Multiscale Quantiles {essHist}

Quantile of the multiscale statistics

Description

Simulate quantiles of the multiscale statistics under any continuous distribution function.

Usage

msQuantile(n, alpha = c(0.5), nsim = 5e3, is.sim = (n < 1e4), 
            intv = genIntv(n), mode = c("Con", "Gen"), ...)

Arguments

n

number of observations.

alpha

significance level; default as 0.5, see also essHistogram. Like quantile, it can also be a vector.

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.

intv

a data frame provides the system of intervals on which the multiscale statistic is defined. The data frame constains the following two columns

left left index of an interval

right right index of an interval

By default, it is set to the sparse interval system proposed by Rivera and Walther (2013), see genIntv and also Li et al. (2016).

mode

"Con" for continuous distribution functions (default)

"Gen" for general (possibly with discontinuous) distribution functions

See Li et al. (2016) for further details.

...

further arguments passed to function quantile.

Details

Empirically, it turns out that the quantile of the multiscale statistic converges fast to that of the limit distribution as the number of observations 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 the R memory for later usage. For memory efficiency, only the last simulation is stored.

Value

A vector of length length(alpha) is returned, the same structure as returned by funtion quantile with option names = FALSE; The values are the (1-alpha)-quantile(s) of the null distribution of the multiscale statistic via Monte Carlo simulation, corresponding to (1-alpha)-confidence level(s). See Li et al. (2016) for further details.

Note

All the printing messages can be disabled by calling suppressMessages.

References

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

Rivera, C., & Walther, G. (2013). Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics. Scand. J. Stat. 40, 752–769.

See Also

checkHistogram, essHistogram, genIntv

Examples

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)