Distribution Fitting and Selection by Maximum Likelihood

Distribution fitting by maximum likelihood estimates the parameters of candidate probability distributions for one variable, ranks them by AIC and BIC, and checks the best fit with goodness-of-fit statistics.

한 변수에 대해 여러 후보 확률분포의 모수를 최대우도로 추정하고 AIC·BIC 로 순위를 매긴 뒤, 적합도 통계량으로 가장 나은 분포를 점검하는 방법입니다.

I have claim amounts that are positive and right-skewed. Which probability distribution describes them best, and what are its parameters?

양수이고 오른쪽으로 치우친 청구 금액 자료가 있어요. 어떤 확률분포가 가장 잘 맞고 모수는 얼마인가요?

Fit several candidate families such as lognormal, gamma and Weibull by maximum likelihood and compare them by AIC, where lower is better and differences under 2 are not decisive. Then check the best family with a goodness-of-fit statistic whose p-value comes from a parametric bootstrap, because the parameters were estimated from the same data.

When to use Distribution Fitting and Selection by Maximum Likelihood

• You need a parametric model of a single variable for simulation, risk or reliability calculations.
• You want to compare several plausible families on the same data.
• Count data may be Poisson, negative binomial or geometric and you want to check overdispersion.

When to use something else

• Observations are right-censored; use parametric survival models instead.
• You only need to test normality; use a normality test such as Shapiro–Wilk instead.

Assumptions and requirements

• Observations are independent and identically distributed.
• The data lie in the support of each candidate family; location is fixed at zero for positive families.
• AIC comparisons are only meaningful among all-continuous or all-discrete candidates.

What the analysis reports

• Parameters with standard errors from the observed information for each family
• Log-likelihood, AIC, BIC, ΔAIC and Akaike weights
• KS, Cramér–von Mises and Anderson–Darling statistics, with a parametric-bootstrap p-value for the best family
• Chi-square goodness of fit for count families
• Q-Q plot of the best continuous fit

How to report it

Among [value] candidate distributions fitted by maximum likelihood, the [family] distribution had the lowest AIC ([value]; next best [family], ΔAIC = [value]), with [parameter] = [value]; AD parametric-bootstrap p = [value].

Related methods

• Shapiro-Wilk Normality Test
• Cramér-von Mises Test
• Empirical Distribution Function with Confidence Band

Run Distribution Fitting and Selection by Maximum Likelihood on Quorant