Generalized p value for multivariate Gaussian stochastic processes in continuous time

Abstract

We construct a Generalized p value for testing statistical hypotheses on the comparison of mean vectors in the sequential observation of two continuous time multidimensional Gaussian processes. The mean vectors depend linearly on two multidimensional parameters and with different conditions about their covariance structures. The invariance of the generalized p value considered is proved under certain linear transformations. We report results of a simulation study showing power and errors probabilities for them. Finally, we apply our results to a real data set.

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Notes

  1. 1.

    For details, see Johnson et al. (1994).

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Acknowledgements

The authors thank the anonymous referees for their helpful comments and suggestions. The first and third authors were supported by MTM2015-65825-P

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Correspondence to Mar Fenoy.

Appendices

Appendix 1

Following an approach similar to that of Gamage Gamage (1997), if hypothesis testing is \(H_0:\delta =\delta _0\) versus \(H_1:\delta \ne \delta _0\), we will replace (in the previous shown material) \(T_{st}\) by \((T_{st}-\delta _0)\), and we obtain

$$\begin{aligned} d_{st}^{\delta _0}= & {} \left( \widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)})^{-1}(\widetilde{T}_{st}-\delta _0\right) \end{aligned}$$
(7.1)
$$\begin{aligned} R_{st}^{\delta _0}= & {} d_{st}^{\delta _0'}(T_{st}-\delta _0)\left( d_{st}^{\delta _0'}(\sigma _1^2\sigma _t^{(1)}+\sigma _2^2 \sigma _s^{(2)})d_{st}^{\delta _0}\right) ^{-1/2}\sim N(0,1)\end{aligned}$$
(7.2)
$$\begin{aligned} W_{st}^{\delta _0}= & {} R_{st}^{\delta _0}\left( \frac{k_{st}^{1^0}}{U_1}+\frac{k_{st}^{2^0}}{U_2}\right) ^{1/2}\end{aligned}$$
(7.3)
$$\begin{aligned} w_{st}^{\delta ^0}= & {} \left( \widetilde{T}_{st}-\delta _0\right) '\left( \widetilde{\eta _1}\sigma _t^{(1)}+\widetilde{\eta _2} \sigma _s^{(2)}\right) \left( \widetilde{T}_{st}-\delta _0\right) , \end{aligned}$$
(7.4)

where

$$\begin{aligned} k_{st}^{1^0}= & {} \widetilde{\eta }_1 d_{st}^{\delta _0'}\sigma _t^{(1)}d_{st}^{\delta _0}\end{aligned}$$
(7.5)
$$\begin{aligned} k_{st}^{2^0}= & {} \widetilde{\eta }_2 d_{st}^{\delta _0'}\sigma _s^{(2)}d_{st}^{\delta _0} \end{aligned}$$
(7.6)

If we consider the following hypotheses

$$\begin{aligned} H_0^l:l'\delta =l'\delta _0\hbox { versus }H_1^l:l'\delta \ne l'\delta _0, \end{aligned}$$

we replace the \(T_{st}\) from earlier by \(l'(T_{st}-\delta _0)\), and we have as a result that

$$\begin{aligned} l'(T_{st}-\delta _0)\sim N\left( 0,l'\left( \sigma _1^2\sigma _t^{(1)}+\sigma _2^2\sigma _s^{(2)}\right) l\right) . \end{aligned}$$

Then we have replace \(\sigma _t^{(1)}\) by \(l'\sigma _t^{(1)}l\), and \(\sigma _s^{(2)}\) by \(l'\sigma _s^{(2)}l\), and we obtain

$$\begin{aligned} R_{st}^{\delta _0,l}= & {} l'(T_{st}-\delta _0)\left( l'(\sigma _1^2\sigma _t^{(1)}+\sigma _2^2\sigma _s^{(2)}) l\right) ^{-1/2}\sim N(0,1)\\ W_{st}^{\delta _0}= & {} l'(T_{st}-\delta _0)\left( l'(\sigma _1^2\sigma _t^{(1)}+\sigma _2^2\sigma _s^{(2)}) l\right) ^{-1/2}\left( \sigma _1^2\frac{\widetilde{\eta }_1}{\eta _1}l'\sigma _t^{(1)}l+\sigma _2^2 \frac{\widetilde{\eta }_2}{\eta _2}l'\sigma _s^{(2)}l\right) ^{1/2}\\ w_{st}^{\delta _0,l}= & {} l'(\widetilde{T}_{st}-\delta _0)\end{aligned}$$

We can rewrite \(W_{st}^{\delta _0,l}\) as following

$$\begin{aligned} W_{st}^{\delta _0,l}=R_{st}^{\delta _0,l}\left( \frac{k_{st}^{1^l}}{U_1}+\frac{k_{st}^{2^l}}{U_2} \right) ^{1/2}, \end{aligned}$$
(7.7)

with

$$\begin{aligned} k_{st}^{1^0}= & {} \widetilde{\eta }_1 l'\sigma _t^{(1)}l \end{aligned}$$
(7.8)
$$\begin{aligned} k_{st}^{2^0}= & {} \widetilde{\eta }_2 l'\sigma _s^{(2)}l. \end{aligned}$$
(7.9)

Let us observe that the conditions \(\delta =\delta _0\) and \(l'\delta =l'\delta _0\) are equivalent, for every non null vector l.

Next, let

$$\begin{aligned} t_l= & {} \frac{l'(\widetilde{T}_{st}-\delta _0)}{\left( l'\left( \widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2 \sigma _s^{(2)}\right) l\right) ^{1/2}} \quad \hbox { and }\end{aligned}$$
(7.10)
$$\begin{aligned} t_l^2= & {} \frac{\left( l'(\widetilde{T}_{st}-\delta _0)\right) ^2}{l'\left( \widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2 \sigma _s^{(2)}\right) l}. \end{aligned}$$
(7.11)

Considering that

$$\begin{aligned} \max _x\frac{(x'd)^2}{x'Bx}=d'B^{-1}d \end{aligned}$$

and that this maximum is attained at \(x=cB^{-1}d\), \(\forall c\ne 0\), then \(t_l^2\) will be the maximum for \(l=c(\widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)})^{-1}\left( \widetilde{T}_{st}-\delta _0\right) \), that is, \(l=d_{st}^{\delta _0}\) maximizes \(t_l^2\), and the maximum is

$$\begin{aligned} \left( \widetilde{T}_{st}-\delta _0\right) '\left( \widetilde{\eta }_1\sigma _t^{(1)}+ \widetilde{\eta }_2\sigma _s^{(2)}\right) ^{-1}\left( \widetilde{T}_{st}-\delta _0\right) =w_{st}^{\delta _0}, \end{aligned}$$
(7.12)

so,

$$\begin{aligned} t_{st}^1(l,\delta _0)=\frac{\left( l'\left( \widetilde{T}_{st}-\delta _0\right) \right) ^2}{ l'\left( \widetilde{\eta }_1\sigma _t^{(1)}+ \widetilde{\eta }_2\sigma _s^{(2)}\right) l}\le w_{st}^{\delta _0},\quad \hbox { }\forall l,\quad \forall \delta _0. \end{aligned}$$
(7.13)

For \(l=d_{st}^{\delta _0}\), we obtain

$$\begin{aligned} \frac{\left( d_{st}^{\delta _0'}\left( \widetilde{T}_{st}-\delta _0\right) \right) ^2}{ d_{st}^{\delta _0'}\left( \widetilde{\eta }_1\sigma _t^{(1)}+ \widetilde{\eta }_2\sigma _s^{(2)}\right) d_{st}^{\delta _0}}=w_{st}^{\delta _0}\end{aligned}$$
(7.14)

As a conclusion, we obtain that the contrasts \(H_0:\delta =0\) and \(H_0':d_{st}'\delta =0\) or equivalently \(H_0': t_{st}^2(0)=0\), where \(t_{st}^2(\delta )=t_{st}^2(d_{st},\delta )\), are equivalent; indeed we accept \(H_0'\) if the observed value \(w_{st}^{0,d_{st}}=d_{st}'\widetilde{T}_{st}\) is small, that is, if \(w_{st}\le c^2\) then

$$\begin{aligned} \frac{\left( l'\widetilde{T}_{st}\right) ^2}{l'\left( \widetilde{\eta }_1\sigma _t^{(1)}+ \widetilde{\eta }_2\sigma _s^{(2)}\right) l}\le c^2, \quad \hbox { }\forall l.\end{aligned}$$
(7.15)

This previous inequality implies that we accept the hypothesis \(l'\delta =0\), for all l, so we accept the hypothesis \(H_0:\delta =0\). On the other hand obviously to accept \(H_0:\delta =0\) implies to accept \(H_0': d_{st}'=0\). Similarly the following contrasts are equivalent:

  1. (1)

    \(H_0: \delta =\delta _0\) against \(H_1:\delta \ne \delta _0\).

  2. (2)

    \(H_0':d_{st}'\delta =d_{st}'\delta _0\) against \(H_1':d_{st}'\delta \ne d_{st}'\delta _0\) or equivalently \(H_0':t_{st}^2(\delta )=t_{st}^2(\delta _0)\).

Thus, we can conclude (for the initial contrast \(\delta _0=0\)) that

$$\begin{aligned} \frac{\left( l'\widetilde{T}_{st}\right) ^2}{l' \left( \widetilde{\eta }_1\sigma _t^{(1)}+ \widetilde{\eta }_2\sigma _s^{(2)}\right) l} \le \frac{\left( d_{st}'\widetilde{T}_{st}\right) ^2}{d_{st}'\left( \widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)}\right) d_{st}}=w_{st}, \hbox { } \quad \forall l. \end{aligned}$$
(7.16)

Appendix 2

Generalized p value for the unilateral case

Now, we consider \(H_0: \delta =0\) versus \(H_1:\delta >0\). As we can see in the Appendix 1, for \(W_{st}\) obtained for the bilateral hypothesis test, we use the direction \(d_{st}=(\widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)})^{-1}\left( \widetilde{T}_{st}-\delta _0\right) \) derived from \(\max _{x\ne 0}\frac{(x'd)^2}{x'Bx}\). Regarding the unilateral test, Park (2010) used the direction provided from \(\max _{x\ge 0}\frac{x'd}{\sqrt{x'Bx}}\) since it seems to be that \(d_{st}\) may be inflated the Type I error rate in the unilateral case, as we can see in a simulation study.

Following the approach presented in Park (2010), we have that

$$\begin{aligned} T_{st}\sim N_k(\delta , \Sigma =\sigma _1^2\sigma _t^{(1)}+\sigma _2^2\sigma _s^{(2)}), \end{aligned}$$

thus \(\widetilde{T}_{st}\) it will be a normal multivariate r.v. with mean \(\delta \) and variance-covariance matrix \(\widehat{\Sigma }=\widetilde{\eta }_1\sigma _t^{(1)}+\widetilde{\eta }_2\sigma _s^{(2)}\). So, we can write the following expressions

$$\begin{aligned} T(l)= & {} \frac{l'(T_{st}-\delta )}{\sqrt{l'\Sigma l}}\end{aligned}$$
(8.1)
$$\begin{aligned} t(l)= & {} \frac{l'(\widetilde{T}_{st}-\delta )}{\sqrt{l'\widehat{\Sigma } l}}. \end{aligned}$$
(8.2)

We define the generalized p value for the one-sided test as follows:

$$\begin{aligned} P\left( \sup _{l\ge 0}T(l)\ge \sup _{l\ge 0}t(l)\right) . \end{aligned}$$
(8.3)

In order to evaluate the previous result, we realized some simulations with R software, as Park (2010, p. 1050) did in his study, using approximations presented in formula (27), (28) and (30). The results for power for one-sided test when \(k=3\) are shown in Table 5.

Table 5 Power for dimension \(k=3\), \(\alpha =0.05\).

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Fenoy, M., Ibarrola, P. & Seoane-Sepúlveda, J.B. Generalized p value for multivariate Gaussian stochastic processes in continuous time. Stat Papers 60, 2013–2030 (2019). https://doi.org/10.1007/s00362-017-0907-7

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Keywords

  • Generalized p value
  • Hypothesis testing
  • Continuous time
  • Multivariate Behrens–Fisher problem

Mathematics Subject Classification

  • Primary: 62M09
  • Secondary: 62H12