|
چکیده
|
This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let $X_1,\ldots,X_n$ be a set of non-negative independent random variables with $X_i$, $i=1,\ldots,p$, having common distribution function $F(\lambda_1 x)$, and $X_j$, $j=p+1,\ldots,n$, having common distribution function $F(\lambda_2 x)$, where $F(\cdot)$ denotes the baseline distribution. Let $X_{i:n}(p,q)$ be the $i$-th smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that $xf'(x)/f(x)$ is decreasing in $x\in\mathcal{R}_{+}$, $\lambda_{1}\leq\lambda_{2}$ and $s_{1}\leq s_{2}$, it is shown that $X_{i:n}^{s_1}(p+k,q-k)$ is larger than $X_{i:n}^{s_2}(p,q)$ according to the likelihood ratio order for any $2\leq i\leq n$ and $k=1,2,\ldots,q$. Some parametric families of distributions are also provided to illustrate the theoretical results.
|