How do you interpret nonparametric and parametric confidence intervals?

Según mi experiencia en seguros de propiedad, es más común el uso de métodos parametricos para estimaciones simples de la pérdida o bien, la de número de siniestros. Por otro lado, los no paramétricos suelen utilizarse en categorizaciones cuando se intenta segmentar el negocio. Algunas buenas referencias teóricas y de uso las encuentran en: Rincón, Luis (2012), Introducción a la teoría del riesgo, Prensa Ciencias - UNAM, Campbell, M. J., & Gardner, M. J. (1988). Calculating Confidence Intervals For Some Non-Parametric Analyses. British Medical Journal (Clinical Research Edition), 296(6634), 1454–1456. https://www.jstor.org/stable/29530803

Parametric and non parametric are statistical test methods so parametric is one that makes assumptions about the parameters of the population distribution from which one's data are drawn like one-Way ANOVA, T-tests while non-parametric test is one that makes no such assumptions like Chi-Square, Mann-Whitney tests.

Parametric confidence intervals assume that the data follows a specific distribution, typically normal, and use this assumption to estimate the interval within which the true population parameter lies with a certain level of confidence. Nonparametric confidence intervals, on the other hand, do not assume a specific data distribution and are thus more versatile, applicable to data of any type or distribution. Interpreting these intervals involves understanding the range they provide: they estimate where the true population parameter (e.g., mean, median) is likely to be found with a given level of confidence, such as 95%, without strict assumptions about the data's underlying distribution.

While the percentile method is covered here for bootstrapping, there are a few other methods to acknowledge. - Standard Error (SE): Builds confidence intervals around the sample estimate using the standard error calculated from bootstrapped samples. - Bias-Corrected (BC): Corrects for bias in the bootstrap distribution before constructing CIs. - Bias-Corrected and Accelerated (BCa): Corrects for both bias and skewness in the bootstrap distribution to provide more accurate CIs. - Studentized Bootstrap: Relies on the distribution of a t-statistic calculated from bootstrapped samples, accommodating varying standard errors across samples. Each method becomes increasingly computationally complex but can be forgivable if the data is small.

depending on the type of parameter, we can calculate the confidence intervals if it is parametric or nonparametric and the nature of data, so if you a normal distribution and large sample size , the formula of mean, standard deviation, sample size n and Z-score is the right decision to construct the confidence interval, while for nonparametric tests with small sample size, using Wilcoxon signed rank test lead us to confidence interval from the test statistic's distribution, statistical software like SPSS, R can provide the confidence intervals

Often, people misinterpret confidence intervals (CIs), viewing them as a probability distribution describing where the true parameter value is likely to lie. But CIs are a frequentist construct that indicate an interval where the true parameter will often fall if the experiment is repeated many times. Thus, post hoc interpretation of CIs can be problematic. There is a correspondence between confidence intervals and a test statistic: A 95% confidence interval is the set of parameters that would not be deemed significantly different from the observed estimate when tested with the test statistic on the 5% level. Like significance testing, interpretation of confidence intervals is most meaningful when the analysis is prespecified

Parametric tests have more inferential powerful compared to nonparametric when the distributional assumptions leading to more precise estimates confidence intervals and allow for a wider range of statistical tests while if the assumptions of parametric test like normality of data and homogeneity of variance are violated , so results can be bias and misleading regarding the nonparametric methods, it doesn't rely any assumptions about the shape of the population distribution and are robust against the presence of outliers, but it may be less powerful than parametric tests.