Package 'gofmalm'

Maximum Likelihood Estimation for Type-II Censored Samples

Description

Estimates distribution parameters under Type-II right- or left-censoring.

Usage

fit_mle(
  data,
  n,
  pdf,
  cdf,
  sf = NULL,
  params,
  censoring = c("right", "left"),
  optim_control = list()
)

Arguments

data	Numeric vector of length r, sorted ascending.
n	Total sample size (n >= r).
pdf	Probability density function.
cdf	Cumulative distribution function.
sf	Survival function. If NULL, computed as 1 - cdf(x, params).
params	Named list or vector of starting parameter values.
censoring	Type of censoring: "right" or "left".
optim_control	Optional control list passed to optim.

Value

A list containing par (estimated parameters), value (negative log-likelihood), convergence status code, and message.

Examples

data_exp <- c(0.1, 0.2, 0.3, 0.4, 0.7, 1.0, 1.4)
fit <- fit_mle(
  data = data_exp, n = 20,
  pdf = function(x, p) dexp(x, rate = p$rate),
  cdf = function(x, p) pexp(x, rate = p$rate),
  params = list(rate = 0.1),
  censoring = "right"
)
fit$par

---

Goodness-of-Fit Testing for Type-II Censored Samples

Description

Performs goodness-of-fit tests for arbitrary user-specified continuous parametric distributions under Type-II right- or left-censoring using the Malmquist and Michael-Schucany transformations.

Usage

gof_censored(
  data,
  n,
  pdf,
  cdf,
  sf = NULL,
  params = NULL,
  estimate = TRUE,
  censoring = c("right", "left"),
  transform = c("both", "malmquist", "ms"),
  statistics = c("all", "direct7", "complete_ad", "composite_mle", "direct_censored"),
  pvalue_method = c("auto", "distribution_free", "parametric_bootstrap"),
  nsim = 2000,
  seed = NULL,
  optim_control = list()
)

Arguments

data	Numeric vector of observed order statistics, length r.
n	Total sample size (n >= length(data)).
pdf	Probability density function f(x, params).
cdf	Cumulative distribution function F(x, params).
sf	Survival function S(x, params). If NULL, computed as 1 - cdf(x, params).
params	Named list/vector of fixed parameter values (if estimate = FALSE) or starting values for MLE (if estimate = TRUE).
estimate	Logical; TRUE for composite hypothesis (unknown parameters estimated via MLE), FALSE for simple hypothesis.
censoring	Type of censoring: "right" or "left".
transform	Transformations to compute: "both", "malmquist", or "ms".
statistics	Subset of statistics to compute: "all", "direct7", "complete_ad", "composite_mle", or "direct_censored".
pvalue_method	Method for p-values: "auto", "distribution_free", or "parametric_bootstrap".
nsim	Number of Monte Carlo or bootstrap simulation replicates.
seed	Optional random seed.
optim_control	Optional control list passed to optim.

Value

An S3 object of class "gofmalm".

Examples

# Simple hypothesis test on Uniform(0,1) with r=5, n=10
data_unif <- c(0.03, 0.06, 0.10, 0.11, 0.13)
res <- gof_censored(
  data = data_unif, n = 10,
  pdf = dunif, cdf = punif,
  estimate = FALSE, pvalue_method = "distribution_free",
  nsim = 500, seed = 123
)
print(res)

---

Compute Distribution-Free p-values for Simple Hypotheses

Description

Obtains p-values for EDF goodness-of-fit statistics via Monte Carlo simulation from Uniform(0,1) order statistics.

Usage

pvalue_distribution_free(
  u_obs,
  n,
  r,
  censoring = c("right", "left"),
  nsim = 2000,
  seed = NULL
)

Arguments

u_obs	Observed order statistics u_1:n < ... < u_r:n.
n	Total sample size.
r	Number of observed failures.
censoring	Type of censoring: "right" or "left".
nsim	Number of Monte Carlo simulation replicates (default 2000).
seed	Optional random seed.

Value

Named numeric vector of empirical p-values for each statistic.

---

Compute Parametric Bootstrap p-values for Composite Hypotheses

Description

Obtains p-values for composite hypothesis tests where parameters are estimated from data.

Usage

pvalue_parametric_bootstrap(
  data,
  n,
  pdf,
  cdf,
  sf = NULL,
  params_hat,
  censoring = c("right", "left"),
  transform = c("both", "malmquist", "ms"),
  nsim = 2000,
  seed = NULL,
  optim_control = list()
)

Arguments

data	Observed data vector of length r.
n	Total sample size.
pdf	Density function.
cdf	Distribution function.
sf	Survival function.
params_hat	Estimated parameter list from observed data.
censoring	Type of censoring.
transform	Transformation method ("ms", "malmquist", or "both").
nsim	Number of bootstrap replicates (default 2000).
seed	Optional random seed.
optim_control	Optional control list passed to optim.

Value

A list containing empirical p-values and number of valid replicates.

---

Composite Hypothesis Procedure for Type-II Censored Samples

Description

Evaluates goodness-of-fit under unknown parameters using MLE estimation, transformation to complete sample, normal residual standardization, and modified CvM/AD statistics.

Usage

stat_composite_mle(
  data,
  n,
  pdf,
  cdf,
  sf = NULL,
  params,
  censoring = c("right", "left"),
  transform = c("malmquist", "ms"),
  optim_control = list()
)

Arguments

data	Numeric vector of observed order statistics.
n	Total sample size (n >= length(data)).
pdf	Probability density function taking (x, params).
cdf	Cumulative distribution function taking (x, params).
sf	Survival function taking (x, params). If NULL, computed as 1 - cdf(x, params).
params	Named list/vector of starting parameter values for MLE.
censoring	Type of censoring: "right" or "left".
transform	Transformation method: "ms" (Michael-Schucany) or "malmquist".
optim_control	Optional control list passed to optim.

Value

A list containing:

stat_W2	Modified Cramér-von Mises statistic (_mW2 or _lW2).
stat_A2	Modified Anderson-Darling statistic (_mA2 or _lA2).
W2	Raw Cramér-von Mises statistic on standardized residuals.
A2	Raw Anderson-Darling statistic on standardized residuals.
theta_hat	Estimated parameter values.
convergence	Optimization convergence code.

Examples

data_exp <- c(0.1, 0.2, 0.3, 0.4, 0.7, 1.0, 1.4)
stat_composite_mle(
  data = data_exp, n = 20,
  pdf = function(x, p) dexp(x, rate = p$rate),
  cdf = function(x, p) pexp(x, rate = p$rate),
  params = list(rate = 0.1),
  censoring = "right",
  transform = "malmquist"
)

---

Compute Direct Censored EDF Statistics (Pettitt & Stephens, 1976)

Description

Computes the untransformed direct censored Cramér-von Mises and Anderson-Darling statistics.

Usage

stat_direct_censored(u_cens, n)

Arguments

u_cens	Numeric vector of order statistics u_1:n < ... < u_r:n in (0, 1).
n	Total sample size (n >= length(u_cens)).

Value

Named numeric vector containing W2_2_rn and A2_2_rn.

Examples

u <- c(0.03, 0.06, 0.10, 0.11, 0.13)
stat_direct_censored(u, n = 10)

---

Compute EDF Statistics for Type-II Censored Samples

Description

Computes the nine EDF test statistics described in Lin, Huang & Balakrishnan (2008).

Usage

stat_edf_censored(u_cens, n, u_ms = NULL, u_malmquist = NULL)

Arguments

u_cens	Numeric vector of order statistics u_1:n < ... < u_r:n in (0, 1).
n	Total sample size (n >= length(u_cens)).
u_ms	Optional pre-computed Michael & Schucany transformed sample.
u_malmquist	Optional pre-computed Malmquist transformed sample.

Value

A named numeric vector of 9 statistics: D_rn, G_rn, T_rn, W2_star_rn, W2_rn, A2_rn, U2_rn, TA2_r, and star_TA2_r.

Examples

u <- c(0.03, 0.06, 0.10, 0.11, 0.13)
stat_edf_censored(u, n = 10)

---

Malmquist (1950) Transformation for Type-II Censored Samples

Description

Converts an r-out-of-n Type-II right- or left-censored uniform sample into an independent complete Uniform(0,1) sample of size r (Lin, Huang & Balakrishnan, 2008).

Usage

transform_malmquist(u_cens, n, censoring = c("right", "left"))

Arguments

u_cens	Numeric vector of length r containing PIT-transformed order statistics.
n	Total sample size (n >= length(u_cens)).
censoring	Type of censoring: "right" or "left".

Value

A list containing:

W	Sorted complete sample W_1:r < ... < W_r:r of size r.
V	Vector of intermediate Beta-distributed random variables V_n, ..., V_{n-r+1}.

Examples

u <- c(0.03, 0.06, 0.10, 0.11, 0.13)
transform_malmquist(u, n = 10, censoring = "right")

---

Michael & Schucany (1979) Transformation

Description

Transforms an r-out-of-n Type-II right-censored uniform sample into a complete sample of size r.

Usage

transform_ms(u_cens, n)

Arguments

u_cens	Numeric vector of order statistics u_1:n < ... < u_r:n in (0, 1).
n	Total sample size (n >= length(u_cens)).

Value

Numeric vector of length r representing the transformed complete sample Z_1:r < ... < Z_r:r.

Examples

u <- c(0.03, 0.06, 0.10, 0.11, 0.13)
transform_ms(u, n = 10)

---