Continuous Fit Distributions

This section contains statistical details for the options in the Continuous Fit menu.

The Normal fitting option estimates the parameters of the normal distribution. The normal distribution is often used to model measures that are symmetric with most of the values falling in the middle of the curve. Select the Normal fitting for any set of data and test how well a normal distribution fits your data.

The parameters for the normal distribution are as follows:

•

μ (the mean) defines the location of the distribution on the x-axis

•

σ (standard deviation) defines the dispersion or spread of the distribution

The standard normal distribution occurs when

and

. The Parameter Estimates table shows estimates of μ and σ, with upper and lower 95% confidence limits.

pdf:

for

;

; 0 < σ

E(x) = μ

Var(x) = σ2

LogNormal

The LogNormal fitting option estimates the parameters μ (scale) and σ (shape) for the two-parameter lognormal distribution. A variable Y is lognormal if and only if

is normal. The data must be greater than zero.

E(x) =

Var(x) =

Weibull, Weibull with Threshold, and Extreme Value

The Weibull distribution has different shapes depending on the values of α (scale) and β (shape). It often provides a good model for estimating the length of life, especially for mechanical devices and in biology.

The pdf for the Weibull and Weibull with Threshold distributions is as follows:

pdf:

for α,β > 0;

E(x) =

Var(x) =

where

is the Gamma function.

The Weibull option sets the threshold parameter (θ) to zero. The Weibull with Threshold option, estimates the threshold parameter (θ) using the value of the minimum observation and estimates α and β using the rest of the observations. If you know what the threshold should be, set it by using the Fix Parameters option. See Fit Distribution Options.

Note: The Distribution platform uses a different estimation technique for the threshold parameter in the Weibull with Threshold distribution than does the Life Distribution platform. The Life Distribution estimation method is recommended for fitting this distribution. See Statistical Details for the Life Distribution Platform in the Reliability and Survival Methods book.

The Extreme Value distribution is equivalent to a two-parameter Weibull (α, β) distribution reparameterized as δ = 1 / β and λ = ln(α).

Exponential

The exponential distribution is especially useful for describing events that randomly occur over time, such as survival data. The exponential distribution might also be useful for modeling elapsed time between the occurrence of non-overlapping events. Examples of non-overlapping events include the time between a user’s computer query and response of the server, the arrival of customers at a service desk, or calls coming in at a switchboard.

The Exponential distribution is a special case of the two-parameter Weibull when β = 1 and α = σ, and also a special case of the Gamma distribution when α = 1.

pdf:

for 0 < σ;

E(x) = σ

Var(x) = σ2

Devore (1995) notes that an exponential distribution is memoryless. Memoryless means that if you check a component after t hours and it is still working, the distribution of additional lifetime (the conditional probability of additional life given that the component has lived until t) is the same as the original distribution.

Gamma

The Gamma fitting option estimates the gamma distribution parameters, α > 0 and σ > 0. The parameter α, called alpha in the fitted gamma report, describes shape or curvature. The parameter σ, called sigma, is the scale parameter of the distribution. A third parameter, θ, called the Threshold, is the lower endpoint parameter. It is set to zero by default, unless there are negative values. You can also set its value by using the Fix Parameters option. See Fit Distribution Options.

pdf:

for

; 0 < α,σ

E(x) = ασ + θ

Var(x) = ασ2

•

The standard gamma distribution has σ = 1. Sigma is called the scale parameter because values other than 1 stretch or compress the distribution along the x-axis.

•

The Chi-square

distribution occurs when σ = 2, α = ν/2, and θ = 0.

•

The exponential distribution is the family of gamma curves that occur when α = 1 and θ = 0.

The standard gamma density function is strictly decreasing when

. When

, the density function begins at zero, increases to a maximum, and then decreases.

Beta

The standard beta distribution is useful for modeling the behavior of random variables that are constrained to fall in the interval 0,1. For example, proportions always fall between 0 and 1. The Beta fitting option estimates two shape parameters, α > 0 and β > 0. There are also θ and σ, which are used to define the lower threshold as θ, and the upper threshold as θ + σ. The beta distribution has values for the interval

only. The θ is estimated as the minimum value, and σ is estimated as the range. The standard beta distribution occurs when θ = 0 and σ = 1.

Set parameters to fixed values by using the Fix Parameters option. The upper threshold must be greater than or equal to the maximum data value, and the lower threshold must be less than or equal to the minimum data value. For details about the Fix Parameters option, see Fit Distribution Options.

pdf:

for

; 0 < σ,α,β

E(x) =

Var(x) =

where

is the Beta function.

Normal Mixtures

The Normal Mixtures option fits a mixture of normal distributions. This flexible distribution is capable of fitting multi-modal data.

Fit a mixture of two or three normal distributions by selecting the Normal 2 Mixture or Normal 3 Mixture options. Alternatively, you can fit a mixture of k normal distributions by selecting the Other option. A separate mean, standard deviation, and proportion of the whole is estimated for each group.

pdf:

E(x) =

Var(x) =

where μi, σi, and πi are the respective mean, standard deviation, and proportion for the ith group, and

is the standard normal pdf.

Smooth Curve

The Smooth Curve option fits a smooth curve using nonparametric density estimation (kernel density estimation). The smooth curve is overlaid on the histogram and a slider appears beneath the plot. Control the amount of smoothing by changing the kernel standard deviation with the slider. The initial Kernel Std estimate is calculated from the standard deviation of the data.

SHASH

The SHASH option fits a sinh-arcsinh (SHASH) distribution. The SHASH distribution is based on a transformation of the normal distribution and includes the normal distribution as a special case. It can be symmetric or asymmetric. The shape is determined by the two shape parameters, γ and δ. For more information about the SHASH distribution, see Jones and Pewsey (2009).

Note: When γ = 0 and δ = 1, the SHASH distribution is equivalent to the normal distribution with location θ and scale σ.

Johnson Su, Johnson Sb, Johnson Sl

The Johnson system of distributions contains three distributions that are all based on a transformed normal distribution. These three distributions are the following:

•

Johnson Su, which is unbounded.

•

Johnson Sb, which has bounds on both tails. The bounds are defined by parameters that can be estimated.

•

Johnson Sl, which is bounded in one tail. The bound is defined by a parameter that can be estimated. The Johnson Sl family contains the family of lognormal distributions.

The S refers to system, the subscript of the range. Although we implement a different method, information about selection criteria for a particular Johnson system can be found in Slifker and Shapiro (1980).

Johnson distributions are popular because of their flexibility. In particular, the Johnson distribution system is noted for its data-fitting capabilities because it supports every possible combination of skewness and kurtosis.

If Z is a standard normal variate, then the system is defined as follows: