Distributions of runs and scans in multistate Markov exchangeable sequences

Abstract

In this paper, we look at the distributions of runs and scans in multistate Markov exchangeable sequences. The joint distributions of runs of several lengths under four types of enumeration schemes are analyzed. We evaluate the upper tail probabilities for ratchet scan statistics exactly. By utilizing the expansion of the generating functions, we propose effective computational tools for the derivation of probability functions. The results presented here provide approaches for the evaluation of the exact distributions of runs and scans in a wide class of practical problems. Finally, we discuss several applications and numerical examples to show how our theoretical results are applied to the investigation of runs and scans, as well as a parameter estimation problem.

Keywords:

• Run
• scan
• Markov exchangeability
• urn model
• system reliability
• startup demonstration test

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62E15
• Secondary 60E05

Acknowledgements

The author wishes to thank the editor and referees for the careful review of our paper and helpful suggestions which led to improved results.

A. Computational technique

The derivation of the reliability of system in Section 5.2 is facilitated by the use of the expansions of the generating function, which is given by (5.1)(5.1) $$R_2=\sum _{j_1=0}^{f_1-1}\cdots \sum _{j_m=0}^{f_m-1}[u_1^{j_1}\cdots u_m^{j_m}]{\varphi }_n({\mathbf{0}}_{1\times m},u_1,\dots ,u_m),$$(5.1) . It is noteworthy that the following formula $$\begin{array}{c}\sum _{j_1=0}^{f_1-1}\cdots \sum _{j_m=0}^{f_m-1}[u_1^{j_1}\cdots u_m^{j_m}]{\varphi }_n({\mathbf{0}}_{1\times m},u_1,\dots ,u_m)=[u_1^{f_1-1}\cdots u_m^{f_m-1}]\frac{{\varphi }_n({\mathbf{0}}_{1\times m},u_1,\dots ,u_m)}{{\prod }_{i=1}^m(1-u_i)}\end{array}$$ will lead to a computationally efficient tool for obtaining the reliability of the system, which can be easily carried out by using the computer algebra systems.

Algorithm

Define the initial function $h^0(u_1,\dots ,u_m);$

Probab (integer f₁,…,f_m)

 {

for (integer i = 1; $i\le m;$ i₊₊) {

 Set $h^i(u_{i+1},\dots ,u_m)={\frac{{\partial }^{f_i-1}}{\partial u_i^{f_i-1}}(\frac{1}{1-u_i}·h^{i-1}(u_i,\dots ,u_m))|}_{u_i=0};$

 }

return $\frac{h^m()}{(f_1-1)!(f_2-1)!\cdots (f_m-1)!};$

 }

We can get the reliability of system (5.1)(5.1) $$R_2=\sum _{j_1=0}^{f_1-1}\cdots \sum _{j_m=0}^{f_m-1}[u_1^{j_1}\cdots u_m^{j_m}]{\varphi }_n({\mathbf{0}}_{1\times m},u_1,\dots ,u_m),$$(5.1) by setting the initial function $h^0(u_1,\dots ,u_m)={\varphi }_n({\mathbf{0}}_{1\times m},u_1,\dots ,u_m).$

The startup demonstration test lengths in Section 5.3 are, of course, determined by the Equations (5.2)(5.2) $$\begin{array}{c}P\left(W_1>n\right)=P\left(N_n^1(k_1;I)=0,S_n^2<r_2,N_n^3(k_3;I)=0\right)\\ =\sum _{j_2=0}^{r_2-1}[u_2^{j_2}]{\varphi }_n(0,1,0,1,u_2,1).\end{array}$$(5.2) and (5.4)(5.4) $$\begin{array}{c}P\left(W_2>n\right)=P\left(S_n^1<r_1,S_n^2<r_2,N_n^3(k_3;I)=0\right)\\ =\sum _{j_1=0}^{r_1-1}\sum _{j_2=0}^{r_2-1}[u_1^{j_1}{u}_2^{j_2} ]{\varphi }_n(1,1,0,u_1,u_2,1).\end{array}$$(5.4) which can be performed by setting the initial functions $h^0(u_1,\dots ,u_m)={\varphi }_n(0,1,0,1,u_2,1)$ and $h^0(u_1,\dots ,u_m)={\varphi }_n(1,1,0,u_1,u_2,1),$ respectively.

We can calculate the upper tail probability for scan statistics, using the Equation (4.3)(4.3) $$P\left(Z_n^0<r_0,\dots ,Z_n^s<r_s\right)=\sum _{j_0=0}^{r_0-1}\cdots \sum _{j_s=0}^{r_s-1}[v_0^{j_0}\cdots v_s^{j_s} g_n]({\mathbf{1}}_{1\times m},v_0,\dots ,v_s),$$(4.3) . Obviously, we have $$\sum _{j_0=0}^{r_0-1}\cdots \sum _{j_s=0}^{r_s-1}[v_0^{j_0}\cdots v_s^{j_s}]g_n({\mathbf{1}}_{1\times m},v_0,\dots ,v_s)=[v_0^{r_0-1}\cdots v_s^{r_s-1}]\frac{g_n({\mathbf{1}}_{1\times m},v_0,\dots ,v_s)}{{\prod }_{i=0}^s(1-v_i)}.$$ Accordingly, the evaluation can be performed through the above Algorithm by setting the initial function $h^0(v_0,v_1,\dots ,v_s)=g_n({\mathbf{1}}_{1\times m},v_0,v_1,\dots ,v_s).$