Package 'renyi'

Generalized Renyi Transform

Description

A Generalization of Aldous Renyi's representation of exponential order statistics

Usage

generalized_renyi_transform(x, eta = NULL, zeta = NULL)

Arguments

x	a vector of independent exponential random variables of the form $X_j = \eta_j Y_j + \zeta_j$ where each $X_j$ is an independent exponential random variable with rate 1
eta	vector of scale parameters implicit in the construction of x: eta[j] = $\eta_j$
zeta	vector of shift parameters implicit in the construction of x: zeta[j] = $\zeta_j$

Details

Maps a vector of shifted and scaled independent exponential random variables to a sequence of standard independent exponential random variables based on the gaps (jumps) between the initial random variables

Value

a list containing two elements

'exps'

a vector of independent standard exponentials where exps[1] is the exponential jump corresponding to min(x) and tail(exps,1) is the exponential jump corresponding to max(x).

'order'

order(x).

References

Christ, R., Hall, I. and Steinsaltz, D. (2024) "The Renyi Outlier Test", arXiv:2411.13542 . Available at: doi:10.48550/arXiv.2411.13542.

Examples

# example code

a <- rchisq(10,1)
b <- rnorm(10)
xx <- a*rexp(10)+b
generalized_renyi_transform(xx, a, b)

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Renyi Outlier Test

Description

A fast, numerically precise outlier test for a vector of exact p-values allowing for prior information

Usage

renyi(u, k = ceiling(0.01 * length(u)), pi = NULL, eta = NULL)

Arguments

u	a vector of p-values
k	a rough upper bound on the number of outliers expected to be present in u
pi	optional vector such that pi[j] is proportional to the probability that u[j] is an outlier. The default, NULL, corresponds to pi = rep_len(1,length(u)).
eta	optional vector proportional to how far outlying we expect u[j] to be given u[j] is an outlier. More precisely, in the common context where each element of u can be thought of as a p-value for testing whether some coefficient $\beta$ in a linear regression model is zero, we assume eta[j] is proportional to $\mathbb{E}\left[\left. \beta_j^2 \right| \beta_j \neq 0\right]$. The default, NULL, corresponds to eta = rep_len(1,length(u)).

Details

The about which p-values are outlying and "how much" of an outlier they are expected to be

Value

a list containing three elements

'p_value'

the p-value returned by the Renyi Outlier Test;

'max_k'

a power of 2 in 2^(0:k) denoting the number of tail p-values that yielded the most significant signal when running the Renyi Outlier Test;

'p_value_k1'

the p-value that would be returned by the Renyi Outlier Test assuming k=1;

'exit_status'

a character string describing any problems that may have been encountered during evaluation, "default is no problems";

'u'

the vector of p-values used by the outlier test after adjusting the u provided for pi and eta.

References

Christ, R., Hall, I. and Steinsaltz, D. (2024) "The Renyi Outlier Test", arXiv:2411.13542 . Available at: doi:10.48550/arXiv.2411.13542.

Examples

# example code

p <- 1e4
u <- runif(p)
u[c(53,88,32)] <- 1e-6 # add a few outliers
renyi(u)$p_value # test for outliers without any prior knowledge
renyi(u,pi=c(rep(1,100),rep(10^-3,p-100)))$p_value # test for outliers with prior knowledge

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Test for Uniformity on [0,1] Using Multiple Statistical Tests

Description

A wraper function that performs multiple statistical tests to assess whether a numeric vector represents independent draws from a uniform distribution on the interval [0,1]. The function combines several complementary approaches including tests based on the Rényi Outlier Test (see renyi), distribution fitting (Kolmogorov-Smirnov), location (t-test), and normality after transformation (Shapiro-Wilk).

Usage

uniformity_tests(u, k = 32)

Arguments

u	A numeric vector of values assumed to be on [0,1]. Each element should represent an independent draw that is being tested for uniformity.
...	optional arguments to be passed to the Rényi Outlier Test, see renyi.

Details

The function applies four different statistical tests:

• Kolmogorov-Smirnov: Compares the empirical distribution of u to the uniform distribution on [0,1]

• Rényi Outlier Test: Tests whether there are outlying small entries of u,see renyi.

• t-test: Transforms u using the inverse normal CDF and tests whether the mean equals 0 (expected value for standard normal)

• Shapiro-Wilk: Tests whether the normal quantile transform Φ⁻¹(u) follows a standard normal distribution

For the Shapiro-Wilk test, if the sample size exceeds 5,000, the function automatically subsamples to 5,000 quantiles to meet the test's sample size limitations.

All tests return p-values where small values (typically < 0.05) suggest evidence against the null hypothesis of uniformity.

Value

A named list containing p-values from four uniformity tests:

shapiro

P-value from Shapiro-Wilk test applied to normal quantile transformed data. Tests whether Φ⁻¹(u) follows standard normal distribution.

renyi

P-value from the Rényi Outlier Test.

t

P-value from one-sample t-test testing if mean of Φ⁻¹(u) equals 0.

ks

P-value from Kolmogorov-Smirnov test comparing the empirical distribution to the uniform distribution.

See Also

ks.test, shapiro.test, t.test, renyi

Examples

# Test truly uniform data
uniform_data <- runif(1000)
results1 <- uniformity_tests(uniform_data)
print(results1)  # Should show large p-values

# Test non-uniform data (beta distribution)
beta_data <- rbeta(1000, 2, 5)
results2 <- uniformity_tests(beta_data)
print(results2)  # Should show small p-values

# Test a data with small u outliers
outlier_data <- c(uniform_data,1e-5,5e-6,1e-6)
results3 <- uniformity_tests(outlier_data)
print(results3)  # Should show small p-values

# Test while passing a different argument k to the Rényi Outlier Test
results4 <- uniformity_tests(outlier_data, k = 4)
print(results4)

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