Statistical Intervals for Maxwell Distributions

Abstract

The problem of constructing statistical intervals for two-parameter Maxwell distribution is considered. An appropriate method of finding the maximum likelihood estimators (MLEs) is proposed. Constructions of confidence intervals, prediction intervals and one-sided tolerance limits based on suitable pivotal quantities are described. Pivotal quantities based on the MLEs, moment estimators and the modified MLEs are proposed and compared the statistical intervals based on them in terms of expected widths. Comparison studies indicate that the statistical intervals based on the MLEs offer little improvements over other interval estimates when sample sizes are small, and all intervals are practically the same even for moderate sample sizes. R functions to compute various intervals are provided and the methods are illustrated using two examples involving real data sets.

Appendices

Appendix A

We here derive an interval based on a sample of size n that would include the location parameter $\mu$ with a specified probability. The derivation below is similar to the one by Hoang-Nguyen-Thuy and Krishnamoorthy (2020) for Rayleigh distributions. Let $X_{(1)}$ denote the smallest order statistic for a sample of size n from a Maxwell$(\mu ,\sigma )$ distribution. Note that the distribution of $X_{(1)}$ is given by

$$\begin{aligned} P(X_{(1)} \le x) = 1-\left( 1-F(x|\mu ,\sigma )\right) ^n, \end{aligned}$$

where $F(x|\mu ,\sigma )$ is the CDF in (3). For a given $P \in (0,1)$, let us determine the value of t so that $P\left( X_{(1)}-t \le \mu \le X_{(1)}\right) = P(\mu \le X_{(1)} \le \mu +t)=P$. Let $G(x|\Gamma (3/2))$ denote the gamma distribution function with the shape parameter 3/2 and the scale parameter 1. For a given P, we need to determine t so that

$$\begin{aligned} P(\mu \le X_{(1)} \le \mu +t)= & {} 1-\left( 1-G(t^2/\sigma ^2|\Gamma (3/2))\right) ^n\\= & {} P. \end{aligned}$$

Solving the above equation for t, we obtain

$$\begin{aligned} t = \sigma \sqrt{G^{-1}\left( 1-(1-P)^{1/n}|3/2\right) }, \end{aligned}$$

where $G^{-1}(q|a)$ is the quantile function of gamma(a, 1). In practice, $\sigma$ is unknown, and so using the moment estimate $\widehat{\sigma }_M$, we estimate t by

$$\begin{aligned} \widehat{t} = \widehat{\sigma }_M\sqrt{G^{-1}\left( 1-(1-P)^{1/n}|3/2\right) }. \end{aligned}$$

Note that we estimated the value of t by replacing $\sigma$ by $\widehat{\sigma }_M$. By choosing $P = .999$, the interval $\left( X_{(1)}-\widehat{t}, \ X_{(1)}\right)$ is expected include with high probability.

Appendix B