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normfit

Normal parameter estimates

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Syntax

[muHat,sigmaHat] = normfit(x)
[muHat,sigmaHat,muCI,sigmaCI] = normfit(x)
[muHat,sigmaHat,muCI,sigmaCI] = normfit(x,alpha)
[___] = normfit(x,alpha,censoring)
[___] = normfit(x,alpha,censoring,freq)
[___] = normfit(x,alpha,censoring,freq,options)

Description

example

[muHat,sigmaHat] = normfit(x)returns estimates of normal distribution parameters (the meanmuHatand standard deviationsigmaHat), given the sample data inx.muHatis the sample mean, andsigmaHatis the square root of the unbiased estimator of the variance.

[muHat,sigmaHat,muCI,sigmaCI] = normfit(x)also returns 95% confidence intervals for the parameter estimates on the mean and standard deviation in the arraysmuCIandsigmaCI, respectively.

example

[muHat,sigmaHat,muCI,sigmaCI] = normfit(x,alpha)specifies the confidence level for the confidence intervals to be100(1–alpha)%.

[___] = normfit(x,alpha,censoring)specifies whether each value inxis right-censored or not. Use the logical vectorcensoringin which 1 indicates observations that are right-censored and 0 indicates observations that are fully observed. With censoring,muHatandsigmaHatare the maximum likelihood estimates (MLEs).

[___] = normfit(x,alpha,censoring,freq)specifies the frequency or weights of observations.

example

[___] = normfit(x,alpha,censoring,freq,options)specifies optimization options for the iterative algorithmnormfitto use to compute MLEs with censoring. Createoptionsby using the functionstatset.

You can pass in[]foralpha,censoring, andfreqto use their default values.

Examples

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Open Live Script

Generate 1000 normal random numbers from the normal distribution with mean 3 and standard deviation 5.

rng('default')% For reproducibilityx = normrnd(3,5,[1000,1]);

Find the parameter estimates and the 99% confidence intervals.

[muHat,sigmaHat,muCI,sigmaCI] = normfit(x,0.01)
muHat = 2.8368
sigmaHat = 4.9948
muCI =2×12.4292 3.2445
sigmaCI =2×14.7218 5.2989

muHatis the sample mean, andsigmaHatis the square root of the unbiased estimator of the variance.muCIandsigmaCIcontain the 99% confidence intervals of the mean and standard deviation parameters, respectively. The first row is the lower bound, and the second row is the upper bound.

Open Live Script

Find the MLEs of a data set with censoring by usingnormfit. Usestatsetto specify the iterative algorithm options thatnormfituses to compute MLEs for censored data, and then find the MLEs again.

Load the sample data.

loadlightbulb

The first column of the data contains the lifetime (in hours) of two types of bulbs. The second column contains the binary variable indicating whether the bulb is fluorescent or incandescent. 1 indicates that the bulb is fluorescent, and 0 indicates that the bulb is incandescent. The third column contains the censorship information, where 0 indicates the bulb is observed until failure, and 1 indicates the item (bulb) is censored.

Find the indices for fluorescent bulbs.

idx = find(lightbulb(:,2) == 0);

Assume that the lifetime follows the normal distribution, and find the MLEs of the normal distribution parameters. The second input argument ofnormfitspecifies the confidence level. Pass in[]to use its default value 0.05. The third input argument specifies the censorship information.

censoring = lightbulb(idx,3) == 1; [muHat1,sigmaHat1] = normfit(lightbulb(idx,1),[],censoring)
muHat1 = 9.4966e+03
sigmaHat1 = 3.0640e+03

Display the default algorithm parameters thatnormfituses to estimate the normal distribution parameters.

statset('normfit')
ans =struct with fields:Display: 'off' MaxFunEvals: 200 MaxIter: 100 TolBnd: 1.0000e-06 TolFun: 1.0000e-08 TolTypeFun: [] TolX: 1.0000e-08 TolTypeX: [] GradObj: [] Jacobian: [] DerivStep: [] FunValCheck: [] Robust: [] RobustWgtFun: [] WgtFun: [] Tune: [] UseParallel: [] UseSubstreams: [] Streams: {} OutputFcn: []

Save the options using a different name. Change how the results are displayed (Display) and the termination tolerance for the objective function (TolFun).

options = statset('normfit'); options.Display ='final'; options.TolFun = 1e-10;

Alternatively, you can specify algorithm parameters by using the name-value pair arguments of the functionstatset.

options = statset('Display','final','TolFun',1e-10);

Find the MLEs with the new algorithm parameters.

[muHat2,sigmaHat2] = normfit(lightbulb(idx,1),[],censoring,[],options)
Successful convergence: Norm of gradient less than OPTIONS.TolFun
muHat2 = 9.4966e+03
sigmaHat2 = 3.0640e+03

normfitdisplays a report on the final iteration.

Open Live Script

The functionnormfitfinds the sample mean and the square root of the unbiased estimator of the variance with no censoring. The sample mean is equal to the MLE of the mean parameter, but the square root of the unbiased estimator of the variance is not equal to the MLE of the standard deviation parameter.

Find the normal distribution parameters by usingnormfit, convert them into MLEs, and then compare the negative log likelihoods of the estimates by usingnormlike.

Generate 100 normal random numbers from the standard normal distribution.

rng('default')% For reproducibilityn = 100; x = normrnd(0,1,[n,1]);

Find the sample mean and the square root of the unbiased estimator of the variance.

[muHat,sigmaHat] = normfit(x)
muHat = 0.1231
sigmaHat = 1.1624

Convert the square root of the unbiased estimator of the variance into the MLE of the standard deviation parameter.

sigmaHat_MLE = sqrt((n-1)/n)*sigmaHat
sigmaHat_MLE = 1.1566

The difference betweensigmaHatandsigmaHat_MLEis negligible for largen.

Alternatively, you can find the MLEs by using the functionmle.

phat = mle(x)
phat =1×20.1231 1.1566

phat(1)andphat(2)are the MLEs of the mean and the standard deviation parameter, respectively.

Confirm that the log likelihood of the MLEs (muHatandsigmaHat_MLE) is greater than the log likelihood of the unbiased estimators (muHatandsigmaHat) by using thenormlikefunction.

logL = -normlike([muHat,sigmaHat],x)
logL = -156.4424
logL_MLE = -normlike([muHat,sigmaHat_MLE],x)
logL_MLE = -156.4399

Input Arguments

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Sample data, specified as a vector.

Data Types:single|double

Significance level for the confidence intervals, specified as a scalar in the range (0,1). The confidence level is100(1—alpha)%, wherealphais the probability that the confidence intervals do not contain the true value.

Example:0.01

Data Types:single|double

Indicator for the censoring of each value inx, specified as a logical vector of the same size asx. Use 1 for observations that are right-censored and 0 for observations that are fully observed.

The default is an array of 0s, meaning that all observations are fully observed.

Data Types:logical

Frequency or weights of observations, specified as a nonnegative vector that is the same size asx. Thefreqinput argument typically contains nonnegative integer counts for the corresponding elements inx, but can contain any nonnegative values.

To obtain the weighted MLEs for a data set with censoring, specify weights of observations, normalized to the number of observations inx.

The default is an array of 1s, meaning one observation per element ofx.

Data Types:single|double

Optimization options, specified as a structure.optionsdetermines the control parameters for the iterative algorithm thatnormfituses to compute MLEs for censored data.

Createoptionsby using the functionstatsetor by creating a structure array containing the fields and values described in this table.

Field Name Value Default Value
Display

Amount of information displayed by the algorithm.

  • 'off'— Displays no information.

  • 'final'— Displays the final output.

 'off'
MaxFunEvals

Maximum number of objective function evaluations allowed, specified as a positive integer.

 200
MaxIter

最大允许的迭代次数,指定为a positive integer.

 100
TolBnd

Lower bound of the standard deviation parameter estimate, specified as a positive scalar.

The bounds for the mean and standard deviation parameter estimates are[–Inf,Inf]and[TolBnd,Inf], respectively.

 1e-6
TolFun

Termination tolerance for the objective function value, specified as a positive scalar.

 1e-8
TolX

Termination tolerance for the parameters, specified as a positive scalar.

 1e-8

您还可以输入statset('normfit')in the Command Window to see the names and default values of the fields thatnormfitaccepts in theoptionsstructure.

Example:statset('Display','final','MaxIter',1000)specifies to display the final information of the iterative algorithm results, and change the maximum number of iterations allowed to 1000.

Data Types:struct

Output Arguments

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Estimate of the mean parameter of the normal distribution, returned as a scalar.

  • With no censoring,muHatis the sample mean.

  • With censoring,muHatis the MLE. To compute the weighted MLE, specify the weights of observations by usingfreq.

Estimate of the standard deviation parameter of the normal distribution, returned as a scalar.

  • With no censoring,sigmaHatis the square root of the unbiased estimator of the variance. To compute the MLE with no censoring, use themlefunction.

  • With censoring,sigmaHatis the MLE. To compute the weighted MLE, specify the weights of observations by usingfreq.

Confidence interval for the mean parameter of the normal distribution, returned as a 2-by-1 column vector containing the lower and upper bounds of the100(1–alpha)% confidence interval.

The first and second rows correspond to the lower and upper bounds of the confidence intervals, respectively.

Confidence interval for the standard deviation parameter of the normal distribution, returned as a 2-by-1 column vector containing the lower and upper bounds of the100(1–alpha)% confidence interval.

The first and second rows correspond to the lower and upper bounds of the confidence intervals, respectively.

Algorithms

To compute the confidence intervals,normfituses the exact method for uncensored data and the Wald method for censored data. The exact method provides exact coverage for uncensored samples based ontand chi-square distributions.

Alternative Functionality

normfitis a function specific to normal distribution. Statistics and Machine Learning Toolbox™ also offers the generic functionsmle,fitdist, andparamciand theDistribution Fitterapp, which support various probability distributions.

  • mlereturns MLEs and the confidence intervals of MLEs for the parameters of various probability distributions. You can specify the probability distribution name or a custom probability density function.

  • Create aNormalDistributionprobability distribution object by fitting the distribution to data using thefitdistfunction or theDistribution Fitterapp. The object propertiesmuandsigmastore the parameter estimates. To obtain the confidence intervals for the parameter estimates, pass the object toparamci.

References

[1] Evans, M., N. Hastings, and B. Peacock.Statistical Distributions. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 1993.

[2] Lawless, J. F.Statistical Models and Methods for Lifetime Data. Hoboken, NJ: Wiley-Interscience, 1982.

[3] Meeker, W. Q., and L. A. Escobar.Statistical Methods for Reliability Data. Hoboken, NJ: John Wiley & Sons, Inc., 1998.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a

See Also

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