# Tests the Coefficients of High Dimensional Generalized Linear Models

### Description

Tests for whole or partial regression coefficient vectors for high dimensional generalized linear models.

### Usage

1 | ```
HDGLM_test(Y, X, beta_0 = NULL, nuisance = NULL, model = "gaussian")
``` |

### Arguments

`Y` |
a vector of observations of length |

`X` |
a design matrix with |

`beta_0` |
a vector with length |

`nuisance` |
an index indicating which coefficients are nuisance parameter. The default is |

`model` |
a character string to describe the model and link function. The default is |

### Value

An object of class "HDGLM_test" is a list containing the following components:

`test_stat` |
the standardized test statistic |

`test_pvalue` |
pvalue of the test against the null hypothesis |

### Note

In global test, the function `"HDGLM_test"`

can deal with the null hypothesis with non-zero coefficients (*β_0*). However, in test with nuisance coefficient,
the function can only deal with the null hypothesis with zero coefficients (*β_0^{(2)}*) in this version.

### Author(s)

Bin Guo

### References

Guo, B. and Chen, S. X. (2015). Tests for High Dimensional Generalized Linear Models.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 | ```
## Example: Linear model
## Global test: if the null hypothesis is true (beta_0=0)
alpha=runif(5,min=0,max=1)
## Generate the data
DGP_0=DGP(80,320,alpha)
result=HDGLM_test(DGP_0$Y,DGP_0$X)
## Pvalue
result$test_pvalue
## Global test: if the alternative hypothesis is true
## (the square of the norm of the first 5 nonzero coefficients to be 0.2)
## Generate the data
DGP_0=DGP(80,320,alpha,sqrt(0.2),5)
result=HDGLM_test(DGP_0$Y,DGP_0$X)
## Pvalue
result$test_pvalue
## Test with nuisance coefficients: if the null hypothesis is true (beta_0^{(2)}=0)
## The first 10 coefficients to be the nuisance coefficients
betanui=runif(10,min=0,max=1)
## Generate the data
DGP_0=DGP(80,320,alpha,0,no=NA,betanui)
result=HDGLM_test(DGP_0$Y,DGP_0$X,nuisance=c(1:10))
## Pvalue
result$test_pvalue
## Test with nuisance coefficients: if the alternative hypothesis is true
## (the square of the norm of the first 5 nonzero coefficients in beta_0^{(2)} to be 2)
## The first 10 coefficients to be the nuisance coefficients
betanui=runif(10,min=0,max=1)
## Generate the data
DGP_0=DGP(80,330,alpha,sqrt(2),no=5,betanui)
result=HDGLM_test(DGP_0$Y,DGP_0$X,nuisance=c(1:10))
## Pvalue
result$test_pvalue
``` |

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