# normfit

Normal parameter estimates

## 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 mean `muHat` and standard deviation `sigmaHat`), given the sample data in `x`. `muHat` is the sample mean, and `sigmaHat` is 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 arrays `muCI` and `sigmaCI`, respectively.```

example

````[muHat,sigmaHat,muCI,sigmaCI] = normfit(x,alpha)` specifies the confidence level for the confidence intervals to be `100(1–alpha)`%.```
````[___] = normfit(x,alpha,censoring)` specifies whether each value in `x` is right-censored or not. Use the logical vector `censoring` in which 1 indicates observations that are right-censored and 0 indicates observations that are fully observed. With censoring, `muHat` and `sigmaHat` are 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 algorithm `normfit` to use to compute MLEs with censoring. Create `options` by using the function `statset`. You can pass in `[]` for `alpha`, `censoring`, and `freq` to use their default values.```

## Examples

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

```rng('default') % For reproducibility x = 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 = 2×1 2.4292 3.2445 ```
```sigmaCI = 2×1 4.7218 5.2989 ```

`muHat` is the sample mean, and `sigmaHat` is the square root of the unbiased estimator of the variance. `muCI` and `sigmaCI` contain 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.

Find the MLEs of a data set with censoring by using `normfit`. Use `statset` to specify the iterative algorithm options that `normfit` uses to compute MLEs for censored data, and then find the MLEs again.

`load lightbulb`

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 of `normfit` specifies 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 that `normfit` uses 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 function `statset`.

`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 ```

`normfit` displays a report on the final iteration.

The function `normfit` finds 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 using `normfit`, convert them into MLEs, and then compare the negative log likelihoods of the estimates by using `normlike`.

Generate 100 normal random numbers from the standard normal distribution.

```rng('default') % For reproducibility n = 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 between `sigmaHat` and `sigmaHat_MLE` is negligible for large `n`.

Alternatively, you can find the MLEs by using the function `mle`.

`phat = mle(x)`
```phat = 1×2 0.1231 1.1566 ```

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

Confirm that the log likelihood of the MLEs (`muHat` and `sigmaHat_MLE`) is greater than the log likelihood of the unbiased estimators (`muHat` and `sigmaHat`) by using the `normlike` function.

`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 is `100(1—alpha)`%, where `alpha` is 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 in `x`, specified as a logical vector of the same size as `x`. 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 as `x`. The `freq` input argument typically contains nonnegative integer counts for the corresponding elements in `x`, 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 in `x`.

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

Data Types: `single` | `double`

Optimization options, specified as a structure. `options` determines the control parameters for the iterative algorithm that `normfit` uses to compute MLEs for censored data.

Create `options` by using the function `statset` or by creating a structure array containing the fields and values described in this table.

Field NameValueDefault 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`

Maximum number of iterations allowed, specified as 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`

You can also enter `statset('normfit')` in the Command Window to see the names and default values of the fields that `normfit` accepts in the `options` structure.

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, `muHat` is the sample mean.

• With censoring, `muHat` is the MLE. To compute the weighted MLE, specify the weights of observations by using `freq`.

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

• With no censoring, `sigmaHat` is the square root of the unbiased estimator of the variance. To compute the MLE with no censoring, use the `mle` function.

• With censoring, `sigmaHat` is the MLE. To compute the weighted MLE, specify the weights of observations by using `freq`.

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 the `100(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 the `100(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, `normfit` uses the exact method for uncensored data and the Wald method for censored data. The exact method provides exact coverage for uncensored samples based on t and chi-square distributions.

## Alternative Functionality

`normfit` is a function specific to normal distribution. Statistics and Machine Learning Toolbox™ also offers the generic functions `mle`, `fitdist`, and `paramci` and the Distribution Fitter app, which support various probability distributions.

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

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

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