Optimal equivalence testing in exponential families

Abstract

We develop uniformly most powerful unbiased (UMPU) two sample equivalence test for a difference of canonical parameters in exponential families. This development involves a non-unique reparametrization. We address this issue via a novel characterization of all possible reparametrizations of interest in terms of a matrix group. Furthermore, our procedure involves an intractable conditional distribution which we reproduce to a high degree of accuracy using saddlepoint approximations. The development of this saddlepoint-based procedure involves a non-unique reparametrization but we show that our procedure is invariant under choice of reparametrization. Our real data example considers the mean-to-variance ratio for normally distributed data. We compare our result to six competing equivalence testing procedures for the mean-to-variance ratio. Only our UMPU method finds evidence of equivalence, which is the expected result. We also perform a Monte Carlo simulation study which shows that our UMPU method outperforms all competing methods by exhibiting an empirical significance level which is not statistically significantly different from the nominal 5% level for all simulation settings.

Appendices

Appendices

1.1 Proof of Theorem 1

As discussed in Sect. 2 a transformation from matrix group M equates to the following changes in canonical parameters and sufficient statistics:

$$\begin{aligned} {\tilde{\gamma }}=A^{T}\gamma =\left[ \begin{array}{c} \psi \\ \psi a^{T}+B^{T}\lambda \end{array} \right] \\ {\tilde{Z}}=A^{-1}Z=\left[ \begin{array} {l} U-aB^{-1}V\\ B^{-1}V \end{array} \right] =\left[ \begin{array}{c} {\tilde{U}}\\ {\tilde{V}} \end{array} \right] . \end{aligned}$$

The associated likelihood function is

$$\begin{aligned} {\mathcal {L}}\left( {\tilde{\gamma }}\right) =\exp \left\{ {\tilde{\gamma }} ^{T}{\tilde{Z}}+c\left( A^{-T}{\tilde{\gamma }}\right) \right\} \text {.} \end{aligned}$$

(13)

The optimal UMPU test for $\psi$ depends upon the following conditional distribution:

$$\begin{aligned} f\left( {\tilde{u}}|{\tilde{v}};\psi \right) \end{aligned}$$

and we show that

$$\begin{aligned} f\left( {\tilde{u}}|{\tilde{v}};\psi \right) =f\left( u|v;\psi \right) . \end{aligned}$$

First note that for a regular exponential family (see for instance Butler 2007, Sec. 5.1.2) we have that

$$\begin{aligned} f\left( u|v;\psi \right)&=\exp \left\{ \psi u-c\left( \psi |v\right) -d\left( z\right) \right\} =\frac{\exp \left( \psi u-d\left( z\right) \right) }{\exp \left\{ c\left( \psi |v\right) \right\} } \end{aligned}$$

(14)

$$\begin{aligned}&=\frac{\exp \left( \psi u-d\left( z\right) \right) }{\int _{\left\{ u:z\in S\right\} }\exp \left( \psi u-d\left( z\right) \right) \mathtt {d}F\left( u\right) } \end{aligned}$$

(15)

where S is the joint support of z and the Riemann-Stieltjes integral in the denominator represents a sum in the discrete case and a surface integral in the continuous case. In a similar fashion, we have that

$$\begin{aligned} f\left( {\tilde{u}}|{\tilde{v}};\psi \right) =\frac{\exp \left( \psi \tilde{u}-d\left( {\tilde{z}}\right) \right) }{\int _{\left\{ {\tilde{u}}:\tilde{z}\in {\tilde{S}}\right\} }\exp \left( \psi {\tilde{u}}-d\left( {\tilde{z}}\right) \right) \mathtt {d}F\left( {\tilde{u}}\right) } \end{aligned}$$

(16)

where ${\tilde{S}}$ is the joint support of ${\tilde{z}}.$

Recall that

$$\begin{aligned} {\tilde{z}}=A^{-1}z \end{aligned}$$

so that

$$\begin{aligned} {\tilde{u}}=A_{\left( 1\right) }^{-1}z \end{aligned}$$

where $A_{\left( 1\right) }^{-1}$ denotes the first row of $A^{-1}$.

Then the right-hand side of (16) can be written as

$$\begin{aligned} \frac{\exp \left( \psi A_{\left( 1\right) }^{-1}z-d\left( A^{-1}z\right) \right) }{\int _{\left\{ A_{\left( 1\right) }^{-1}z\text { }:\text { } A^{-1}z\in A^{-1}S\right\} }\exp \left( \psi A_{\left( 1\right) } ^{-1}z-d\left( A^{-1}z\right) \right) \mathtt {d}F\left( A_{\left( 1\right) }^{-1}z\right) }. \end{aligned}$$

Now consider the change variable for this integral in which we replace z with Az and to get equivalent expression

$$\begin{aligned} \frac{\exp \left( \psi A_{\left( 1\right) }^{-1}Az-d\left( A^{-1}Az\right) \right) }{\int _{\left\{ A_{\left( 1\right) }^{-1}Az\text { }:\text { } A^{-1}Az\in A^{-1}AS\right\} }\exp \left( \psi A_{\left( 1\right) } ^{-1}Az-d\left( A^{-1}Az\right) \right) \mathtt {d}F\left( A_{\left( 1\right) }^{-1}Az\right) }. \end{aligned}$$

(17)

But note that

$$\begin{aligned} A_{\left( 1\right) }^{-1}A=\left[ \begin{array}{cc} 1&0_{1\times \left( p-1\right) } \end{array} \right] \end{aligned}$$

so then $f\left( {\tilde{u}}|{\tilde{v}};\psi \right) =f\left( u|v;\psi \right)$.

1.2 Proof of Theorem 2

Consider the log-likelihood function for likelihood (13) under a transformation A in matrix group M;

$$\begin{aligned} \ell \left( {\tilde{\gamma }}\right) ={\tilde{\gamma }}^{T}{\tilde{Z}}+c\left( A^{-T}{\tilde{\gamma }}\right) . \end{aligned}$$

(18)

The log-likelihood for primary reparametrization in (5) is

$$\begin{aligned} \ell \left( \gamma \right) =\gamma ^{T}Z+c\left( \gamma \right) \text {.} \end{aligned}$$

The score equations for this log-likelihood are

$$\begin{aligned} Z+\nabla c\left( \gamma \right) =0 \end{aligned}$$

where $\nabla$ is the gradient symbol which represents the first partial derivative of $c\left( \cdot \right)$ w.r.t. each element in $\gamma$. MLE ${\hat{\gamma }}$ is the solution to above score equation which means that

$$\begin{aligned} Z+\nabla c\left( {\hat{\gamma }}\right) =0 \end{aligned}$$

and

$$\begin{aligned} \widehat{{\tilde{\gamma }}}=A^{T}{\hat{\gamma }} \end{aligned}$$

due to the invariance of MLEs. A similar argument shows that the conditional MLEs follow the same idea, that is

$$\begin{aligned} \widehat{{\tilde{\lambda }}}\left( \psi \right) =A_{\left( p-1\right) } ^{T}\left[ \begin{array}{c} \psi \\ {\hat{\lambda }}\left( \psi \right) \end{array} \right] \end{aligned}$$

where $A_{\left( p-1\right) }^{T}$ denotes the last $p-1$ rows of $A^{T}$.

Next we consider the likelihood quantities in Skovgaard’s CDF approximation ( 6) and show that they are invariant under the interest parameter $\psi$ preserving reparametrizations induced by transformations in matrix group M.

First we consider likelihood ratio

$$\begin{aligned} \frac{{\mathcal {L}}\left( \psi ,{\hat{\lambda }}\left( \psi \right) \right) }{{\mathcal {L}}\left( {\hat{\psi }},{\hat{\lambda }}\right) } \end{aligned}$$

which appears in the expression for the w parameter in (6) . We first show that the maximized likelihood is invariant under reparametrizations in M. To see this note that

$$\begin{aligned} {\mathcal {L}}\left( \widehat{{\tilde{\gamma }}}\right)&=\exp \left\{ \widehat{{\tilde{\gamma }}}^{T}{\tilde{Z}}+c\left( A^{-T}\widehat{{\tilde{\gamma }} }\right) \right\} =\exp \left\{ \left( A^{T}{\hat{\gamma }}\right) ^{T} {\tilde{Z}}+c\left( A^{-T}A^{T}{\hat{\gamma }}\right) \right\} \\&=\exp \left\{ {\hat{\gamma }}^{T}Z+c\left( {\hat{\gamma }}\right) \right\} ={\mathcal {L}}\left( {\hat{\gamma }}\right) \text {.} \end{aligned}$$

The profile likelihood has similar invariance properties;

$$\begin{aligned} {\mathcal {L}}\left( \psi ,\widehat{{\tilde{\lambda }}}\left( \psi \right) \right)&=\exp \left\{ \widehat{{\tilde{\gamma }}_{p}^{T}}{\tilde{Z}}+c\left( A^{-T}\widehat{{\tilde{\gamma }}_{p}}\right) \right\} =\exp \left\{ \left( A^{T}{\hat{\gamma }}_{p}\right) ^{T}{\tilde{Z}}+c\left( A^{-T}A^{T}\hat{\gamma }_{p}\right) \right\} \\&=\exp \left\{ {\hat{\gamma }}_{p}^{T}Z+c\left( {\hat{\gamma }}_{p}\right) \right\} ={\mathcal {L}}\left( \psi ,{\hat{\lambda }}\left( \psi \right) \right) \end{aligned}$$

where $\widehat{{\tilde{\gamma }}_{p}}$ and ${\hat{\gamma }}_{p}$ are augmented MLE vectors defined as

$$\begin{aligned} \widehat{{\tilde{\gamma }}_{p}}=\left[ \begin{array}{c} \psi \\ \widehat{{\tilde{\lambda }}}\left( \psi \right) \end{array} \right] \text { and }{\hat{\gamma }}_{p}=\left[ \begin{array}{c} \psi \\ {\hat{\lambda }}\left( \psi \right) \end{array} \right] \text {.} \end{aligned}$$

Next consider the invariance of the ratio of determinants for the full and partial Fisher information matrices;

$$\begin{aligned} \frac{\left| j\left( {\hat{\psi }},{\hat{\lambda }}\right) \right| }{\left| j_{\mathbf {\lambda \lambda }}\left( \psi ,{\hat{\lambda }}\left( \psi \right) \right) \right| } \end{aligned}$$

which appears in the expression for the u parameter in (6).

Using results from (Harville 1997, Sec.15.7) the derivative of $\ell \left( {\tilde{\gamma }}\right)$ w.r.t. ${\tilde{\gamma }}$ can be expressed in the following way:

$$\begin{aligned} \frac{\partial \ell \left( {\tilde{\gamma }}\right) }{\partial {\tilde{\gamma }} }=A^{-1}\left[ Z+\nabla c\left( A^{-T}{\tilde{\gamma }}\right) \right] \end{aligned}$$

so that the Hessian matrix for $\ell \left( {\tilde{\gamma }}\right)$ is given as

$$\begin{aligned} \frac{\partial ^{2}\ell \left( {\tilde{\gamma }}\right) }{\partial \tilde{\gamma }\partial {\tilde{\gamma }}^{T}}=A^{-1}\left[ Hc\left( A^{-T}\tilde{\gamma }\right) \right] A^{-T} \end{aligned}$$

where $Hc\left( \gamma \right)$ is the Hessian matrix for $c\left( \gamma \right)$ and is given as

$$\begin{aligned} Hc\left( \gamma \right) =\frac{\partial ^{2}\ell \left( \gamma \right) }{\partial \gamma \partial \gamma ^{T}}. \end{aligned}$$

Therefore

$$\begin{aligned} \frac{\partial ^{2}\ell \left( {\tilde{\gamma }}\right) }{\partial \tilde{\gamma }\partial {\tilde{\gamma }}^{T}}=A^{-1}\frac{\partial ^{2}\ell \left( \tilde{\gamma }\right) }{\partial {\tilde{\gamma }}\partial {\tilde{\gamma }}^{T}}A^{-T} \end{aligned}$$

and

$$\begin{aligned} j\left( \psi ,{\tilde{\lambda }}\right) =A^{-1}j\left( \psi ,\lambda \right) A^{-T}. \end{aligned}$$

To consider partial information matrix for ${\tilde{\lambda }}$ we first partition the full information matrix for $\gamma$ in the following way:

$$\begin{aligned} j\left( \psi ,\lambda \right) =\left[ \begin{array}{cc} j_{\psi \psi }\left( \psi ,\lambda \right) &{} j_{\psi \lambda }\left( \psi ,\lambda \right) \\ j_{\lambda \psi }\left( \psi ,\lambda \right) &{} j_{\lambda \lambda }\left( \psi ,\lambda \right) \end{array} \right] . \end{aligned}$$

Note that

$$\begin{aligned} j\left( \psi ,{\tilde{\lambda }}\right)&=A^{-1}j\left( \psi ,\lambda \right) A^{-T}\\&=\left[ \begin{array}{cc} 1 &{} -aB^{-1}\\ 0_{\left( p-1\right) \times 1} &{} B^{-1} \end{array} \right] \left[ \begin{array}{cc} j_{\psi \psi }\left( \psi ,\lambda \right) &{} j_{\psi \lambda }\left( \psi ,\lambda \right) \\ j_{\lambda \psi }\left( \psi ,\lambda \right) &{} j_{\lambda \lambda }\left( \psi ,\lambda \right) \end{array} \right] \left[ \begin{array}{cc} 1 &{} 0_{1\times \left( p-1\right) }\\ -aB^{-T} &{} B^{-T} \end{array} \right] \\&=\left[ \begin{array}{cc} j_{\psi \psi }\left( \psi ,{\tilde{\lambda }}\right) &{} j_{\psi {\tilde{\lambda }} }\left( \psi ,{\tilde{\lambda }}\right) \\ j_{{\tilde{\lambda }}\psi }\left( \psi ,{\tilde{\lambda }}\right) &{} j_{\tilde{\lambda }{\tilde{\lambda }}}\left( \psi ,{\tilde{\lambda }}\right) \end{array} \right] \end{aligned}$$

so that

$$\begin{aligned} j_{{\tilde{\lambda }}{\tilde{\lambda }}}\left( \psi ,{\tilde{\lambda }}\right) =B^{-1}j_{\lambda \lambda }\left( \psi ,\lambda \right) B^{-T}. \end{aligned}$$

Consider now the following ratio of determinants of the full and partial Fisher information matrices for a secondary reparametrization;

$$\begin{aligned} \frac{\left| j\left( \psi ,{\tilde{\lambda }}\right) \right| }{\left| j_{{\tilde{\lambda }}{\tilde{\lambda }}}\left( \psi ,\tilde{\lambda }\right) \right| }=\frac{\left| A^{-1}j\left( \psi ,\lambda \right) A^{-T}\right| }{\left| B^{-1}j_{\lambda \lambda }\left( \psi ,\lambda \right) B^{-T}\right| }=\frac{\left| j\left( \psi ,\lambda \right) \right| }{\left| j_{{\tilde{\lambda }}{\tilde{\lambda }} }\left( \psi ,\lambda \right) \right| }. \end{aligned}$$

This result has a number of ramifications including the invariance of the approximate asymptotic conditional variance of ${\hat{\psi }}$ given that $V=v$ which is given in (Butler 2007, Sec. 5.4.5) as

$$\begin{aligned} j_{\psi \psi \cdot \lambda }^{-1}&=\frac{\left| j_{\lambda \lambda }\left( \psi ,{\hat{\lambda }}\left( \psi \right) \right) \right| }{\left| j\left( \psi ,{\hat{\lambda }}\left( \psi \right) \right) \right| }\\&=\left. j_{\psi \psi }\left( \psi ,\lambda \right) -j_{\psi \lambda }\left( \psi ,\lambda \right) j_{\lambda \lambda }^{-1}\left( \psi ,\lambda \right) j_{\lambda \psi }\left( \psi ,\lambda \right) \right] _{\psi ,\hat{\lambda }\left( \psi \right) }. \end{aligned}$$

This asymptotic conditional variance is used to define the conditionally studentized statistic for $\psi$ in Sect. 5.1.