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In Confidence regions when the Fisher information is zero - Matteo Bottai (2003) the effect of a singular information matrix on the confidence regions from statistical tests is getting looked at.

In the paper Bottai describes some models he uses to perform simulations. I would like to duplicate these simulations but I get stuck on how to generate the data. He says:

Model 1: Gaussian nonlinear regression. The data consist of $n$ independent observations $Y_i=(X_i,Z_i)$ of a bivariate vector $Y = (X, Z)$, with density: $$f(y) = f(x)f(z | x)$$ where $f(x)$ is fixed and known and $$f(z|x) = \phi(z-e^{\theta x}+\theta x)$$ where $\theta$ is unknown and lies in some open interval and $\phi(\cdot)$ denotes the density of a standard normal variable.

A bit further he denotes:

Figure 2 shows typical loglikelihoods obtained by pseudorandomly generating four samples from Model 1, where $n = 10000$, $\theta =0$ and $X$ is generated from a $Un(- 1, 2)$ distribution. Note that, despite the large sample size, two of the loglikelihood functions are bimodal.

Generating uniformly distributed data is no problem in R. But how can I generate this $Y$ (or the $Z_i$).

Also, I guess $f(x)$ can be chosen at random? Would something like $f(x) = \phi(x)$ be valid?

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  • $\begingroup$ You have not correctly transcribed the model. Please fix the question $\endgroup$ – Glen_b -Reinstate Monica Apr 14 '17 at 1:50
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  1. Note that you said $Y=(X,Z)$, so as soon as you have pairs of $X$ and $Z$ values, you have your $Y$ simply by putting them together.

  2. You already have $X$ values, since they were drawn from a uniform.

  3. note that $f(z|x) = \phi(z-\mu(x))$ specifies a normal density with mean $\mu(x)$ and standard deviation $1$.

  4. As a result you proceed as follows (after you specify your $\theta$), repeating many times:

    a. generate $X_i$ as you already have it (i.e. from a uniform); denote the observed generated value as $x_i$.

    b. generate $Z_i\sim N(\mu(x_i),1)$, call the observed generated value $z_i$.

    c. call the pair $(x_i,z_i)$ "$y_i$".

Also, I guess $f(x)$ can be chosen at random? Would something like $f(x) = \phi(x)$ be valid?

No, for the example $f(x)$ was specified to be uniform just above.

plots of z vs x for theta = 0, 0.8, 1.5 and -1; the first shows no relationship (random scatter about z=1), the other three show varying amounts of curved relationship according to the nonlinear function with random scatter about it

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