# UMPU Tests and Confidence Intervals, Fuzzy and Randomized, in R

## The Binomial Distribution

The `ump` package for R (see package info on the main page for this site for how to get it) has a function `umpu.binom` that calculates the critical function φ(x, α, θ) for the UMPU randomized (two-tailed) test, where x is the data, α is the significance level, and θ is the parameter value assumed by the null hypothesis.

In particular, the data x is assumed to be Binomial(n, θ) distributed, where θ is the success probability.

### Fuzzy and Randomized Decisions

For example, suppose the data is 17 successes out of 25 trials and the null hypothesis is θ = 0.5. The randomized test for α = 0.05 rejects the null hypothesis with probability (click the Submit button to see the result).

You can edit this box to change the numbers and re-submit to do a different analysis.

Any one of the numbers, except the sample size 25, can be made a vector, for example, the following code gives the values of the critical function for all possible x values and checks that the `umpu.binom` actually calculates what it claims to (the last two lines of output should each repeat the same number).

### Fuzzy Confidence Intervals

If we make θ a vector in the preceding example we have a so-called fuzzy confidence interval.

Suppose, as in the first example, the data is 17 successes out of 25 trials and we want a 95% confidence interval (so α = 0.05). The fuzzy confidence interval corresponding to the UMPU test has a membership function calculated and plotted by

A somewhat clearer visualization of the edges of the fuzzy interval is given by the following code.

This code also prints the so-called support and core of the interval (four significant figure approximations thereto).

### Fuzzy and Randomized P-values

#### Fuzzy Sets and Distribution Functions

If we make α a vector we get the so-called fuzzy P-value.

Suppose, as in the first example, the data is 17 successes out of 25 trials and we want an α = 0.05 level UMPU two-tailed test of the null hypothesis θ = 0.5. The following code calculates and plots the (membership function of) the fuzzy P-value.

The function plotted here has another interpretation. It can also be considered to be the distribution function of an abstract random variable, the so-called randomized P-value.

In either case the function is continuous and nondecreasing and its domain is the whole parameter space, the interval [0, 1]. The plot only shows the interesting part. The function is zero to the left of the plot region and one to the right.

#### Probability Density Functions

The derivative of the distribution function is the density function. We happen to know the distribution function is piecewise linear. Hence the density is piecewise constant (a step function).