Add test / se for estimated parameters in PLN

[mahendra-mariadassou]

Implement score test score and/or Wald test to test significance of model parameters. Stick with univariate tests for the time being.

[mahendra-mariadassou]

As of af761b5, there are now fisher and standard_error S3 methods to compute a variational approximation of the Fisher information matrix and the corresponding standard errors for the model parameters.
Additional (empirical) work is needed to ensure that the corresponding tests / confidence intervals have proper Type I error / coverage

[jchiquet]

For the record, a conversation below between @mahendra-mariadassou and Stephane. It implies that

• we must clarify the documentation about Fisher and se of the coefficient
• we must double check the "divided-by-n" thing
• we should keep in mind the related Issue vcov is traditionally associated with the covariance of the coefficients, not with the residuals covariance #30 to have a relevant interface in PLNmodels through vcov

Fisher returns I(\theta) and for iid data, we usually have I_n(\theta) = n*I(\theta)
As a result, it must be multiplied by n before it is reversed (or divided by n). by n after inverting it) to have the asymptotic variance of \hat{\theta}. If you just reverse, you'll have the variance of \sqrt{n}
\hat{\theta}.
But if you just want to make univariate confidence intervals, you can use standard_error which directly returns the standard deviation of each of the coefficients of \theta. The function multiplies fisher by n, Conversely, retrieves the diagonal and reorganizes it into a matrix of
so that standard_error(myPLN)[i, j] returns the standard deviation of the estimator from coefficients(myPLN)[i, j]

[jchiquet]

Since version 1.0.0, variance estimation can be made for PLNfit either with jackknife or bootstrap. The variational approximation being too rough, we suppressed the direct interface to variational-based Fisher information matrix for estimating the variance (and the standard error).