# CD-plot and adjusted deviance test Construction of CD-plot and adjusted deviance test. The confidence bands are also adjusted for post-selection inference. ```r dhatL2(data, g, M = 6, Mmax = NULL, smooth = TRUE, criterion = "AIC", hist.u = TRUE, breaks = 20, ylim = c(0, 2.5), range = c(min(data),max(data)), sigma = 2) ``` ## Arguments - `data`: A vector of data. See details. - `g`: The postulated model from which we want to assess if deviations occur. - `M`: The desired size of the polynomial basis to be used. - `Mmax`: The maximum size of the polynomial basis from which `M` was selected (the default is `20`). See details. - `smooth`: A logical argument indicating if a denoised solution should be implemented. The default is `FALSE`, meaning that the full solution should be implemented. See details. - `criterion`: If `smooth=TRUE`, the criterion with respect to which the denoising process should be implemented. The two possibilities are `"AIC"` or `"BIC"`. See details. - `hist.u`: A logical argument indicating if the CD-plot should be displayed or not. The default is `TRUE`. - `breaks`: If `hist.u=TRUE`, the number of breaks of the CD-plot. The default is `20`. - `ylim`: If `hist.u=TRUE`, the range of the y-axis of the CD-plot. - `range`: Range of the data/search region considered. - `sigma`: The significance level (in sigmas) with respect to which the confidence bands should be constructed. See details. ## Details The argument `data` collects the data for which we want to test if its distribution deviates from the one of the postulated model specified in the argument `g`. In Algeri, 2019, the sample specified under `data` corresponds to the source-free sample in the background calibration phase and to the physics sample in the signal search phase. The value `M` selected determines the smoothness of the estimated comparison density, with smaller values of `M` leading to smoother estimates. The deviance test is used to select the value `M` which leads to the most significant deviation from the postulated model. The default value for `Mmax` is set to `20`. Notice that numerical issues may arise for larger values of `Mmax`. If `smooth=TRUE` the largest coefficient estimates are selected according to either the AIC or BIC criterion as described in Algeri, 2019 and Mukhopadhyay, 2017. If `Mmax>1` and/or `smooth=TRUE`, post-selection Bonferroni's correction is automatically implemented to both the deviance test p-value and the confidence bands. The desired level of significance can be expressed as one minus the cdf of a standard normal evaluated at `sigma` (see Algeri, 2019). ## Returns - **Deviance**: Value of the deviance test statistic. - **Dev_pvalue**: Unadjusted p-value of the deviance test. - **Dev_adj_pvalue**: Post-selection Bonferroni adjusted p-value of the deviance test. - **kstar**: Number of coefficients selected by the denoising process. If `smooth=FALSE`, `kstar=M`. - **dhat**: Function corresponding to the estimated comparison density in the u domain. - **dhat.x**: Function corresponding to the estimated comparison density in the x domain. - **SE**: Function corresponding to the estimated standard errors of the comparison density in the u domain. - **LBf1**: Function corresponding to the lower bound of the confidence bands under in u domain. - **UBf1**: Function corresponding to the upper bound of the confidence bands in u domain. - **f**: Function corresponding to the estimated density of the data. - **u**: Vector of values corresponding to the cdf of the model specified in `g` evaluated at the vector `data`. - **LP**: Estimates of the coefficients. - **G**: Cumulative density function of the postulated model specified in the argument `g`. ## References S. Algeri, 2019. Detecting new signals under background mismodelling. . S. Mukhopadhyay, 2017. Large-scale mode identification and data-driven sciences. Electronic Journal of Statistics 11 (2017), no. 1, 215--240. ## Author(s) Sara Algeri ## See Also `Legj`,`BestM`,`denoise`. ## Examples ```r #generaing data x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] #create suitable postulated quantile function of data G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} #Choose best M Mmax=20 range=c(10,20) m<-BestM(data=xx,g, Mmax,range) # vectorize postulated quantile function g<-Vectorize(g) u<-g(xx) #M has to be sufficient big, otherwise dhatL2 function will crush. #So,here we set m eqaul 6 as an example m<-6 comp.density<-dhatL2(data=xx,g, M=m, Mmax=Mmax,smooth=FALSE,criterion="AIC",hist.u=TRUE,breaks=20, ylim=c(0,2.5),range=range,sigma=2) ```