Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The Infinitesimal Jackknife and Moment Structure Analysis Using Higher Order Moments

  • 315 Accesses


Mean corrected higher order sample moments are asymptotically normally distributed. It is shown that both in the literature and popular software the estimates of their asymptotic covariance matrices are incorrect. An introduction to the infinitesimal jackknife is given and it is shown how to use it to correctly estimate the asymptotic covariance matrices of higher order sample moments. Another advantage in using the infinitesimal jackknife is the ease with which it may be used when stacking or sub-setting estimators. The estimates given are used to test the goodness of fit of a non-linear factor analysis model. A computationally accelerated form for infinitesimal jackknife estimates is given.

This is a preview of subscription content, log in to check access.


  1. Bentler, P. M. (2012). EQS 6 structural equations program manual. Encino, CA: Multivariate Software.

  2. Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.

  3. Cragg, J. G. (1997). Using higher order moments to estimate the simple errors-in-variables model. The Rand Journal of Economics, S71–S91. Special Issue in honor of Richard E. Quandt.

  4. Cramer, H. (1946). Mathematical methods of statistics. Princeton, NJ: Princeton University Press.

  5. Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York, NY: Chapman & Hall.

  6. Hausman, J. A., Newey, W. K., Ichimura, H., & Powell, J. L. (1991). Identification and estimation of polynomial errors-in-variables models. Journal of Econometrics, 50, 273–295.

  7. Jaeckel, L. (1972). The infinitesimal jackknife, Memorandum #MM 72–1215-11. Murray Hill, NJ: Bell Laboratories.

  8. Jennrich, R. I. (2008). Nonparametric estimation of standard errors in covariance structure analysis. Psychometrika, 73, 579–594.

  9. Jennrich, R. I., & Satorra, A. (2013). Continuous orthogonal complement functions and distribution-free tests in moment structure analysis. Psychometrika, 78, 545–552.

  10. Mooijaart, A. (1985). Factor analysis of non-normal variables. Psychometrika, 50, 323–342.

  11. Mooijaart, A., & Bentler, P. M. (2010). An alternative approach for non-linear latent variable models. Structural Equation Modeling, 17, 357–373.

  12. Neyman, J. (1937). Remarks on a paper by E. C. Rhodes. Journal of the Royal Statistical Society, 100, 50–57.

  13. Ozaki, K., Toyoda, H., Iwama, N., Kubo, S., & Ando, J. (2011). Using non-normal SEM to resolve the ACDE model in the classical twin design. Behavioral Genetics, 41, 329–339.

Download references


The authors would like to thank the review team and in particular reviewer 2 who motivated the inclusion of the offline material. The research of the second author is supported by Grant EC02011-28875 from the Spanish Ministry of Science and Innovation.

Author information

Correspondence to Robert Jennrich.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 61 KB)



\(\nu _k\) has the form (2)

Note that for any two column vectors \(u\) and \(v\), \(u\otimes v\) is a permutation of \(v\otimes u\). Using this repeatedly if \(u_1,\ldots ,u_k\) is a permutation of \(v_1,\ldots ,v_k\)

$$\begin{aligned} u_1\otimes \cdots \otimes u_k=P(v_1\otimes \cdots \otimes v_k), \end{aligned}$$

for some permutation matrix \(P\).


$$\begin{aligned} \xi =(x-\mu )\otimes \cdots \otimes (x-\mu ), \end{aligned}$$

where the product is of order \(k\). This can be expanded as a sum of \(m\) terms of the form

$$\begin{aligned} \xi _i=u_{1}\otimes \cdots \otimes u_{k}, \end{aligned}$$

where each \(u_j\) is either \(x\) or \(\mu \). This can be written in the form

$$\begin{aligned} \xi _i=P(x\otimes \cdots \otimes x\otimes \mu \otimes \cdots \otimes \mu ), \end{aligned}$$

where \(P\) is a permutation matrix, the Kronecker product is of order \(k\), and the sub-product of the \(x\)’s is of order \(i\).

$$\begin{aligned} E(\xi _i)=P(E(x\otimes \cdots \otimes x)\otimes \mu \otimes \cdots \otimes \mu ). \end{aligned}$$

This is a continuously differentiable function \(h_i\) of the expected values of \(x\) and \(x\otimes \cdots \otimes x\). Thus

$$\begin{aligned} E(\xi _i)=h_i\left( \int t_i(x)\mathrm{{d}}F(x)\right) , \end{aligned}$$

where \(t_1(x)=x\) and for \(i>1\)

$$\begin{aligned} t_i(x)=\left( \begin{array}{c} x\\ x\otimes \cdots \otimes x \end{array}\right) \end{aligned}$$

and the Kronecker product has \(i\) terms.


$$\begin{aligned} h\left( \begin{array}{c} u_1\\ \vdots \\ u_m \end{array}\right) =h_1(u_1)+\cdots +h_m(u_m). \end{aligned}$$

Then \(h\) is continuously differentiable. Let

$$\begin{aligned} t(x)=\left( \begin{array}{c} t_1(x)\\ \vdots \\ t_m(x) \end{array}\right) . \end{aligned}$$

Using \(h\) and \(t\)

$$\begin{aligned} E(\xi )&= E(\xi _1)+\cdots +E(\xi _m)\\&= h_1\left( \int t_1(x)\mathrm{{d}}F(x)\right) \!+\!\cdots \!+\! h_m \left( \int t_m(x)\mathrm{{d}}F(x)\right) \\&= h\left( \left( \begin{array}{c} \int t_1(x)\mathrm{{d}}F(x))\\ \vdots \\ \int t_m(x)\mathrm{{d}}F(x) \end{array}\right) \right) \\&= h\left( \int \left( \begin{array}{c} t_1(x)\\ \vdots \\ t_m(x) \end{array}\right) \mathrm{{d}}F(x)\right) \\&= h\left( \int t(x)\mathrm{{d}}F(x)\right) . \end{aligned}$$


$$\begin{aligned} \nu _k=E(\xi )=h\left( \int t(x)\mathrm{{d}}F(x)\right) \end{aligned}$$

has the form (2).

Computing the Matrix \(C_k\)

Let \(a\in R^m\) and \(b\in R^n\). Then \(b\otimes a\) is a permutation of \(a\otimes b\) and

$$\begin{aligned} P_{mn}(a\otimes b)=b\otimes a, \end{aligned}$$

for some permutation matrix \(P_{mn}\). Note that

$$\begin{aligned} P_{m1}a=P_{m1}(1\otimes a)=a\otimes 1=a \end{aligned}$$


$$\begin{aligned} P_{m1}=I_m \end{aligned}$$

Let \(u_1,\ldots ,u_m\) be the coordinate vectors in \(R^m\) and \(v_1,\ldots ,v_n\) be the coordinate vectors in \(R^n\). Then the vectors \(u_i\otimes v_j\) with \(i=1,\ldots ,m\) and \(j=1,\ldots ,n\) are a basis for \(R^m\times R^n\). Order this basis lexicographically. Since

$$\begin{aligned} P_{mn}(u_i\otimes v_j) =v_j\otimes u_i \end{aligned}$$

the columns of \(P_{mn}\) are \(v_j\otimes u_i\) in lexicographic order on \(i\) and \(j\). All of the components of \(v_j\otimes u_i\) are zero except for its \(r=(j-1)m+i\) component which is one. Thus the computation of \(P_{mn}\) is inexpensive.

In the notation of Section 4 note that the sum of all distinct permutations of

$$\begin{aligned} (x_i-\bar{x})\otimes (x_j-\bar{x})\otimes \cdots \otimes (x_j-\bar{x}) \end{aligned}$$

is equal to the sum on \(\ell =0,\ldots ,k-1\) of

$$\begin{aligned} P_{p^{k-\ell } p^\ell }(x_i-\bar{x})\otimes (x_j-\bar{x})\otimes \cdots \otimes (x_j-\bar{x}). \end{aligned}$$


$$\begin{aligned} C_k=\sum \limits _{\ell =0}^{k-1}P_{p^{k-\ell } p^\ell }. \end{aligned}$$


$$\begin{aligned} C_k((x_i-\bar{x})\otimes (x_j-\bar{x})\otimes \cdots \otimes (x_j-\bar{x})) \end{aligned}$$

is the sum of all distinct permutations of \((x_i-\bar{x})\otimes (x_j-\bar{x})\otimes \cdots \otimes (x_j-\bar{x})\) and the mean on \(j\) of these is

$$\begin{aligned} C_k((x_i-\bar{x})\otimes q_{k-1}), \end{aligned}$$

as asserted in Section 3.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jennrich, R., Satorra, A. The Infinitesimal Jackknife and Moment Structure Analysis Using Higher Order Moments. Psychometrika 81, 90–101 (2016). https://doi.org/10.1007/s11336-014-9426-9

Download citation


  • Kronecker products
  • Asymptotically normal
  • Influence functions
  • Pseudo-values
  • Goodness of fit
  • Acceleration