Order Statistics Based on a Combined Simple Random Sample from a Finite Population and Applications to Inference

Abstract

In this paper, we study probability distributions of order statistics from a set obtained by combining several simple random samples (SRS) selected from the same finite population. Each simple random sample is taken using without replacement selection procedure and does not contain any ties. On the other hand, in the combined sample, the same observation may appear more than once since each SRS is selected from the same finite population. Consequently, the number of the distinct observations in the combined sample is a discrete random variable. We provide the probability mass function of this discrete random variable. Next, using the order statistics in the combined SRSs, we construct confidence intervals for the quantiles and outer-inner confidence intervals for the quantile interval of a finite population. Finally, we also present a prediction interval for a future observation from the same finite population.

Appendices

Appendix:

Proof 1 (Proof of Theorem 1).

Suppose j = 0. Then, there is no tie in the combined sample. The number of combined samples with no ties can be obtained by selecting n₁ units out of N in the first sample and then n₂ units out of the remaining N − n₁ in the second sample, that is,

$$ \begin{array}{@{}rcl@{}} \text{ Number of sampled}~S_{1:2}~\text{with no ties}=\left( \begin{array}{c}{N} \\ {n_{1}}\end{array}\right)\left( \begin{array}{c} {N-n_{1}} \\ {n_{2}}\end{array}\right). \end{array} $$

Since the samples are selected independently, we readily have

$$ \begin{array}{@{}rcl@{}} P(j=0)= \frac{\left( \begin{array}{c} {N} \\ {n_{1}}\end{array}\right)\left( \begin{array}{c}{N-n_{1}} \\ {n_{2}}\end{array}\right)}{ \left( \begin{array}{c} {N} \\ {n_{1}}\end{array}\right) \left( \begin{array}{c}{N} \\ {n_{2}}\end{array}\right)}. \end{array} $$

Now, suppose 0 < j ≤ n₁. We must have j tied pairs in the combined sample S_1,2, and suppose that j specific units in the population are tied in both samples. Then, there are

$$ \begin{array}{@{}rcl@{}} \left( \begin{array}{c} {N-j} \\ {n_{1}-j}\end{array}\right)\left( \begin{array}{c} {N-n_{1}} \\ {n_{2}-j}\end{array}\right) \end{array} $$

different ways to select the remaining untied units in both samples. Since j units in the population can be selected in $\left (\begin {array}{c} {N} \\ {j}\end {array}\right )$ different ways, using the product rule we then obtain

$$ \begin{array}{@{}rcl@{}} P(j~\text{tied pairs in } S_{1:2})= \frac{\left( \begin{array}{c}{N} \\ {j}\end{array}\right) \left( \begin{array}{c} {N-j} \\ {n_{1}-j}\end{array}\right)\left( \begin{array}{c} {N-n_{1})} \\ {n_{2}-j}\end{array}\right)}{\left( \begin{array}{c} {N} \\ {n_{1}}\end{array}\right) \left( \begin{array}{c} {N} \\ {n_{2}}\end{array}\right)}, \end{array} $$

which completes the proof. □

Proof 2 (Proof of Theorem 2).

This is a special case of Theorem 3 and so there is no need to present its proof. □

Proof 3 (Proof of Theorem 3).

By conditioning on u_K = u, we can write

$$ \begin{array}{@{}rcl@{}} P(Z_{(i:u)}=x_{t}|u_{K}=u)= f_{(i:u)}(x_{t}), \end{array} $$

where f_(i:u) is the pmf of the i-th order statistic from a sample of size u selected without replacement from the population $\mathcal {P}$. If i < n_K, the unconditional distribution follows from the joint distribution as

$$ \begin{array}{@{}rcl@{}} P(Z_{(i:n_{T})}=x_{t})= \sum\limits_{u=n_{K}}^{n_{T}} f_{(i:u)}(x_{t}) P(u_{K}=u), \quad i=1, \ldots, n_{k}-1, \end{array} $$

where P(u_K ≥ i) = 1 if i < n_k. If i ≥ n_K, the i-th order statistic is observed only if there are at least i distinct observations in the combined sample S_1:K. Hence we must use the truncated distribution of U_k to marginalize the distribution of $Z_{(i:n_{T})}$ to get

$$ \begin{array}{@{}rcl@{}} P(Z_{(i:n_{T})}=x_{t})= \sum\limits_{u=n_{K}}^{n_{T}} \frac{f_{(i:u)}(x_{t}) P(u_{K}=u)}{P(u_{K} \ge i)}, \quad i=n_{K}, \ldots, n_{T}. \end{array} $$

The proof then gets completed by combining these two pieces together. □

R-function

############################# # This function is used by StepKF ### tieF=function(KV,N,n,m){ ret<-rep(0,width(KV)) ki=1 for(k in KV){ if(k > min(n,m)) {print("k must be less than minimum of n and m");return} ret[ki]= choose(N,k)*(choose(N-k,n-k)*choose(N-n,m-k))/(choose (N,n)*choose(N,m)) ki=ki+1 } return(ret) }

######################################## # This function computes the probability mass function # of sample size of distinct observations in the # K-th combined sample # P1: probability mass function of sample size in (K-1)-st step # NV: Sample size vector, ordered from smallest to largest StepKF=function(NV){ K=width(NV) # width of sample size vector P1= matrix(c(NV[1],1),ncol=2) for (k in (2:K)){ ML=min(P1[,1],NV[k]) MU=sum(NV[1:k]) # maximum value of distinct observations in step k Ck=NV[k]:MU # The range of sample size of distinct P2=matrix(0,ncol=2,nrow=width(Ck)) # updated pmf of sample size # in the K-th step P2[,1]=Ck # Values of sample size of distinct obs in step k pd=dim(P1) dv=P1[,1] # Values of sample size of distinct obs in step k-1 d=width(dv) newind=1:width(Ck) # this is used to locate the sample size in the range # in k-th step oldind=1:d # this is used to locate the sample size in the range # in (k-1)-th step for(dd in dv){ Jk=0:min(dd,NV[k]) # The tie vector in the K-th step given that # (k-1)-st step had dd distinct observations JkD=sort(Jk,decreasing=TRUE) dold=which(dd==dv) MV=sort(c(P1[dold,1],NV[k])) TPkV=tieF(JkD,N,MV[1],MV[2]) # this computes the probability having #JkD ties when we combine the distinct observations # in step k-1 with sample n_k for(jkn in JkD){ dnew=which(Ck==(dd+NV[k]-jkn)) P2[dnew,2]=P2[dnew,2]+P1[dold,2]*tieF(jkn,N,MV[1],MV[2]) # this updates # the probability of sample size in step k } } P1=P2 } colnames(P1)=c("u_K","P(U_K)") return(P1) }