mvnQuadratic: Inference for Normal Means after Aggregate Testing
In ammeir2/PSAT: Inference after Selection with Aggregate Testing

Description Usage Arguments Details Value See Also

mvnQuadratic is used to estimate a normal means model that was selected based on a single quadratic aggregate test of the form:

y' K y > c > 0

mvnQuadratic(y, sigma, testMat = "wald", threshold = NULL,
  pval_threshold = 0.05, contrasts = NULL, estimate_type = c("mle",
  "naive"), pvalue_type = c("hybrid", "polyhedral", "naive"),
  ci_type = c("switch", "polyhedral", "naive"), confidence_level = 0.95,
  verbose = TRUE, control = psatControl())

`y`	the observed normal vector.
`sigma`	the covariance matrix of `y`.
`testMat`	the (positive semi-definite) test matrix K used in the aggregate test
`threshold`	the threshold c > 0 used in the aggregate test.
`pval_threshold`	the signficance level of the aggregate test. Overrided by `threshold` if both are provided.
`contrasts`	an optional matrix of contrasts to be tested: must have number of columns identical to the length of `y`. If left as `NULL`, the coorinates of `y` will be tested by default.
`estimate_type`	the types of point estimates to compute and report. The first estimator listed will be used as the default method.
`pvalue_type`	a vector of methods with which to compute the p-values. The first method listed will be used as the default method.
`ci_type`	a vector of confidence interval computation methods to be used. The first method listed will be will be used as the default method.
`confidence_level`	the confidence level for constructing confidencei intervals.
`verbose`	whether to report on the progress of the computation.
`control`	an object of type `psatControl`.

The function is used to perform inference for normal mean vectors that were selected based on a single quadratic aggregate test. To be exact, suppose that y ~ N(μ,Σ) and that we are interested in estimating μ only if we can determine that μ\neq 0 using an aggregate test of the form:

y' K y > c > 0

for some predetermined constant c. If testMat is set to the default value of "wald", then K = Σ^{-1}. If wald test is used, it is recommended to specify testMat as "wald" because this setting makes some of computations more efficient. Otherwise, testMat must be a positive definite matrix of an appropriate dimension.

If estimate_type includes the string "mle" then mvnQuadratic will compute the conditional maximum likelihood estimator for the mean vector, which is typically a shrinkage estimator. If testMat = "wald" then the computation is performed via an efficient line-search method. Otherwise, the computation is performed via the Nelder-Mead method where the probability of selection is approximated using the liu function.

The threshold parameter specifies the constant c>0 which is used to threshold the aggregate test. It takes precedence over pval_threshold if both are specified. We use the liu function to compute the the threshold if a non-Wald test is used.

mvnQuadratic offers several options for computing p-values. The "global-null" method relies on comparing the magnitude of y to samples from the truncated global-null distribution. This method is powerful when μ is sparse and its non-zero coordinates are not very large. The "polyhedral" method is exact when the observed data is approximately normal and is quite robust to model misspecification. It tends to be more powerful than the 'global-null' method when the magnitude of μ is large. The "hybrid" method combines the strengths of the "global-null" and "polyhedral" methods, possessing good power regardless of the sparsity or magnitude of μ. However it is less robust to the misspecification of the distribution of y than the "polyhedral" method. The confidence interval methods are similar to the p-values ones, with the Regime switching confidence intervals ("switch") serving a simialr purpose as the "hybrid" method.

An object of class mvnQuadratic.

getCI, getPval, coef.mvnQuadratic, plot.mvnQuadratic, psatGLM.