17.1.11.3 Algorithms (Distribution Fit)
Contents
- 1 Distributions and Maximum Likelihood Estimation(MLE)
- 2 Goodness of Fit
- 3 Mean Test
Use the Distribution Fit to fit a distribution to a variable.
There are seven distributions can be used to fit a given variable. We calculate the Maximum Likelihood Estimation(MLE) as parameters estimators. For some continuous distributions, we not only give Confidence Limit but also offer Goodness of Fit test.
Distributions and Maximum Likelihood Estimation(MLE)
Normal Distribution
![\frac{1}{\sqrt{2\pi \sigma^2}}\exp [-\frac{(x-\mu)^2}{2\sigma^2}]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-7185152b716035521a23a68b77203be4.png?v=0 "\frac{1}{\sqrt{2\pi \sigma^2}}\exp [-\frac{(x-\mu)^2}{2\sigma^2}]")
where /math-49b0e58e1dd5014c08e10261bb1d461b.png?v=0 "-\infty <x, \mu<\infty")
and /math-528254e4a948602b02ee9f265f38740f.png?v=0 "0 < \sigma")
. With /math-76fdc225d2f2211e1519863b802d6709.png?v=0 "E(X)=\mu")
and /math-14c206e7f7d57210bef18b81baaafc8a.png?v=0 "Var(X)=\sigma^2")
.
Maximum Likelihood Estimation(MLE)
Parameters
/math-da64b80aec46ed0c4069115594fd16fd.png?v=0 "\hat{\mu} = \bar{X}_n")
-
/math-d93c5053992c5fef361caa8fbfd6d1e0.png?v=0 "\hat{\sigma} = \sqrt{\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X}_n)^2}")
.
Confidence Intervals
The confidence interval for /math-c9faf6ead2cd2c2187bd943488de1d0a.png?v=0 "\mu")
and /math-a2ab7d71a0f07f388ff823293c147d21.png?v=0 "\sigma")
are:
![\left[ \hat{\mu} - z \hat{\mu}_{se}, \hat{\mu} + z\hat{\mu}_{se} \right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-ce2e8d2b0c42adb8837e69b4adc2606e.png?v=0 "\left[ \hat{\mu} - z \hat{\mu}_{se}, \hat{\mu} + z\hat{\mu}_{se} \right]")
![\left[ \frac{\hat{\sigma}}{\exp \left[ (z \hat{\sigma}_{se})/\hat{\sigma} \right]},\hat{\sigma}\exp \left[ (z \hat{\sigma}_{se})/\hat{\sigma} \right] \right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-0535dfa29e418e5d578e22cbf35fee81.png?v=0 "\left[ \frac{\hat{\sigma}}{\exp \left[ (z \hat{\sigma}_{se})/\hat{\sigma} \right]},\hat{\sigma}\exp \left[ (z \hat{\sigma}_{se})/\hat{\sigma} \right] \right]")
where /math-fbade9e36a3f36d3d676c1b808451dd7.png?v=0 "z")
is the /math-c30bc392f593aa6950e0b2c260534e68.png?v=0 "0.975")
critical value for the standard normal distribution in which /math-cde2231b7ed96e7c94d4d02b1a0715d5.png?v=0 "95\%")
is the confidence level. And /math-149dace5c8ac18f6b902df281d6b9e62.png?v=0 "\hat{\mu}_{se}")
is standard error for while /math-237d6e4519dd1de11a647716b37457f7.png?v=0 "\hat{\sigma}_{se}")
is for .
LogNormal Distribution
![\frac{1}{x\sqrt{2\pi \sigma^2}} exp\left[ -\frac{(\ln(x)-\mu)^2}{2\sigma^2}\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-3445b6d79ccc732f12131d5a2ea72f25.png?v=0 "\frac{1}{x\sqrt{2\pi \sigma^2}} exp\left[ -\frac{(\ln(x)-\mu)^2}{2\sigma^2}\right]")
,
where /math-10c7c1bfff8e4bcf3303c2f64074d424.png?v=0 "0 \leq x, -\infty < \mu < \infty")
and /math-f655a612a67bcad76b1eef0ec3941f9e.png?v=0 "0 < \sigma")
. With /math-5bba238f59c9a8caeb43e786d3702613.png?v=0 "E(X)=exp(\mu + \sigma^2/2)")
and /math-43de677b47e0ac3bb214060a9e5b3d94.png?v=0 "Var(X)=exp(2(\mu + \sigma^2)) -exp(2\mu + \sigma^2 )")
.
Maximum Likelihood Estimation(MLE)
Parameters
/math-3ce36b54ba5e954002575e762522c35b.png?v=0 "\hat{\mu} = ln\left(\bar{X}_n \right)")
-
/math-e7912f2225843710a3931acacc584d0a.png?v=0 "\hat{\sigma} =ln\left(\sqrt{\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X}_n)^2} \right)")
.
Confidence Interval
The confidence interval for and are:
![\left[ \hat{\mu} - z \hat{\mu}_{se}, \hat{\mu} + z \hat{\mu}_{se} \right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-9f42c3e58301ad6809667e7761678ba3.png?v=0 "\left[ \hat{\mu} - z \hat{\mu}_{se}, \hat{\mu} + z \hat{\mu}_{se} \right]")
where is the critical value for the standard normal distribution in which is the confidence level. And is standard error for while is for .
Weibull Distribution
![\frac{\beta}{\alpha^\beta}x^{\beta -1} exp\left[ -\left(\frac{x}{\alpha}\right)^\beta\right],](/origin-help/en/images/Algorithm(Distribution_Fit)/math-930e3969310050c9b005c63c3e55c587.png?v=0 "\frac{\beta}{\alpha^\beta}x^{\beta -1} exp\left[ -\left(\frac{x}{\alpha}\right)^\beta\right],")
where /math-89d8caf054af6cb4d5b227ed3c6f371c.png?v=0 "\alpha , \beta > 0")
. With /math-88632a2aee115afccf082f1aa0c5df81.png?v=0 "E(X)=\alpha \Gamma \left(1+ \frac{1}{\beta}\right)")
and /math-bc0d001ccd32f6d9b8d24d2b2e15bbc2.png?v=0 "Var(X)=\alpha ^2 \{ \Gamma \left(1+\frac{2}{\beta}\right) -\Gamma ^2 \left(1+\frac{1}{\beta} \right) \}")
.
Maximum Likelihood Estimation(MLE)
Origin calls a NAG function nag_estim_weibull (g07bec), for the MLE of statistics of weibull distribution. Please refer to related NAG document, for more details on the algorithm.
Exponential Distribution
![\frac{1}{\sigma} exp\left[ -\frac{x}{\sigma}\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-93211aca5cc18a198a16fa64a16d62bc.png?v=0 "\frac{1}{\sigma} exp\left[ -\frac{x}{\sigma}\right]")
,
where and . With /math-083da1d1605e96215a0ef4087aafdd8b.png?v=0 "E(X)=\sigma")
and .
Maximum Likelihood Estimation(MLE)
Parameters
/math-c96984e0d1a4829be70e0ff58aec9097.png?v=0 "\hat{\sigma} = \bar{X}_n")
Confidence Interval
The confidence interval for is:
where is the critical value for the standard normal distribution in which is the confidence level. And is standard error for .
Gamma Distribution
/math-353d8a8e32ffc83a95afced239dffabb.png?v=0 "\frac{1}{\Gamma(\alpha)\sigma^\alpha}x^{\alpha -1} exp(-x/\sigma),")
where /math-14750740bcb034accaf628fa17200f81.png?v=0 "\alpha , \sigma > 0")
. With /math-bf04b662a73d46d27594a8f55662b4a0.png?v=0 "E(X)=\alpha \sigma")
and /math-535eac68cc84673d74c4a8f87ab7ed63.png?v=0 "Var(X)=\alpha \sigma ^2")
.
Maximum Likelihood Estimation(MLE)
Parameters
It's not easy to calculate MLE of /math-7b7f9dbfea05c83784f8b85149852f08.png?v=0 "\alpha")
and by hand. But with Newton-Raphson method, we can easily get what we want. In order to obtain good root of likelihood equation, we need to offer a proper initial estimator, which can be given by:
/math-441cf37268328fc4942de17036c5049b.png?v=0 "\alpha_0 = \frac{3-s+\sqrt{(s-3)^2+24s}}{12s}$,where $s = \ln \left(\frac{1}{n}\sum_{i=1}^{n}x_i \right) - \frac{1}{n}\sum_{i=1}^{n}\ln (x_i).")
Confidence Interval
The confidence interval for and /math-2554a2bb846cffd697389e5dc8912759.png?v=0 "\theta")
are:
![\left[ \hat{\alpha} - z \hat{\alpha}_{se}, \hat{\alpha} + z\hat{\alpha}_{se} \right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-0db75f463c71582cbdca0e76f50541cc.png?v=0 "\left[ \hat{\alpha} - z \hat{\alpha}_{se}, \hat{\alpha} + z\hat{\alpha}_{se} \right]")
![\left[ \frac{\hat{\theta}}{\exp \left[ (z \hat{\theta}_{se})/\hat{\theta} \right]},\hat{\theta}\exp \left[ (z \hat{\theta}_{se})/\hat{\theta} \right] \right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-7d5a0986370f7911ef882a0e600e072e.png?v=0 "\left[ \frac{\hat{\theta}}{\exp \left[ (z \hat{\theta}_{se})/\hat{\theta} \right]},\hat{\theta}\exp \left[ (z \hat{\theta}_{se})/\hat{\theta} \right] \right]")
where is the critical value for the standard normal distribution in which is the confidence level. And /math-3a0c8cf56b8e3103e12d08f326b58db2.png?v=0 "\hat{\alpha}_{se}")
is standard error for while /math-49c2974272e9431d0dca0e2f27076a33.png?v=0 "\hat{\theta}_{se}")
is for .
Binomial Distribution
/math-313a3bb44b8fb85a48fe2601a729ae2c.png?v=0 "\left( \begin{matrix} n \\ x \end{matrix}\right)
p^x (1-p)^{n-x},")
where /math-8f4ea9a0cdd8693a22df6edd418d412b.png?v=0 "0 \leq p \leq 1")
and /math-7de81659bbfe6e4c1426ff94c5d1aa6c.png?v=0 "x=0,1,2,...,n")
. With /math-2c0251c0d9c6721f02a4aff7f0bf9e1a.png?v=0 "E(X)=np")
and /math-dff83f60b9fb185f3ef6ac4b3d6a1dc4.png?v=0 "Var(X)=np(1-p)")
.
Given a number of success /math-9dd4e461268c8034f5c8564e155c67a6.png?v=0 "x")
and sample size /math-7b8b965ad4bca0e41ab51de7b31363a1.png?v=0 "n")
Maximum Likelihood Estimation(MLE)
Parameters
/math-27844e8d6f75a01cafa467a5fb783f6a.png?v=0 "\hat{p} = x/n")
Confidence Interval
![\left[\frac{1}{1+z^2/n}\left(\hat{p}+\frac{z^2}{2n} - z \sqrt{\frac{1}{n}\hat{p}(1-\hat{p})+\frac{z^2}{4n^2}}\right),\frac{1}{1+z^2/n}\left(\hat{p}+\frac{z^2}{2n} + z \sqrt{\frac{1}{n}\hat{p}(1-\hat{p})+\frac{z^2}{4n^2}}\right)\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-f3d50d49217ea8b0292780d0368d1468.png?v=0 "\left[\frac{1}{1+z^2/n}\left(\hat{p}+\frac{z^2}{2n} - z \sqrt{\frac{1}{n}\hat{p}(1-\hat{p})+\frac{z^2}{4n^2}}\right),\frac{1}{1+z^2/n}\left(\hat{p}+\frac{z^2}{2n} + z \sqrt{\frac{1}{n}\hat{p}(1-\hat{p})+\frac{z^2}{4n^2}}\right)\right]")
where is the critical value for the standard normal distribution in which is the confidence level.
Possion Distribution
/math-bcbf3ebb2bb81a74b70f00c2720a5ab8.png?v=0 "e^{-\lambda}\frac{{\lambda}^x}{x!},")
where /math-cb7443ecb882dbc563b5ef6a7882deb7.png?v=0 "x=1,2,...,n")
. With /math-062fecedb9cebe34535a0ab182b49123.png?v=0 "E(X)=Var(X)=\lambda")
.
Maximum Likelihood Estimation(MLE)
Parameters
/math-16b37f87978936f10afe731963846c28.png?v=0 "\hat{\lambda} = \frac{1}{n}\sum_{k=1}^{n}x_k")
.
Confidence Interval
The confidence interval for /math-c6a6eb61fd9c6c913da73b3642ca147d.png?v=0 "\lambda")
are:
![\left[ \hat{\lambda} - z \sqrt{\hat{\lambda}}, \hat{\lambda} + z \sqrt{\hat{\lambda}} \right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-c48a8dbb31f94b38dff287f355edda4d.png?v=0 "\left[ \hat{\lambda} - z \sqrt{\hat{\lambda}}, \hat{\lambda} + z \sqrt{\hat{\lambda}} \right]")
where is the critical value for the standard normal distribution in which is the confidence level.
Goodness of Fit
Kolmogorov-Smirnov
Origin calls a NAG function nag_1_sample_ks_test (g08cbc) , to compute the statistics. Please refer to related NAG document, for more details on the algorithm.
Kolmogorov-Smirnov(Modified)
- Modified Kolmogorov-Smirnov Statistic
The modified Kolmogorov-Smirnov statisticis a modification of the Kolmogorov-Smirnov Statistic based on different distribution.
- P-value
The p-value for the Kolmogorov-Smirnov statistic is computed based on critical values table below, provided by D’Agostino and Stephens (1986). If the value of D is between two probability levels, then linear interpolation is used to estimate the p-value.
Here /math-e4f7629b574dd05d19a2107c87806700.png?v=0 "D_n")
is the Kolmogorov-Smirnov statistic
Normal/Lognormal Distribution
- Modified Kolmogorov-Smirnov Statistic:
/math-c7904070be3a74d551bb739c1e1302aa.png?v=0 "D=D_n\left(\sqrt{N}-0.01+\frac{0.85}{\sqrt{N}}\right)")
- Critical Values Table
| D | <0.775 | 0.775 | 0.819 | 0.895 | 0.995 | 1.035 | >1.035 |
|---|---|---|---|---|---|---|---|
| P-Value | >=0.15 | 0.15 | 0.10 | 0.05 | 0.025 | 0.01 | <=0.01 |
Weibull distribution
- Modified Kolmogorov-Smirnov Statistic:
/math-840a6ff93f6a973714de5aefc582e522.png?v=0 "D=D_n\sqrt{N}")
- Critical Values Table
| D | <1.372 | 1.372 | 1.477 | 1.577 | 1.671 | >1.671 |
|---|---|---|---|---|---|---|
| P-Value | >=0.1 | 0.1 | 0.05 | 0.025 | 0.01 | <=0.01 |
Exponential Distribution
- Modified Kolmogorov-Smirnov Statistic:
/math-51f418b7ba803de514cf1c2175443db6.png?v=0 "D=\left(D_n-\frac{0.2}{N}\right)\left(\sqrt{N}+0.26+\frac{0.5}{\sqrt{N}}\right)")
- Critical Values Table
| D | <0.926 | 0.926 | 0.995 | 1.094 | 1.184 | 1.298 | >1.298 |
|---|---|---|---|---|---|---|---|
| P-Value | >=0.15 | 0.15 | 0.10 | 0.05 | 0.025 | 0.01 | <=0.01 |
Gamma Distribution
- Modified Kolmogorov-Smirnov Statistic:
/math-b40216015b4648e578387227d5b2f4fa.png?v=0 "D=D_n\left(\sqrt{N}+\frac{0.3}{\sqrt{N}}\right)")
- Critical Values Table
| D | <0.74 | 0.74 | 0.780 | 0.800 | 0.858 | 0.928 | 0.990 | 1.069 | 1.13 | >1.13 |
|---|---|---|---|---|---|---|---|---|---|---|
| P-Value | >=0.25 | 0.25 | 0.20 | 0.15 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | <=0.005 |
Anderson-Darling
- Anderson-Darling Statistics
![z=-N-\sum_{i=1}^n\frac{(2i-1)}{N}\left[lnF(Y_i)+ln(1-F(Y_{N+1-i})\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-d461965280138227f5da76f80d9689bd.png?v=0 "z=-N-\sum_{i=1}^n\frac{(2i-1)}{N}\left[lnF(Y_i)+ln(1-F(Y_{N+1-i})\right]")
- where
/math-800618943025315f869e4e1f09471012.png?v=0 "F")
is the cumulative distribution function of the specified distribution/math-762c0f38c71ffa6e93f0a1daab55ab04.png?v=0 "Y_i")
are ordered data points: /math-c185391949923d8da4f48411bbf33384.png?v=0 "Y_{1} \leq Y_2 \leq ... \leq Y_{n-1} \leq Y_n")
- P-value
- The p-value for the Adjusted Anderson-Darling statistics is computed based on critical values table below, provided by D’Agostino and Stephens (1986). If the value of
/math-2b95531f1ae4208e8e5f0b27495271cf.png?v=0 "z^{*}")
is between two probability levels, then linear interpolation is used to estimate the p-value.
- The p-value for the Adjusted Anderson-Darling statistics is computed based on critical values table below, provided by D’Agostino and Stephens (1986). If the value of
Normal/Lognormal Distribution
- Adjusted Anderson-Darling Statistics
/math-6bf71347f7d61098150247a2bfda2084.png?v=0 "z^*=z\left(1 + \frac{0.75}{N}+\frac{2.25}{N^2}\right)")
- P-value
/math-2a2b67918133d13a2817e65f322caf5c.png?v=0 "p=\begin{cases}
1-e^{-13.436+101.14z^{*}-223.73z^{*2}}, z^{*} \leq 0.2\\
1-e^{-8.318+42.796z^{*}-59.938z^{*2}}, 0.2 < z^{*} \leq 0.34\\
e^{0.9177-4.279z^{*}-1.38z^{*2}}, 0.34 < z^{*} \leq 0.6\\
e^{1.2937-5.709z^{*}+0.0186z^{*2}}, z^{*} \geq 153.467
\end{cases}")
Weibull distribution
- Adjusted Anderson-Darling Statistics
-
/math-bfd9a1456c6783613b0d73f4575e32e8.png?v=0 "z^{*}=\left(1+\frac{0.2}{N}\right)")
- Critical Values Table
|
|
<0.474 | 0.474 | 0.637 | 0.757 | 0.877 | 1.038 | >1.038 |
|---|---|---|---|---|---|---|---|
| P-Value | >=0.25 | 0.25 | 0.10 | 0.05 | 0.025 | 0.01 | <=0.01 |
Exponential Distribution
- Adjusted Anderson-Darling Statistics
-
/math-32701776e64a36ed14a21051a388193f.png?v=0 "z^{*}=z\left(1+\frac{0.6}{N}\right)")
- P-value
/math-1c9ae14bd39f8568fd2bb50af62206c8.png?v=0 "p=\begin{cases}
1-e^{-12.2204+67.459z^{*}-110.3z^{*2}}, z^{*} \leq 0.26\\
1-e^{-6.1327+20.218z^{*}-18.663z^{*2}}, 0.26 < z^{*} \leq 0.51\\
e^{0.9209-3.353z^{*}-0.3z^{*2}}, 0.51 < z^{*} \leq 0.95\\
e^{0.731-3.009z^{*}+0.15z^{*2}}, 0.95 < z^{*} \leq 10.03\\
0, z^{*} \geq 10.03
\end{cases}")
Gamma Distribution
- Critical Values Table
|
|
<0.486 | 0.486 | 0.657 | 0.786 | 0.917 | 1.092 | 1.227 | >1.227 |
|---|---|---|---|---|---|---|---|---|
| P-Value | >=0.25 | 0.25 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | <=0.005 |
/math-2e66a001f9537949375c83eeda12ae32.png?v=0 "0 < \alpha \leq 1")
|
|
<0.473 | 0.473 | 0.637 | 0.759 | 0.883 | 1.048 | 1.173 | >1.173 |
|---|---|---|---|---|---|---|---|---|
| P-Value | >=0.25 | 0.25 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | <=0.005 |
/math-06bc584d4dbbd570df241691d426a54e.png?v=0 "1 < \alpha \leq 8")
|
|
<0.470 | 0.470 | 0.631 | 0.752 | 0.873 | 1.035 | 1.159 | >1.159 |
|---|---|---|---|---|---|---|---|---|
| P-Value | >=0.25 | 0.25 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | <=0.005 |
/math-5b341847c85146df515c5a692ae03852.png?v=0 "\alpha \geq 8")
Mean Test
Z-Test
Test Statistics
/math-a074c6fdd2993c2507a84df9287219ea.png?v=0 "t=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}")
where
-
/math-ee501779fc71e7a3604af73ef039893a.png?v=0 "\bar{x}: \frac{1}{n}\sum_{i=1}^n x_i")
-
/math-3a84341711ab951b21f1b4e7295baa0e.png?v=0 "\mu_0")
: The specified test mean - : The specified standard deviation
-
P-Value
The /math-44c29edb103a2872f519ad0c9a0fdaaa.png?v=0 "P")
, is returned based on an approximate Normal test statistics /math-21c2e59531c8710156d34a3c30ac81d5.png?v=0 "Z")
.
Confidence Intervals
For the specified significance level, the confidence interval for the sample mean is:
| Null Hypothesis | Confidence Interval |
|---|---|
/math-b23316a7b376f00824a8871ab845a72c.png?v=0 "H_0:z=z_0\,\!")
|
![\left[\bar{x}-Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}}),\bar{x}+Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}})\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-f00e0ff52a4e10d48b0756a75004ebf8.png?v=0 "\left[\bar{x}-Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}}),\bar{x}+Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}})\right]")
|
/math-04d6877b0a052f28b993d0c15ec02677.png?v=0 "H_0:z \le z_0")
|
![\left[\bar{x}-Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}}), \infty\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-f273cb8d3e5319c4d25bd2f9c9b5363a.png?v=0 "\left[\bar{x}-Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}}), \infty\right]")
|
/math-73ec6779b590c748b1e6037a7fd77d38.png?v=0 "H_0:z \ge z_0")
|
![\left[-\infty, \bar{x}+Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}})\right]](/origin-help/en/images/Algorithm(Distribution_Fit)/math-1f042506a10f53d770c812ead58ccd3b.png?v=0 "\left[-\infty, \bar{x}+Z_{\frac{\sigma}{2}}(\frac{\sigma}{\sqrt{n}})\right]")
|