Interval estimation

The most prevalent forms of interval estimation are confidence intervals (a frequentist method) and credible intervals (a Bayesian method). Less common forms include likelihood intervals, fiducial intervals, tolerance intervals, and prediction intervals. For a non-statistical method, interval estimates can be deduced from fuzzy logic.

Types

Confidence intervals

Confidence intervals are used to estimate the parameter of interest from a sampled data set, commonly the mean or standard deviation. A confidence interval states there is a 100γ% confidence that the parameter of interest is within a lower and upper bound. A common misconception of confidence intervals is 100γ% of the data set fits within or above/below the bounds, this is referred to as a tolerance interval, which is discussed below.

There are multiple methods used to build a confidence interval, the correct choice depends on the data being analyzed. For a normal distribution with a known variance, one uses the z-table to create an interval where a confidence level of 100γ% can be obtained centered around the sample mean from a data set of n measurements. For a binomial distribution, confidence intervals can be approximated using the Wald approximate method, Jeffreys interval, and Clopper–Pearson interval. The Jeffrey method can also be used to approximate intervals for a Poisson distribution. If the underlying distribution is unknown, one can utilize bootstrapping to create bounds about the median of the data set.

Credible intervals

As opposed to a confidence interval, a credible interval requires a prior assumption, modifying the assumption utilizing a Bayes factor, and determining a posterior distribution. Utilizing the posterior distribution, one can determine a 100γ% probability the parameter of interest is included, as opposed to the confidence interval where one can be 100γ% confident that an estimate is included within an interval.

$${\displaystyle \text{Posterior}\propto \text{Likelihood}\times \text{Prior}}$$

While a prior assumption is helpful towards providing more data towards building an interval, it removes the objectivity of a confidence interval. A prior will be used to inform a posterior, if unchallenged this prior can lead to incorrect predictions.

The credible interval's bounds are variable, unlike the confidence interval. There are multiple methods to determine where the correct upper and lower limits should be located. Common techniques to adjust the bounds of the interval include highest posterior density interval (HPDI), equal-tailed interval, or choosing the center the interval around the mean.

Less common forms

Likelihood-based

Utilizes the principles of a likelihood function to estimate the parameter of interest. Utilizing the likelihood-based method, confidence intervals can be found for exponential, Weibull, and lognormal means. Additionally, likelihood-based approaches can give confidence intervals for the standard deviation. It is also possible to create a prediction interval by combining the likelihood function and the future random variable.

Fiducial

Fiducial inference utilizes a data set, carefully removes the noise and recovers a distribution estimator, generalized fiducial distribution (GFD). Without the use of Bayes' theorem, there is no assumption of a prior, much like confidence intervals. Fiducial inference is a less common form of statistical inference. The founder, R.A. Fisher, who had been developing inverse probability methods, had his own questions about the validity of the process. While fiducial inference was developed in the early twentieth century, the late twentieth century believed that the method was inferior to the frequentist and Bayesian approaches but held an important place in historical context for statistical inference. However, modern-day approaches have generalized the fiducial interval into generalized fiducial inference (GFI), which can be used to estimate discrete and continuous data sets.

Tolerance

Tolerance intervals use collected data set population to obtain an interval, within tolerance limits, containing 100γ% values. Examples typically used to describe tolerance intervals include manufacturing. In this context, a percentage of an existing product set is evaluated to ensure that a percentage of the population is included within tolerance limits. When creating tolerance intervals, the bounds can be written in terms of an upper and lower tolerance limit, utilizing the sample mean, ${\displaystyle \mu }$, and the sample standard deviation, s.

${\displaystyle (l_b,u_b)=\mu \pm k_{2}s}$ for two-sided intervals

for two-sided intervals

And in the case of one-sided intervals where the tolerance is required only above or below a critical value,

$${\displaystyle l_b=\mu -k_{1}s}$$ $${\displaystyle u_b=\mu +k_{1}s}$$

${\displaystyle k_i}$ varies by distribution and the number of sides, i, in the interval estimate. In a normal distribution, ${\displaystyle k_2}$ can be expressed as

$${\displaystyle k_2=z_{\alpha /2}\sqrt{\frac{\nu }{{\chi }_{1-\alpha ,\nu }^2}\left(1+\frac{1}{N}\right)}}$$

where ${\displaystyle {\chi }_{1-\alpha ,\nu }^2}$ is the critical value of the chi-square distribution utilizing ${\displaystyle \nu }$ degrees of freedom that is exceeded with probability ${\displaystyle \alpha }$, and ${\displaystyle z_{\alpha /2}}$ is the critical values obtained from the normal distribution.

Prediction

A prediction interval estimates the interval containing future samples with some confidence, γ. Prediction intervals can be used for both Bayesian and frequentist contexts. These intervals are typically used in regression data sets, but prediction intervals are not used for extrapolation beyond the previous data's experimentally controlled parameters.

Fuzzy logic

Fuzzy logic is used to handle decision-making in a non-binary fashion for artificial intelligence, medical decisions, and other fields. In general, it takes inputs, maps them through fuzzy inference systems, and produces an output decision. This process involves fuzzification, fuzzy logic rule evaluation, and defuzzification. When looking at fuzzy logic rule evaluation, membership functions convert our non-binary input information into tangible variables. These membership functions are essential to predict the uncertainty of the system.