A $k$-Sample Model in Order Statistics

Abstract

Presented here is a new $k$-sample model in order statistics, in which $k$ random samples of equal size are first ordered within themselves in the usual manner, and are then ordered among themselves by considering the size of the maximum value in each sample. The distribution functions and several probabilities relevant to the model are derived. Consider $k$ random samples of size $n (X_{11}, X_{21}, \cdots, X_{n1}), (X_{12}, X_{22}, \cdots, X_{n2}), \cdots, (X_{1k}, X_{2k}, \cdots, X_{nk})$, where the $X_{ir}$ are independent and identically distributed according to the absolutely continuous distribution function $F(x)$. Arrange each sample in decreasing order, and let $Z_{1r}$ be the greatest random variable in the sample $(X_{1r}, X_{2r}, \cdots, X_{nr})$, let $Z_{2r}$ be the second greatest random variable in the same sample, and so on. The $k$ samples, after ordering, are then $(Z_{11}, Z_{21}, \cdots, Z_{n1}), (Z_{12}, Z_{22}, \cdots, Z_{n2}), \cdots, (Z_{1k}, Z_{2k}, \cdots, Z_{nk})$. Form the set $S = \{Z_{11}, Z_{12}, \cdots, Z_{1k}\}$, so that the elements of $S$ are the greatest random variables in each of the $k$ samples. Order the random variables in $S$, and let $Y_{11}$ denote the greatest, $Y_{12}$ the second greatest, and so on to $Y_{1k}$. Then for each point in the sample space, $Z_{1r}$ corresponds to some $Y_{1j}$. Define $Y_{ij}$ as the $i$th ranked random variable from the same sample as the $Z_{1r}$ mentioned above. In other words, for each point in the sample space where $Z_{1r}$ corresponds to $Y_{1j}$, the sample $(Z_{1r}, Z_{2r}, \cdots, Z_{nr})$ will be denoted by $(Y_{1j}, Y_{2j}, \cdots, Y_{nj})$. Since $Z_{1r} > Z_{2r} > \cdots > Z_{nr}$, it follows that $Y_{1j} > Y_{2j} > \cdots > Y_{nj}$. $Y_{ij}$ is called the $i$th order statistic in the $j$th sample. The number $i$ is called the rank of $Y_{ij}$ within the sample, and the number $j$ is called the rank of the sample. The above model is useful in flood frequency analysis, when it is desired to combine flood records at several independent stations in an attempt to study rare floods. Also the above model can be applied in the following situation. A large number $(nk)$ of displays have been entered in a science fair, at which $k$ prizes are to be awarded. The judge feels it is impractical to consider all $nk$ displays at once, and so he randomly divides the displays into $k$ groups of $n$ elements in each group. He then judges each group separately, ordering the $n$ displays in each group according to excellence. When each group has thus been ordered, he then considers the $k$ best displays, one from each group, and awards the 1st prize, 2nd prize, $\cdots, k$th prize to these $k$ displays according to their relative excellence. $Y_{ij}$ represents the display that was $i$th best in its group where the best display in that group won $j$th prize. The distribution function of $Y_{ij}$ is found in Section 2. A method of comparing $Y_{ij}$ with $X$, an additional random variable with the same distribution function $F(x)$, is given by the equation for $P(X > Y_{ij})$, derived in Section 3. Section 4 contains $P(Y_{i_1j_1} > Y_{i_2j_2})$ where $Y_{i_1j_1}$ and $Y_{i_2j_2}$ are two ordered random variables from the same array of $nk$ random variables. The results of Section 4 permit the comparison of any two of the ordered random variables. The comparison of $Y_{1,k}$ with $Y_{2,1}$ is related to the probability of $Y_{1,k}$ exceeding $\max_j Y_{2,j}$ which was considered by Cohn, Mosteller, Pratt, and Tatsuoka (1960). In Section 5 the expected value of $Y_{ij}$ is discussed. The fairness of the above method of awarding prizes is examined in Section 6.