# msQuantile: Quantile of the multiscale statistics In essHist: The Essential Histogram

## Description

Simulate quantiles of the multiscale statistics under the null hypothesis.

## Usage

 `1` ```msQuantile(n, alpha = c(0.1), nsim = 5000, verbose = TRUE, is.sim = (n < 1e4), ...) ```

## Arguments

 `n` number of observations `alpha` significance level; the (1-`alpha`)-quantile of the null distribution of the multiscale statistic via Monte Carlo simulation `nsim` numer of Monte Carlo simulations `is.sim` logical. If `TRUE` (default if `n` < 10,000) the quantile is determined via Monte Carlo simulations, which might take a long time; otherwise (default if `n` >= 10,000) it uses the quantile with `n` = 10,000, which has been precomputed and stored. `verbose` logical. If `TRUE` (default) it prints some details about the computation; otherwise nothing is printed. `...` further arguments passed to function `quantile`.

## Details

Empirically, it turns out that the (1-`alpha`)-quantile of the multiscale statistic converges fast to that of the limit distribution as the number of samples `n` increases. Thus, for the sake of computational efficiency, the quantile with `n` = 10,000 are used by default for that with `n` > 10,000, which has already been precomputed and stored. Of course, for arbitrary sample size `n`, one can always simulate the quantile by setting `is.sim = TRUE`, and use the precomputed value by setting `is.sim = FALSE`. For a given sample size `n`, simulations are once computed, and then automatically recorded in main memory for later usage.

## Value

A vector of length `length(alpha)` is returned, the same structure as returned by funtion `quantile`. See Li et al. (2016) for further details.

## References

Li, H., Munk, A., Sieling, H., and Walther, G. (2016). The essential histogram. arXiv:1612.07216.

`essHistogram`
 ```1 2 3 4 5 6 7``` ```n = 100 # number of observations nsim = 100 # number of simulations alpha = c(0.1, 0.9) # significance level q = msQuantile(n, alpha, nsim) print(q) ```