Nonparametric maximum likelihood estimation of the distribution function using ranked-set sampling

Abstract

Kvam and Samaniego (J Am Stat Assoc 89: 526–537, 1994) derived an estimator that they billed as the nonparametric maximum likelihood estimator (MLE) of the distribution function based on a ranked-set sample. However, we show here that the likelihood used by Kvam and Samaniego (1994) is different from the probability of seeing the observed sample under perfect rankings. By appealing to results on order statistics from a discrete distribution, we write down a likelihood that matches the probability of seeing the observed sample. We maximize this likelihood by using the EM algorithm, and we show that the resulting MLE avoids certain unintuitive behavior exhibited by the Kvam and Samaniego (1994) estimator. We find that the new MLE outperforms both the Kvam and Samaniego (1994) estimator and the unbiased estimator due to Stokes and Sager (J Am Stat Assoc 83: 374– 381, 1988) in terms of integrated mean squared error under perfect rankings.

Appendix

We will demonstrate here that, for the three estimators considered in Sect. 4, $MSE_{{\hat{F}}}(x)$ depends on x only through F(x) if F(x) is continuous and if the rankings are done using the fraction-of-random-rankings model. It then follows that the IMSE and RIMSE values from Sect. 4 are distribution-free for continuous distributions. The perfect rankings case of this result is implicit in Sect. 4 of Kvam and Samaniego (1993), though not stated in a formal way.

Let $Y_1,\ldots ,Y_N$ be the independent judgment order statistics to be included in the sample, with the associated ranks and set sizes being $s_1,\ldots ,s_N$ and $m_1,\ldots ,m_N$. Using the fact that the three estimators considered in Sect. 4 are step functions with ${\hat{F}}(x)$ values that depend only on the ranks of the ordered sample values and on which ordered values x falls between, we have that

$$\begin{aligned} MSE_{{\hat{F}}}(x) = \sum _{\pi \in S_N} \sum _{i=0}^N P\left( Y_{\pi (1)}< \cdots< Y_{\pi (N)}, Y_{\pi (i)}< x < Y_{\pi (i+1)}\right) \left( {\hat{F}}(\pi ,i) - F(x)\right) ^2, \end{aligned}$$

where $S_N$ is the permutation group on the integers $\{1,\ldots ,N\}$, $Y_{\pi (0)}$ and $Y_{\pi (N+1)}$ are given by $Y_{\pi (0)} \equiv -\infty$ and $Y_{\pi (N+1)} \equiv \infty$, and ${\hat{F}}(\pi ,i)$ is the estimated value for F(x) on the interval $x \in [Y_{\pi (i)},Y_{\pi (i+1)})$ when the judgment order statistics are ordered as $Y_{\pi (1)}< \cdots < Y_{\pi (N)}$. In other words, $MSE_{{\hat{F}}}(x)$ can be obtained by running through all possible values for ${\hat{F}}(x)$ and weighting the corresponding squared errors by the probability of the particular ${\hat{F}}(x)$ value occurring. In Sect. 4, we did this for a specific example in Table 2 and the associated discussion.

By the probability integral transform, if Y is a random draw from the distribution with distribution function F(x), then $F(Y) \sim \text{ Uniform }(0,1)$. Similarly, if $Y_{r:m}$ is an rth order statistic from a set of size m, then $F(Y_{r:m}) \sim \text{ Beta }(r,m+1-r)$. Under the fraction-of-random-rankings model, $Y_i$ is either a true order statistic or, with probability $\lambda$, a random draw from the parent distribution. Thus, the values $F(Y_1),\ldots ,F(Y_N)$ are independently distributed, with $F(Y_i)$ being a mixture of the $\text{ Uniform }(0,1)$ and $\text{ Beta }(s_i,m_i+1-s_i)$ distributions where the components get weights $1-\lambda$ and $\lambda$. Applying $F(\cdot )$ to each part of the inequality in the expression for $MSE_{{\hat{F}}}(x)$ and using the fact that, by continuity, the $F(Y_i)$ values are all distinct with probability 1, we have that

$$\begin{aligned} MSE_{{\hat{F}}}(x)= & {} \sum _{\pi \in S_N} \sum _{i=0}^N P\left( F(Y_{\pi (1)})< \cdots< F(Y_{\pi (N)}),\right. \\{} & {} \left. F(Y_{\pi (i)})< F(x)< F(Y_{\pi (i+1)})\right) \left( {\hat{F}}(\pi ,i) - F(x)\right) ^2, \end{aligned}$$

where the only dependence on x is through the two instances of F(x).