## 4 Conclusions  
In this paper, a new statistical method for positional accuracy quality control has been proposed, which can be applied to non-normal errors and to any number of dimensions.  
The new control method is based on the use of a multinomial distribution that categorizes the cases of errors according to metric tolerances. This method of defining a quality control is very versatile and allows the control of positional errors in 1D, 2D, and 3D cases. In addition, depending on the number of tolerances considered, a greater degree of similarity could be established between the observed distribution of errors and any desired probabilistic model. The major advantages of this method are as follows:  
(1) it can be applied to any kind of error model (parametric or nonparametric),  
(2) it can be applied to a mix of error models (e.g., in a 3D case, the X and Y errors can be normally distributed and the Z errors can be non-normally distributed),  
(3) it can be applied to any kind of geometry (e.g., points, line strings, etc.),  
(4) it can be applied to cases of any dimension (1D, 2D, 3D,... _n_D),  
(5) the method allows quantitative and qualitative aspects to be jointly controlled by means of proportions in established categories.  
The method has been proposed in a generic way and has been developed for the case of two tolerances (three categories of errors). The most complex aspect of the method is that it is an exact test, so the calculation of the _p_-value is not done by means of approximations but by calculating the sum of a finite set of solutions. In the appendix, an example is presented to show how to calculate the _p_-value using the R statistical program. This test is very flexible because the specification limits, the probabilities for each category and even the dimension number can be established by the user.  
To demonstrate the applicability, ALS data from three different study cases (infrastructure, urban and natural) have been used, and 1D and 3D errors have been considered. Three quality controls with different tolerances were applied. In two of the quality control cases, the tolerances were defined from a Gaussian model (with and without bias), and in the third quality control case, the tolerances were defined from the quantiles of the observed error distribution.  
The results of the three study cases and of the three quality controls show that if tolerances are well established for in the control of positional accuracy, problems of excess rejection or of excess acceptance can occur. For the case of non-normal data, the use of quantiles is suggested to establish the values of the metric tolerances. The quality controls developed by simulation for the three study cases and for different sample sizes show that this method is capable of ensuring the desired significance level. In addition, the sensitivity analysis carried out by modifying the tolerances indicates that the statistical power of the new proposed control method is adequate since correct results are obtained, and they are quite stable for sample sizes greater than 100.  
Taking into account the results according to the variability of the study cases, the proposed method is very promising to improve the quality control of ALS data and other non-normally distributed spatial datasets, which is the usual case when we deal with in geotechnologies, such as terrestrial laser scanning or photogrammetry.  
Data curation, D.G.-A. and P.R.-G.; Investigation, F.J.-A.-L. and [PERSON]-A.; Methodology, F.J.-A.-L., [PERSON]-A. and P.R.-G.; Software, J.R.-A.; Writing--Original draft, [PERSON]-A.-L., [PERSON]-A., [PERSON]-A. and P.R.-[PERSON]. Writing--Review & editing, [PERSON] and P.R.[PERSON].  
This research was partially funded by grant CTM2015-68276-R of the Spanish Ministry on Science and Innovation and the European Regional Development Funds.  
The authors declare no conflict of interest.