Distribution fitting by maximum likelihood

### 20.4.23 Distribution fitting by maximum likelihood

The fitdistr command finds the parameters for a distribution of a specified type that best fits a set of samples.

- fitdistr takes two arguments:
  - L, a list of presumably independent and identically distributed samples.
  - distr, a distribution type, which can be one of:
    - normal or normald, for a normal distribution.
    - exp, exponential of exponentiald, for an exponential distribution.
    - poisson, for a Poisson distribution.
    - geometric, for a geometric distribution.
    - gammad, for a gamma distribution.
    - betad, for a beta distribution.
    - cauchy or cauchyd, for a Cauchy distribution.
    - weibull or weibulld for a Weibull distribution.
- fitdistr(L,distr) returns the name of the specified type of distribution with parameters that fit L most closely according to the method of maximum likelihood.

#### Examples

fitdistr(randvector(1000,weibulld,1/2,1),weibull)

weibulld | ⎛ ⎝ | 0.517079036032,1.05683817484 | ⎞ ⎠

X:=randvar(normal,stddev=9.5):; Y:=randvar(normal,stddev=1.5):; S:=sample(eval(X/Y,0),1000):; Z:=fitdistr(S,cauchy)

cauchyd | ⎛ ⎝ | 0.347058460176,6.55905486387 | ⎞ ⎠

histogram(select(x->(x>-100 and x<100),S)); plot(Z(x),x=-100..100,display=red+line_width_2)

kolmogorovt(S,Z)

| ⎡ ⎣ | “D=”,0.0161467485236,“K=”,0.510605021406,“1-kolmogorovd(K)=”,0.956753826255 | ⎤ ⎦

The Kolmogorov-Smirnov test indicates that the samples from S are drawn from Z with high probability.

You can fit a lognormal distribution to samples x1,x2,…,xn by fitting a normal distribution to the sample logarithms log x1,logx2,…,logxn because log-likelihood functions are the same. For example, generate some samples according to the lognormal rule with parameters µ=5 and σ2=2:

X:=randvar(normal,mean=5,variance=2):; S:=sample(eval(exp(X),0),1000):;

Then fit the normal distribution to logS:

Y:=fitdistr(log(S),normal)

normald(5.04754808715,1.42751619912) |  |  |  |  |  |  |  |  |  |

The mean of Y is about 5.05 and the variance is about 2.04. Now the variable Z=eY has the sought lognormal distribution.