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Quantiles of a test statistic

qStat.Rd

Quantile of a test statistic.

Usage

qStat(p, n,
         type = c("Greenwood", "Jackson", "logLRGPD", "logLRLomax",
                  "logLRGEV", "logLRFrechet"),
          outNorm = FALSE)

Arguments

p

Numeric vector of probabilities. Very small values (p < 0.01) or very large ones (p > 0.99) will be truncated as 0.00 or 1.00 to maintain a realistic level of precision.

n

Sample size.

type

The type of statistic, see Details.

outNorm

Logical. If TRUE the output is normalized in a such fashion that its distribution is the asymptotic one (i.e. standard normal in practice). When FALSE, the quantiles are given in the true scale of the statistic: \(\textrm{CV}^2\), Jackson. For LR statistics this argument has no impact.

Value

A vector of quantiles.

Details

The function provides an approximation of the distribution for several statistics.

  • For "Greenwood", the statistic is Greenwood's statistic. The distribution is that of the squared coefficient of variation \(\textrm{CV}^2\) of a sample of size n from the exponential distribution as computed by CV2.

  • For "Jackson", the statistic is Jackson's statistic, see Jackson.

  • For "logLRGPD" and "logLRLomax", the statistic is the log of the likelihood ratio of a sample from the exponential distribution. The log-likelihoods are for an exponential distribution compared to a GPD with non-zero shape, or to a GPD with positive shape (equivalently, a Lomax distribution).

  • For "logLRGEV" and "logLRFrechet", the statistic is the log of the likelihood ratio of a sample from the Gumbel distribution. The log-likelihoods are for a Gumbel distribution compared to a GEV with non-zero shape, or to a GEV with positive shape (equivalently, a Fréchet distribution).

The log of Likelihood Ratios are multiplied by 2, so that they compare to a chi-square statistic with one degree of freedom.

Note

The precision of the result given is limited, and is about two-digits. This function is not intended to be used as such and is only provided for information.

Author

Yves Deville

Examples

res <- qStat(n = 40, type = "Greenwood")
#> Warning: 'p' contains values smaller than 0.005. Truncated.
#> Warning: 'p' contains values larger than 0.995. Truncated.
plot(res$q, res$p, type = "o")