Hypothesis Testing on A Nuisance Parameter

Posted on May 11, 2021 0 Comments

Tags: Hypothesis Testing, Two-phase Regression, Time Series

This note is for DAVIES, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74(1), 33–43.

Test a hypothesis in the presence of a nuisance parameter $\theta$, which enters the model only under the alternative. In other words, $\theta$ is meaningless under the null hypothesis.

Examples:

• Two phase regression: $Y_i = \beta_0 + \beta(X_i - \theta)_ +$
• $\E(X_j)=\xi\sin(j\theta) + \xi_2\cos(j\theta)$. The hypothesis is $\xi_1=\xi_2=0$ and the alternatives is that at least one of $\xi_1$ and $\xi_2$ is nonzero.

Because $\theta$ cannot be estimated under the hypothesis, traditional large sample theory is not applicable. However, if $\theta$ were known it would be easy to find an appropriate test.

Isn’t it possible to derive a test without the information of $\theta$.

Suppose $S(\theta)$ is the appropriate test statistic, with large values corresponding to the alternative being true. The test statistic suggested for the case when $\theta$ is unknown is

\[M=\sup\{S(\theta): \cL\le \theta\le \cU\}\,,\]

where $[\cL,\cU]$ is the range of possible values of $\theta$.

The problem is to find the significance probability of the resulting test.

• Davies (1977): studied the case when $S(\theta)$ had a normal distribution for each value of $\theta$ and gave a sharp bound on the significance probability. The bound was calculated from the autocorrelation function of $S(\theta)$.

Quick Calculation of Significance: Normal Case

Suppose that an appropriate test, if $\theta$ was known would be reject the hypothesis for large values of $S(\theta)$ where, for each $\theta$, $S(\theta)$ has a standard normal distribution under the hypothesis. Suppose further that $S(\theta)$ is continuous on $[\cL, \cU]$ with a continuous derivative except possibly for a finite number of jumps in the derivative, and forms a Gaussian process.

Chi-squared case

For each value of $\theta\in [\cL,\cU]$, the test statistic appropriate for that value of $\theta$ is of the form

\[S(\theta) = Z_1^2(\theta) + \cdots + Z_s^2(\theta)\,,\]

where the ${Z_i(\theta)}$ are continuous with continuous first derivatives, except possibily for a finite number of jumps in the derivatives, and form a vector Gaussian process.

Simulations

Observe $X_1,\ldots,X_n$, a sequence of independent normal random variables with unit variance and expectations given by

\[E(X_i) = \begin{cases} a + bt_i & (t_i < \theta)\\ a + bt_i & \xi(t_i-\theta) & (t_i \ge \theta)\,. \end{cases}\]

Suppose $t_i$ to be centered so that $\sum t_i=0$. Test the hypothesis that $\xi_0$ against the alternative $\xi=0$.

Using $C(\alpha)$ principles (??) one can find an appropriate test statistic for the case when $\theta$ is unknown:

\[S(\theta) = \sum_1(X_i-\hat a -\hat bt_i)(t_i-\theta)/V^{1/2}\,,\]

where

\[\hat a = \sum X_i/n, \hat b = \sum t_iX_i/n\,, V=s_1s_2/s_0 + s_3s_4/n\\ s_0=\sum t_i^2\,, s_1=\sum_1t_i(t_i-\theta)\,, s_2=\sum_2t_i(t_i-\theta)\,, s_3=\sum_1(t_i-\theta)\,, s_4=\sum_2(t_i-\theta)\,,\]

$\Sigma_1$ is over $t_i > \theta$ and $\Sigma_2$ is over $t_i < \theta$.