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.
![\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")
.
/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}")
.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 .
![\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 )")
.
/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)")
.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 .
![\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) \}")
.
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.
![\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 .
/math-c96984e0d1a4829be70e0ff58aec9097.png?v=0 "\hat{\sigma} = \bar{X}_n")
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 .
/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")
.
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).")
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 .
/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")
/math-27844e8d6f75a01cafa467a5fb783f6a.png?v=0 "\hat{p} = x/n")
![\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.
/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")
.
/math-16b37f87978936f10afe731963846c28.png?v=0 "\hat{\lambda} = \frac{1}{n}\sum_{k=1}^{n}x_k")
.
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.
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.
The modified Kolmogorov-Smirnov statisticis a modification of the Kolmogorov-Smirnov Statistic based on different distribution.
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
/math-c7904070be3a74d551bb739c1e1302aa.png?v=0 "D=D_n\left(\sqrt{N}-0.01+\frac{0.85}{\sqrt{N}}\right)")
| 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 |
/math-840a6ff93f6a973714de5aefc582e522.png?v=0 "D=D_n\sqrt{N}")
| 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 |
/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)")
| 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 |
/math-b40216015b4648e578387227d5b2f4fa.png?v=0 "D=D_n\left(\sqrt{N}+\frac{0.3}{\sqrt{N}}\right)")
| 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 |
![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]")
/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")
/math-2b95531f1ae4208e8e5f0b27495271cf.png?v=0 "z^{*}")
is between two probability levels, then linear interpolation is used to estimate the p-value./math-6bf71347f7d61098150247a2bfda2084.png?v=0 "z^*=z\left(1 + \frac{0.75}{N}+\frac{2.25}{N^2}\right)")
/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}")
/math-bfd9a1456c6783613b0d73f4575e32e8.png?v=0 "z^{*}=\left(1+\frac{0.2}{N}\right)")
|
|
<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 |
/math-32701776e64a36ed14a21051a388193f.png?v=0 "z^{*}=z\left(1+\frac{0.6}{N}\right)")
/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}")
|
|
<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 |
|
|
<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 |
|
|
<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-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 deviationThe /math-44c29edb103a2872f519ad0c9a0fdaaa.png?v=0 "P")
, is returned based on an approximate Normal test statistics /math-21c2e59531c8710156d34a3c30ac81d5.png?v=0 "Z")
.
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]")
|
/math-2e66a001f9537949375c83eeda12ae32.png?v=0 "0 < \alpha \leq 1")
/math-06bc584d4dbbd570df241691d426a54e.png?v=0 "1 < \alpha \leq 8")
/math-5b341847c85146df515c5a692ae03852.png?v=0 "\alpha \geq 8")