 of [2] except that in was overlooked. □
Fisher and Cornish (1960) [28] showed the accuracy available using a few terms for the quantile expansions for the chi-square (or gamma), Student’s t, and F distributions. Similar results can be given for the accuracy of their Edgeworth expansions in approximating their distributions.
6. An Extension to Theorem 1
Here we remove the condition in Theorem 1 that and give the extra terms referred to but not given on p49 of [19]. We use of Theorem 1, and the shorthand where Suppose that for , the rth-order cumulants of (8) can be expanded as
There is a key difference with Theorem 1: there, was treated as an algebraic expression. But now we must view each of them as a function of w. So, we assume that the distribution of is determined by w.
The derivatives of of Theorem 1 are given by Leibniz’s rule for the derivatives of a product:
Theorem 4.
Proof
and are functions of w. By (10),
where by (30), has th component Consider the Taylor series expansion
say. Substituting into (10) gives (31) with
Also so that (32) holds with
□
The Edgeworth expansion (13) holds if are replaced by .
7. Discussion
Approximations to the distributions of estimates is of vital importance in statistical inference. Asymptotic normality uses just the first term of the Edgeworth expansion. That approximation can be greatly improved with further terms. When the estimate is a sample mean, basic results were given by [29] and [30] with major advances by [31,32,33], Corollary 20.4 of [9], and many others. For an application to the jackknife, see [34]. See [35] for some historical references. For an application to the bootstrap, see [26]. For an application to transport, see [36]. For an application to medical research, see [27]. For an application to econometrics, see [37]. For an extension to order stats for a finite population, see [38]. For a first-order application to inference on networks, see [39]. For more historical references and a recent application to option and derivative pricing, see [40].
Extensions to stationary sequences were given by [41,42] For a derivation of the Edgeworth expansion for a sample mean from the Gram–Charlier expansion, see [5,43] for the univariate and vector cases. These showed for the first time that the coefficients in these expansions were Bell polynomials in the cumulants.
The first extension from a sample mean for univariate estimates was by [28,44] They assumed that the rth cumulant of the estimate was where is a constant. However, in applications they assumed that was a Type A estimate, and collected terms. It was not until [14] that explicit results were given a univariate Type A estimate. Major advances were made in [3]. This gave explicit results for the terms in the Edgeworth expansion of a Type A or B estimate using Bell polynomials, as outlined in Section 1. It also allowed for expansions about asymptotically normal random variables. The advantage of this approach in greatly reducing the number of terms in each was illustrated in [7].
For univariate estimates, Cornish and Fisher (1937) [44] also showed how to invert the Edgeworth expansion to obtain an expansion for the distribution quantiles. This was extended to Type A estimates in [14]. For extensions to transformations of multivariate estimates, like , see [4,45,46]. An application to the amplitude and phase of the mean of a complex sample is given in [47].
Turning now to smooth functions of a Type A estimate, the first univariate results were given by [2] for parametric problems and [22] for nonparametric problems. These built on a deep result of [19]. This is why if is a Type A (or B) estimate of w, then a smooth function of , say , is a Type A (or B) estimate of .
The extension from a vector to a matrix estimate is just a matter of relabelling: a single sum becomes a double sum. The first examples of this we know of are in [17,24]. The extension to a complex scalar or vector or matrix w was given in these same papers. The first of these three papers applied it to the multi-tone problem in electrical engineering, and the other papers to channel capacity problems where is a weighted mean of complex matrix random variables, and n is no longer a sample size, but the number of transmitters or receivers.
A different type of extension can be obtained by identifying a sample mean from a distribution with its empirical distribution , and with , a smooth functional of , such as the bivariate correlation. is a Type A estimate of , and its cumulant coefficient can be read off those of . In this way one obtains the Edgeworth expansion for See [15] and its references for one or more weighted samples. An extension to samples from a linear process was given by [18].
A caveat on the use of an Edgeworth expansion is that including more terms makes it more inaccurate in the tails. This is where the tilted expansions, also known as saddlepoint, or small sample expansions, become essential. Results for the density of for a sample mean, were given in Section 5 of [12] and Section 6 of [13]. For a discussion and more references on tilting, mainly for a sample mean, see [11]. Ref. [3] shows how the cumulant coefficients given in this paper can be used to obtain the tilted expansion for the distribution and density of any Type A estimate.
8. Conclusions
Let represent a Type A estimate of an unknown parameter w belonging to . Its cumulant coefficients, as defined by (1), serve as the foundational elements for the Edgeworth expansion (2) as a series , where n is typically the sample size.
The necessary coefficients for the rth term, , are provided in (6) and (7). Consider a smooth function mapping to , which in turn is regarded as a Type A estimate of in . Consider a smooth function mapping to , which in turn is regarded as a Type A estimate of mapping to . This paper presents the cumulant coefficients for in terms of those of and the derivatives of . Substituting these coefficients into (2) yields the Edgeworth expansion of up to .
The tilted Edgeworth expansion for , crucial for tail accuracy, was previously delineated in [3] in terms of its cumulant coefficients. By incorporating those of as presented here, we derive the tilted Edgeworth expansion for .
In many practical statistical estimation problems, simulations serve as a favored method for approximating distributions. However, their limitation lies in their inability to comprehensively represent the entire parametric landscape.
We have showcased some applications in electrical engineering. For instance, ref. [17] offered numerical comparisons of the initial three approximations to channel capacity for multiple arrays with multiple frequencies and delay spread. Given , this permitted an expansion for the percentile. There exist myriad other potential applications across electrical engineering and allied fields.
Lastly, we outline potential future research avenues. Chain rules applied to can yield the cumulant coefficients of its Studentised form, paving the way for expansions in the coverage probability of confidence regions and enhancements in their accuracy. These coefficients find applications in bias reduction, Bayesian inference, confidence regions, and power analysis. While the Edgeworth expansion can sometimes yield negative values in distribution tails, tilted expansions circumvent this issue. Alternatively, selecting y in such that is offers another approach. For the diversity of y choices allows for potential reductions to or smaller. Introducing replacements such as further expands the range of options.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article and Appendix A.
Acknowledgments
I wish to thank the reviewers for their careful reading of the manuscript. Their suggestions have greatly improved the paper.
Conflicts of Interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Appendix A. Some Comments on the References
Here, we give some comments and corrections to some of our papers.
Withers (1982) [2]: To the expression for on p59 add where
This correction does not effect applications in which is unbiased, as in [2,23].
In the expression on p60 for , should be .
On p61, 4 lines before Table 1, replace by .
On p67, add to For see Section 4.
On p68 in (A3), replace by . Changing to the simpler notation of [15], denote the expressions for and given on pages 58–59 by , and So, the expressions on pages 59–60 become
We now illustrate how the results on p.60 were obtained. Let denote when is replaced by its Studentised form Then
The first few derivatives of at w, and of , are
Substitution into (A1) yields . The other given on p.60 are obtained similarly.
Withers (1984) [14]:
p393: In the 5-line expression for , replace by , and by .
p394: In (3.4), replace by
p394: In line 2 of Section 4, ‘of Section 2’ should be ‘of Section 3’.
The following corrigendum for a printer’s error appeared in Withers, C.S. J. R. Stat. Soc. B, 1986, 48, p258:
The expression should be added to the last line on p393.
That also gives and for the last line on p393.
Withers (1987) [21]:
p2371: (2.4): need not converge. We only require an asymptotic expansion. The same is true for (3.2) p2375.
p2371, 3rd to last paragraph: Replace ‘Appendix C, which also’ with ‘Appendix D. Appendix C’
p2372, Example 2.2, line 2: Replace with , the derivative of . In line 3 and in Example 3.1,
p2377 line 2: Replace (1.2) with (2.4)
p2377 line 3: Replace with
p2377 line 9: Replace ‘Section 2’ with ‘Section 3’ Since of Example 3.2 p2376 is unaffected.
p2378: These expression for are correct if is unbiased. In that case, the terms on p2378 with a 1 in the top line are 0 so that has only terms where However, if is biased, then these expression for did not allow for contributions from replacing by in the cumulant coefficients of (3.2). These are corrected in Withers, C.S. and Nadarajah, S. (Submitted), Bias-reduced estimates for parametric problems.
p2379 Appendix D: Add at start: For see (3.4) of
Withers, C.S. and Nadarajah, S., Journal of Multivariate Analysis, 2013, 118, 138–147.
Withers (1988) [23]:
p729: In the 10th line from the bottom, replace “their range ” with “their range ”
p732 line 9: should be .
p734: In the expression for in the 5th to last line, replace with
p737: In line 11, “Sections 1 and 2 of Withers (1983a)” should read “Sections 1 and 2 of Withers (1983b)”.
p741: In the 4th equation from the bottom, at the end of the line, replace with
Withers and Nadarajah (2008) [15]:
p743 para 2, line 4: Replace ’about zero.’ qith ’about zero when G puts mass 1 at x.’
p754, p756: Replace with . Different samples can have different weights.
p754, 2nd-to-last line: The first term on RHS, , should be .
p755, line 6: There is a typesetting error in the first of the 2 lines for . Replace the first line with
p756: The 3rd and 4th lines after (8), should be
Withers and Nadarajah (2009) [43]:
p272. Line 3: Convergence of is not needed, since is a finite sum.
on LHS(1.1) should be .
p273 last paragraph: Also, is only meaningful if X is dimension-free.
p275. (2.8) is correct, but since , (2.8) can also be written as
In the 5th line of Section 3, insert after , ‘at ’.
The first line of (3.1) should read
(3.2) can be written where is the integral part of x.
p276: In the expression for should be .
p277: In the 2nd-to-last line, should be .
p278: In the expression for , the first term should be doubled. In the expression for , should be .
Withers and Nadarajah (2010a) [16]:
p3: In the 5th- and 6th-to-last lines, replace with
p5: 2 lines above Theorem 2.2, replace “third moments” with “third central moments”
p7, lines 2–3: delete “and its Studentised version”
p7, lines 3–4: delete “or ”
p7, line 7–10: delete from “So, a one-sided” to “by
p9: Move “Set on the last 2 lines of p9 and 1st line of p10 to just before “Set” on p9 line 9.
p10, lines 14–15: replace “ where” by “.” and move the rest of the sentence, “” to the line after (6.1) p9, preceded by the word “Set”
Withers and Nadarajah (2010b) [3]:
p1129, line 7: replace by
To the 9th-to-last line we can add
From p1130 line 6 to the end of Section 5: Replace s with p, the dimension of .
p1130 line 7 is clearer, we replace line 8 with
p1130 line 9: Replace with
p1130, 5th- and 6th-to-last lines: for example where
p1132: A note on Corollary 3.2. For the duality of and see p176 of McCullagh, P., Tensor methods in statistics. Chapman and Hall, London, 1987.
p1133: In line 14, replace with
Withers and Nadarajah (2014a) [5]:
p81: In (2.14), replace and with and .
p81: The 2nd line after (2.15) should read
The next line is correct:
p82: In (2.20), replace with .
p85: In Withers, C.S. and Nadarajah, S. (2009), replace ‘via’ with ‘in terms of’.
Withers and Nadarajah (2014b) [7]:
p676. Multiply RHS of (1.13) by . That is, replace it by
p699: In the editing of the original paper of 64 pages down to 21 pages, some details had to be removed. Here, are some more details for Theorem 1.2 after (1.24)
where the needed for are as follows.
p702 Section 2: In the 3rd equation of Theorem 1, should be . p704: Disregard Table 3.
Withers and Nadarajah (2015) [8]:
In (22) and the formulas for that follow, replace by
As stated this gives For example,
In the first reference, [1], replace J.J. Alfredo with J.A. Jimenez.
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