continuous

An R project for numerical solution of the Kiefer-Weiss problem

Continuous Koopman-Darmois families

This directory contains the R code for sampling from a normal population.

A detailed discussion of the problem and the description of the algorithms is provided in Novikov, A., Novikov, A., Farkhshatov, F. 2023. Numerical solution of Kiefer-Weiss problems when sampling from continuous exponential families, Sequential Analysis 42 (02), pages 189-209

Content description

• The file normal.R contains all the functions providing the user interface for the Kiefer-Weiss and related problems. List of functions can be seen below.

• The file usage.R is a usage example of these functions.

• The notebook modified_kw_performance.ipynb contains a demonstration of usage of the program code for solution of the modified Kiefer-Weiss problem when sampling from a normal distribution, visualization and related calculations.

normal.R functions list

modified_kw calculates the optimal test for the modified Kiefer-Weiss problem.

Takes arguments l0, l1, th0, th1, th, H=0, precision=0.01, print_output=TRUE.

• l0, l1, th0, th1, th are input parameters of sequential test. Currently, only the case th0 < th < th1 is supported.

• Input variable H is the desired maximum number of steps the test can take. It will be reduced if step H - 1 has no continuation. In this case or if H was set to 0, it will be redefined as the biggest number n such that n - 1 has continuation. The final adjusted H can be obtained as test[["info"]]$H.

• precision is the grid size for numerical integration.

• print_output (boolean parameter) Print continuation intervals at each step of test evaluation.

Returns an R list test with two fields:

• info - test parameters l0, l1, th0, th1, th, H, precision,
• data - calculated test values.

Let H be test[["info"]]$H. For n from 1 to H - 1 test$data[[n]] is an R list with the following fields

• n - step number,
• grid - grid on the continuation interval at step n,
• val- stored values for construction of optimal test, calculated at each point of grid.

For the casen=H

• n - step number,
• const - the acceptance constant,
• grid - only one value which equals const field,
• val- equals NA.

Structure example of the modified_kw output:

{
  "info": {
    "l0": 621.0, 
    "l1": 1042.0, 
    "th0": -0.1, 
    "th1": 0.1, 
    "th": -0.016, 
    "H": 846, 
    "precision": 0.05},
  "data": [
    {"n": 1, "grid": "example", "val": "example"},
    {"n": 2, "grid": "example", "val": "example"},
    {"n": 3, "grid": "example", "val": "example"},
    ... 
    {"n": 846, "const":  "example", "grid": "example", "val": "NA"}
  ]
  
}

operating_characteristic calculates the operating characteristic curve of a sequential test

average_sample_number calculates the average sample number of a sequential test

sample_number_quantile calculates the quantile of a given level of the sample number distribution of a sequential test

calculate_g_values calculates values for functions operating_characteristic, average_sample_number, sample_number_quantile

calc_delta

The following functions are used for calculation process of modified_kw

d

bound

h_theoretical

b_normal

step_n_has_continuation

find_first_continuation_interval

find_continuation_interval

L

calculate_integral

calc_center_bound

calc_left_bound

calc_right_bound

intgr_trapezoidal

intgr