Package 'multChernoff'

Critical value x such that $P(2 LRT > x) \le p$

Description

The LRT is the log-likelihood ratio test statistic, which can be written as

$LRT = n \, KL(\hat{p} \| p).$

By the Wilks' theorem, for a fixed k-dimensional probability vector, it holds that

$2 \times LRT \to_{d} \chi^2_{k-1}.$

This function returns a finite-sample counterpart to qchisq(p, k-1, lower.tail=FALSE). The LRT is also extended to multiple independent multinomial trials. For example, for a $(k_1, n_1)$-trial and a $(k_2, n_2)$-trial, we have

$LRT = n_1 \, KL(\hat{p}_1 \| p_1) + n_2 \, KL(\hat{p}_2 \| p_2).$

Usage

criticalValue(k, n, p = 0.05, verbose = FALSE)

Arguments

k	number of categories (a vector for independent multinomial draws)
n	sample size (a vector for independent multinomial draws)
p	significance level (e.g., 0.05)
verbose	draw the minimizer if TRUE

Value

A finite-sample critical value $x$ such that the bound on $P(2 \times LRT > x)$ is at most p.

Note

For independent multinomial samples, k and n must be of the same length.

See Also

tailProbBound, mgfBound

Examples

n <- 1:40
crit <- sapply(n, function(.n) criticalValue(20, .n, p=0.01))
plot(n, crit)
# chi-squared asymptotic by Wilks' theorem
abline(h=qchisq(0.01, df=20-1, lower.tail = FALSE))
criticalValue(10, 40, p=0.05)
# two independent multinomial trials (k=3, n=4) and (k=12, n=20)
criticalValue(c(3, 4), c(12, 20), p=0.05)

---

An upper bound on the moment generating function of LRT

Description

The LRT is the log-likelihood ratio test statistic, which can be written as

$LRT = n \, KL(\hat{p} \| p),$

namely the Kullback-Leibler divergence from the empirical probabilities to the true probabilities multiplied by the sample size. $G(\lambda; k,n)$ is a polynomial in $\lambda$ such that

$MGF(\lambda; LRT) := E_{p}[\exp(\lambda \times LRT)] \leq G(\lambda; k,n)$

holds for every $\lambda \in [0,1]$ and every $p$.

Usage

mgfBound(k, n, lambda)

Arguments

k	number of categories
n	sample size
lambda	number between 0 and 1

Value

A numeric upper bound on the MGF of LRT evaluated at lambda.

See Also

tailProbBound, criticalValue

Examples

mgfBound(k = 5, n = 20, lambda = 0.5)
mgfBound(k = 5, n = 20, lambda = 0)  # always 1 at lambda = 0

---

Tail bound on P(2 LRT > x).

Description

The LRT is the log-likelihood ratio test statistic, which can be written as

$LRT = n \, KL(\hat{p} \| p).$

By the Wilks' theorem, for a fixed k-dimensional probability vector, it holds that

$2 \times LRT \to_{d} \chi^2_{k-1}.$

This function returns a finite-sample counterpart to pchisq(x, k-1, lower.tail=FALSE). The LRT is also extended to multiple independent multinomial trials. For example, for a $(k_1, n_1)$-trial and a $(k_2, n_2)$-trial, we have

$LRT = n_1 \, KL(\hat{p}_1 \| p_1) + n_2 \, KL(\hat{p}_2 \| p_2).$

Usage

tailProbBound(x, k, n, verbose = FALSE)

Arguments

x	the value of $2 \times LRT$
k	number of categories (a vector for independent multinomial draws)
n	sample size (a vector for independent multinomial draws)
verbose	draw the minimizer if TRUE

Value

An upper bound on P(2 LRT > x), which can be used as a conservative p-value.

Note

For independent multinomial samples, k and n must be of the same length.

See Also

criticalValue, mgfBound

Examples

tailProbBound(20, 7, 50)
pchisq(20, 6, lower.tail=FALSE) # compare with the standard chi-square asymptotic
# two independent multinomial trials (k=3, n=4) and (k=12, n=20)
tailProbBound(12, c(3, 4), c(12, 20))
pchisq(12, 5, lower.tail=FALSE) # compare with the standard chi-square asymptotic

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