Universal inference with composite likelihoods

Let $\mathbf{X}_{n}=\left(\bm{X}_{i}\right)_{i=1}^{n}$ be a sequence of $n$ IID random variables with the same DGP as $\bm{X}$, and split $\mathbf{X}_{n}$ into two subsamples $\mathbf{X}_{n}^{1}=\left(\bm{X}_{i}^{1}\right)_{i=1}^{n_{1}}$ and $\mathbf{X}_{n}^{2}=\left(\bm{X}_{i}^{2}\right)_{i=1}^{n_{2}}$ of sizes $n_{1}$ and $n_{2}$, respectively, where $n=n_{1}+n_{2}$. We assume that $\bm{X}$ has a DGP that is characterized by the PDF $p\left(\bm{x};\bm{\theta}_{0}\right)$, for some $\bm{\theta}_{0}\in\Theta$, and we let $\Pr_{\bm{\theta}_{0}}$ be its corresponding probability measure. Let $\hat{\bm{\theta}}_{n}^{1}$ and $\hat{\bm{\theta}}_{n}^{2}$ be a pair of generic estimators of $\bm{\theta}_{0}$, using only $\mathbf{X}_{n}^{1}$ or $\mathbf{X}_{n}^{2}$, respectively.

For $k\in\left\{1,2\right\}$, we let

$$L_{\bm{\sigma},\bm{\tau}}\left(\bm{\theta};\mathbf{X}_{n}^{k}\right)=\prod_{i=1}^{n_{k}}p_{\bm{\sigma},\bm{\tau}}\left(\bm{X}_{i}^{k}\right)$$

be the CL function of $\mathbf{X}_{n}^{k}$, as a function of $\bm{\theta}$. We write the split sample CL ratio statistics (spCLRSs) and the swapped sample CL ratio statistic (swCLRS) as

$$U_{\bm{\sigma},\bm{\tau}}^{k}\left(\bm{\theta};\mathbf{X}_{n}\right)=L_{\bm{\sigma},\bm{\tau}}\left(\hat{\bm{\theta}}^{3-k};\mathbf{X}_{n}^{k}\right)/L_{\bm{\sigma},\bm{\tau}}\left(\bm{\theta};\mathbf{X}_{n}^{k}\right)\text{,}$$

for each $k\in\left\{1,2\right\}$, and

$$\bar{U}_{\bm{\sigma},\bm{\tau}}\left(\bm{\theta};\mathbf{X}_{n}\right)=\left\{U_{\bm{\sigma},\bm{\tau}}^{1}\left(\bm{\theta};\mathbf{X}_{n}\right)+U_{\bm{\sigma},\bm{\tau}}^{2}\left(\bm{\theta};\mathbf{X}_{n}\right)\right\}/2\text{,}$$

respectively.

For $\alpha\in\left(0,1\right)$, let

$$\mathcal{C}^{\alpha}\left(\mathbf{X}_{n}\right)=\left\{\bm{\theta}\in\Theta:U_{\bm{\sigma},\bm{\tau}}^{1}\left(\bm{\theta};\mathbf{X}_{n}\right)\leq 1/\alpha\right\}\text{ and }\bar{\mathcal{C}}^{\alpha}\left(\mathbf{X}_{n}\right)=\left\{\bm{\theta}\in\Theta:\bar{U}_{\bm{\sigma},\bm{\tau}}\left(\bm{\theta};\mathbf{X}_{n}\right)\leq 1/\alpha\right\}$$

be confidence sets based on the spCLRS and the swCLRS, respectively. We have the following result regarding the validity of $\mathcal{C}^{\alpha}\left(\mathbf{X}_{n}\right)$ and $\bar{\mathcal{C}}^{\alpha}\left(\mathbf{X}_{n}\right)$ (all theoretical results in this work are proved in Nguyen, 2020).

We now consider the testing of null and alternative hypotheses

$$\text{H}_{0}:\bm{\theta}\in\Theta_{0}\text{ and }\text{H}_{1}:\bm{\theta}\in\Theta_{1}\text{,}$$

where $\Theta_{0},\Theta_{1}\subseteq\Theta$. Let

$$\mathbb{M}\left(\mathbf{X}_{n}^{k}\right)=\left\{\bm{\theta}\in\Theta_{0}:L_{\bm{\sigma},\bm{\tau}}\left(\bm{\theta};\mathbf{X}_{n}^{k}\right)=\max_{\bm{\vartheta}\in\Theta_{0}}L_{\bm{\sigma},\bm{\tau}}\left(\bm{\vartheta};\mathbf{X}_{n}^{k}\right)\right\}$$

be the set of maximizers of the CL function $L_{\bm{\sigma},\bm{\tau}}\left(\bm{\theta};\mathbf{X}_{n}^{k}\right)$, for each $k\in\left\{1,2\right\}$, and write $\tilde{\bm{\theta}}_{n}^{k}\in\mathbb{M}\left(\mathbf{X}_{n}^{k}\right).$ We then write the sample splitting and sample swapping test statistics as

$$V_{\bm{\sigma},\bm{\tau}}^{k}\left(\mathbf{X}_{n}\right)=U_{\bm{\sigma},\bm{\tau}}^{k}\left(\tilde{\bm{\theta}}_{n}^{k}\right)\text{, and }\bar{V}_{\bm{\sigma},\bm{\tau}}\left(\mathbf{X}_{n}\right)=\left\{U_{\bm{\sigma},\bm{\tau}}^{1}\left(\tilde{\bm{\theta}}_{n}^{1}\right)+U_{\bm{\sigma},\bm{\tau}}^{2}\left(\tilde{\bm{\theta}}_{n}^{2}\right)\right\}/2\text{,}$$

respectively. Further, define the split sample CL ratio test (spCLRT) and the swapped sample CL ratio test (swCLRT) by the rejection rules: reject $\text{H}_{0}$ if $V_{\bm{\sigma},\bm{\tau}}^{1}\left(\mathbf{X}_{n}\right)\geq 1/\alpha$ or if $\bar{V}_{\bm{\sigma},\bm{\tau}}\left(\mathbf{X}_{n}\right)\geq 1/\alpha$, respectively. We have the following result regarding the finite sample-validity of the tests.

We first consider a simulation study regarding data generated from the bivariate exponential distribution of Arnold et al. (1999, Sec. 4.4). Here the random variable $\bm{X}^{\top}=\left(X_{1},X_{2}\right)$ has joint PDF

$$p\left(\bm{x};\theta\right)=\kappa\left(\theta\right)\exp\left\{-x_{1}-x_{2}-\theta x_{1}x_{2}\right\}\text{,}$$

where $\theta\geq 0$ is the parameter of interest, and $\kappa\left(\theta\right)=\theta\exp\left\{-1/\theta\right\}/\int_{1/\theta}^{\infty}w^{-1}\exp\left(-w\right)\text{d}w$ is an intractable normalization constant. However, the conditional PDFs of $X_{k}|X_{3-k}=x_{3-k}$, for $k\in\left\{1,2\right\}$, can be specified by

$$p\left(x_{k}|x_{3-k};\theta\right)=f_{\text{Exp}}\left(x_{k};1+\theta x_{3-k}\right)\text{,}$$

where $f_{\text{Exp}}\left(x;\lambda\right)=\lambda\exp\left(-\lambda x\right)$ is the PDF of the exponential distribution with rate $\lambda>0$. Thus, we can conduct inference regarding this DGP by considering ICLs of the form

$$p_{\bm{\sigma},\bm{\tau}}\left(\bm{x};\theta\right)=\left[p\left(x_{1}|x_{2};\theta\right)\right]^{1/2}\left[p\left(x_{2}|x_{1};\theta\right)\right]^{1/2}\text{,}$$

where $\bm{\sigma}=\mathbf{0}$ and $\bm{\tau}=\left(1/2\right)\mathbf{1}$.

For data $\mathbf{X}_{n}$ with identical DGP to $\bm{X}$, characterized by $\theta_{0}\in\left\{1,5,10\right\}$, where $n_{1}=n_{2}\in\left\{100,1000,10000\right\}$, we consider the use of the spCLRS and swCLRS confidence sets at the $\alpha=0.05$ level. Here, each confidence set is constructed using the maximum composite likelihood estimator (MCLE).

For each pair $\left(n_{1},\theta\right)$, we replicate the simulation $r=100$ times and compute the coverage proportion (CP) and average size (AS) of the confidence intervals for the two set constructions. Here, CP and AS are computed as $r^{-1}\sum_{j=1}^{r}\left\llbracket\theta_{0}\in\mathcal{C}_{j}\right\rrbracket$ and $r^{-1}\sum_{j=1}^{r}\text{diam}\left(\mathcal{C}_{j}\right)$, where $\mathcal{C}_{j}$ is a stand-in for a confidence set constructed from the $r\text{th}$ replicate, $\left\llbracket\cdot\right\rrbracket$ are Iverson brackets, and $\text{diam}\left(\cdot\right)$ is the metric set diameter operator.

The results are presented in Table 1(a). We observe that CP was near perfect, with only one scenario yielding a confidence set that did not contain $\theta_{0}$. This supports Proposition 1, although it indicates that the confidence sets are fairly conservative. We observe that AS is decreasing in $n_{1}$, as expected, and increasing in $\theta_{0}$. We also find that the swCLRS sets are smaller than the spCLRS sets, which suggests a more efficient use of the data.

We now consider the bivariate distribution of Sarabia et al. (2007), which is specified by the PDF

$$p\left(\bm{x};\bm{\theta}\right)=\frac{\kappa\left(c\right)}{2\pi\sigma_{1}\sigma_{2}x_{1}x_{2}}\exp\left\{-\frac{1}{2}\left[\left(\frac{\log x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\frac{\log x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}+c\left(\frac{\log x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}\left(\frac{\log x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\}\text{,}$$

(1)

where $\bm{\theta}^{\top}=\left(\mu_{1},\sigma_{1}^{2},\mu_{2},\sigma_{2}^{2},c\right)$, with $\mu_{1},\mu_{2}\in\mathbb{R}$, $\sigma_{1}^{2},\sigma_{2}^{2}>0$, and $c\geq 0$. Here, $\kappa\left(c\right)=\sqrt{2c}/U\left(1/2,1,\left(2c\right)^{-1}\right)$, where $U\left(a,b,z\right)$ is the confluence hypergeometric function, defined as per Abramowitz and Stegun (1972, Eqn. 13.2.5). Like in the previous example, the normalizing constant of the joint PDF makes it intractable. However, we may again specify the conditional PDFs of $X_{k}|X_{3-k}=x_{3-k}$, for $k\in\left\{1,2\right\}$, by

$$p\left(x_{k}|x_{3-k};\bm{\theta}\right)=f_{\text{LN}}\left(x_{k};\mu_{k},\sigma_{k}^{2}/\left\{1+c\left(\frac{\log x_{3-k}-\mu_{3-k}}{\sigma_{3-k}}\right)^{2}\right\}\right)\text{,}$$

where

$$f_{\text{LN}}\left(x;\mu,\sigma^{2}\right)=\frac{1}{x\sqrt{2\pi\sigma^{2}}}\exp\left\{-\frac{1}{2}\left(\frac{\log x-\mu}{\sigma}\right)^{2}\right\}$$

is the PDF of a log-normal distribution with location and scale parameters $\mu\in\mathbb{R}$ and $\sigma^{2}>0$, respectively. We can use the conditional PDFs to conduct CL inference via the ICLs of the form

$$p_{\bm{\sigma},\bm{\tau}}\left(\bm{x};\bm{\theta}\right)=\left[p\left(x_{1}|x_{2};\bm{\theta}\right)\right]^{1/2}\left[p\left(x_{2}|x_{1};\bm{\theta}\right)\right]^{1/2}\text{,}$$

where $\bm{\sigma}=\mathbf{0}$ and $\bm{\tau}=\left(1/2\right)\mathbf{1}$.

We simulate data $\mathbf{X}_{n}$, $n_{1}=n_{2}\in\left\{100,1000,10000\right\}$ from DGPs that are characterized by the PDF (1), with parameter vector $\bm{\theta}_{0}=\left(2,1,2,1,c_{0}\right)$, where $c_{0}\in\left\{0,1,5\right\}$. For each pair $\left(n_{1},c_{0}\right)$, we use the spCLRT and swCLRT to test the hypotheses $\text{H}_{0}:c_{0}=0$ versus $\text{H}_{1}:c_{0}>0$, at the $\alpha=0.05$ level. We repeat each simulation pair $r=100$ times and compute the proportion of times the null hypothesis was rejected. Here, we again make use of the MCLE.

The results are reported in Table 1(b). Notice that no false rejections were made when $c_{0}=0$, thus the size of the test is conservatively controlled, as predicted by Proposition 2. We also see that the tests become increasingly powerful as $c_{0}$ increases and as $n_{1}$ increases, as would be expected. There is some evidence that the swCLRT is more powerful than the spCLRT, conforming to observations from the previous study.