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HISTOGRAM Statement |
The following sections provide information on the families of parametric distributions that you can fit with the HISTOGRAM statement. Properties of these distributions are discussed by Johnson et al. (1994, 1995).
where and
lower threshold parameter (lower endpoint parameter)
scale parameter
shape parameter
shape parameter
h = width of histogram interval
Note: This notation is consistent with that of other
distributions that you can fit with the HISTOGRAM statement.
However, many texts, including Johnson et al. (1995),
write the beta density function as
The two notations are related as follows:The range of the beta distribution is bounded below by a threshold parameter and above by . If you specify a fitted beta curve using the BETA option, must be less than the minimum data value, and must be greater than the maximum data value. You can specify and with the THETA= and SIGMA= beta-options in parentheses after the keyword BETA. By default, and .If you specify THETA=EST and SIGMA=EST, maximum likelihood estimates are computed for and .
In addition, you can specify and with the ALPHA= and BETA= beta-options, respectively. By default, the procedure calculates maximum likelihood estimates for and . For example, to fit a beta density curve to a set of data bounded below by 32 and above by 212 with maximum likelihood estimates for and , use the following statement:
histogram length / beta(theta=32 sigma=180);The beta distributions are also referred to as Pearson Type I or II distributions. These include the power-function distribution (), the arc-sine distribution (), and the generalized arc-sine distributions (, ).
You can use the DATA step function BETAINV to compute beta quantiles and the DATA step function PROBBETA to compute beta probabilities.
where
threshold parameter
scale parameter
h = width of histogram interval
The threshold parameter must be less than or
equal to the minimum data value.
You can specify with
the THRESHOLD= exponential-option. By default,
.
If you specify THETA=EST, a maximum likelihood estimate
is computed for .In addition, you can specify with
the SCALE= exponential-option.
By default, the procedure calculates a maximum likelihood
estimate for . Note that some authors define the
scale parameter as .The exponential distribution is a special case of both the gamma distribution (with ) and the Weibull distribution (with c=1). A related distribution is the extreme value distribution. If Y = exp(-X) has an exponential distribution, then X has an extreme value distribution.
where
threshold parameter
scale parameter
shape parameter
h = width of histogram interval
The threshold parameter must be less than the minimum
data value. You can specify
with the THRESHOLD= gamma-option.
By default, .
If you specify THETA=EST, a maximum likelihood estimate
is computed for .In addition, you can specify
and with the SCALE= and ALPHA=
gamma-options. By default, the procedure
calculates maximum likelihood estimates for
and .
The gamma distributions are also referred to as
Pearson Type III distributions, and they include
the chi-square, exponential, and Erlang distributions.
The probability density function for the chi-square
distribution is
You can use the DATA step function GAMINV to compute gamma quantiles and the DATA step function PROBGAM to compute gamma probabilities.
where
threshold parameter
scale parameter
shape parameter
shape parameter
h = width of histogram interval
The S_{B} distribution is bounded below by the parameter
and above by the value .The parameter
must be less than the minimum data value. You can
specify with the THETA= S_{B}-option,
or you can request that be estimated
with the THETA = EST S_{B}-option.
The default value for is zero.
The sum must be greater than the maximum
data value.
The default value for is one.
You can specify with the SIGMA= S_{B}-option,
or you can request that be estimated
with the SIGMA = EST S_{B}-option.
By default, the method of percentiles given by Slifker and Shapiro (1980) is used to estimate the parameters. This method is based on four data percentiles, denoted by x_{-3z}, x_{-z}, x_{z}, and x_{3z}, which correspond to the four equally spaced percentiles of a standard normal distribution, denoted by -3z, -z, z, and 3z, under the transformation
The following values are computed from the data percentiles:
If you specify FITMETHOD = MOMENTS (in parentheses after the SB option) the method of moments is used to estimate the parameters. If you specify FITMETHOD = MLE (in pareqntheses after the SB option) the method of maximum likelihood is used to estimate the parameters. Note that maximum likelihood estimates may not always exist. Refer to Bowman and Shenton (1983) for discussion of methods for fitting Johnson distributions.
where
location parameter
scale parameter
shape parameter
shape parameter
h = width of histogram interval
You can specify the parameters with the
THETA=, SIGMA=, DELTA=, and GAMMA= S_{U}-options,
which are enclosed in parentheses after the SU option.
If you do not specify these parameters, they are estimated.
By default, the method of percentiles given by Slifker and Shapiro (1980) is used to estimate the parameters. This method is based on four data percentiles, denoted by x_{-3z}, x_{-z}, x_{z}, and x_{3z}, which correspond to the four equally spaced percentiles of a standard normal distribution, denoted by -3z, -z, z, and 3z, under the transformation
The following values are computed from the data percentiles:
If you specify FITMETHOD = MOMENTS (in parentheses after the SU option) the method of moments is used to estimate the parameters. If you specify FITMETHOD = MLE (in parentheses after the SU option) the method of maximum likelihood is used to estimate the parameters. Note that maximum likelihood estimates may not always exist. Refer to Bowman and Shenton (1983) for discussion of methods for fitting Johnson distributions.
where
threshold parameter
scale parameter
shape parameter
h = width of histogram interval
The threshold parameter must be less than the minimum data value. You can specify with the THRESHOLD= lognormal-option. By default, . If you specify THETA=EST, a maximum likelihood estimate is computed for .You can specify and with the SCALE= and SHAPE= lognormal-options, respectively. By default, the procedure calculates maximum likelihood estimates for these parameters.
Note: The lognormal distribution is also referred to as the S_{L} distribution in the Johnson system of distributions.
Note: This book uses to denote the shape parameter of the lognormal distribution, whereas is used to denote the scale parameter of the beta, exponential, gamma, normal, and Weibull distributions. The use of to denote the lognormal shape parameter is based on the fact that has a standard normal distribution if X is lognormally distributed.
You can specify and with the MU= and SIGMA= normal-options, respectively. By default, the procedure estimates with the sample mean and with the sample standard deviation.
where mean standard deviation h = width of histogram interval
You can use the DATA step function PROBIT to compute normal quantiles and the DATA step function PROBNORM to compute probabilities.
Note: The normal distribution is also referred to as the S_{N} distribution in the Johnson system of distributions.
where threshold parameter scale parameter c = shape parameter (c >0) h = width of histogram interval
The threshold parameter must be less than the minimum data value. You can specify with the THRESHOLD= Weibull-option. By default, . If you specify THETA=EST, a maximum likelihood estimate is computed for .You can specify and c with the SCALE= and SHAPE= Weibull-options, respectively. By default, the procedure calculates maximum likelihood estimates for and c.
The exponential distribution is a special case of the Weibull distribution where c=1.
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