In this paper, joint distributions of number of success runs of length k and number of failure runs of length k' are obtained by using combinatorial techniques including lattice path approach under Pólya-Eggenberger model. Some of its particular cases, for different values of the parameters, are derived. Sooner and later waiting time problems and joint distributions of number of success runs of various types until first occurrence of consecutive success runs of specified length under the model are obtained. The sooner and later waiting time problems for Bernoulli trials (see Ebneshahrashoob and Sobel [3]) and joint distributions of the type discussed by Uchiada and Aki [11] are shown as particular cases. Assuming L_{n} and S_{n} to be the lengths of longest and smallest success runs, respectively, in a sample of size n drawn by Pólya-Eggenberger sampling scheme, the joint distributions of L_{n} and S_{n} as well as distribution of M=max(L_{n},F_{n})_{n}, where F_{n} is the length of longest failure run, are also obtained.
SEN, K., AGARWAL, M. and CHAKRABORTY, S., Lattice path approach to generalized Pólya-Eggenberger model of order k, J. Statist. Planning Inference 102 (2002), 467-476.
'Lattice path approach to generalized Pólya-Eggenberger model of order ' () 102 Planning Inference : 467 -476.
SEN, K. and GOYAL, B., Distributions of the numbers of success runs of length k and failure runs of length k 1 J. Indian Statist. Assoc. 38 (2000), 1-22.
'Distributions of the numbers of success runs of length k and failure runs of length k1 ' () 38 J. Indian Statist. Assoc. : 1 -22.
LING, K. D., On binomial distributions of order k, Statist. Probab. Lett. 6 (1988), 247-250. MR 89e:60022
'On binomial distributions of order ' () 6 Probab. Lett. : 247 -250.
FELLER, W., An introduction to probability theory and its applications, Vol. 1, 3rd edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 37 # 3604
An introduction to probability theory and its applications , ().
GIBBONS, J.D., Nonparametric statistical inference, Second edition, Statistics: Textbooks and Monographs, 65, Marcel Dekker, Inc., New York, 1985. MR 86m:62067
Nonparametric statistical inference , ().
GOLDSTEIN, L., Poisson approximation and DNA sequence matching, Comm. Statist. Theory Methods 19 (1990), 4167-4179.
'Poisson approximation and DNA sequence matching ' () 19 Comm. Statist. Theory Methods : 4167 -4179.
SCHUSTER, E. F., Exchangeability and recursion in the conditional distribution theory of numbers and length of runs, Runs and patterns in probability: selected papers, Math. Appl. 283, Kluwer Acad. Publ., Dordrecht, 1994, 91-118. MR 95m:60018
'Exchangeability and recursion in the conditional distribution theory of numbers and length of runs ' () 283 Runs and patterns in probability: selected papers, Math. Appl. : 91 -118.
UCHIADA, M. and AKI, S., Sooner and later waiting time problems in a two-state Markov chain, Ann. Inst. Statist. Math. 47 (1995), 415-433. MR 97c:60173
'Sooner and later waiting time problems in a two-state Markov chain ' () 47 Ann. Inst. Statist. Math. : 415 -433.
RIORDAN, J., An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 20 #3077. Reprinted: Princeton University Press, Princeton, NJ, 1980. MR 81e:05002
An introduction to combinatorial analysis , ().
AKI, S. and HIRANO, K., Discrete distributions related to the succession events in a two-state Markov chain, Statistical sciences and data analysis (Tokyo, 1991), VSP International Science Publishers, Utrecht, 1993, 467-474. MR 96b:62022
Discrete distributions related to the succession events in a two-state Markov chain, Statistical sciences and data analysis (Tokyo, 1991) , () 467 -474.
AKI, S. and HIRANO, K'., Joint distributions of numbers of success-runs and failures until the first consecutive k-successes, Ann. Inst. Statist. Math. 47 (1995), 225-235. MR 96k:62031
'Joint distributions of numbers of success-runs and failures until the first consecutive k-successes ' () 47 Ann. Inst. Statist. Math. : 225 -235.
EBNESHAHRASHOOB, M. and SOBEL, M., Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett. 9 (1990), 5-11. MR 91e:60027
'Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas ' () 9 Statist. Probab. Lett. : 5 -11.