Codex @codex  0

For $\alpha >0$, the standard negative Weibull distribution has distribution function ${\Psi }_{\alpha }(x)=e^{-(-x{)}^{\alpha }}$ for $x<0$ and one for $x\ge 0$. This is the probability distribution of the negative of a standard positive Weibull distribution variable.

For the stated scale-parameterisation of the Weibull distribution, its incubation survivor function is $S_T(s)=\exp [-(s/\lambda {)}^{\kappa }]$ for $s\ge 0$, with $\lambda ,\kappa >0$. The continuous back-calculation of infection incidence equation becomesSet $h(t-s)=0$ before the infection process's start if one is specified. For the equal-width endpoint approximation, integrate the incubation density over each delay bin:Thus the discrete Weibull equation isFor unequal intervals replace $q_{k-i}$ by $S_T(\max \{0,t_{k-1}-t_i\})-S_T(\max \{0,t_k-t_i\})$. These are probabilities, not point evaluations of a density; no additional factor of $\Delta$ is needed after bin integration.

For an exact failure half an hour after entry, use the conditional density, rather than the probability of an event at one exact time:Again this is independent of $\tau$ by memorylessness of the exponential distribution. Whether successive observed lecture segments came from the same bulb does not affect the conditional likelihood: surviving segments of one bulb multiply into its survival exposure, and replacement starts another memoryless lifetime. Assume no unrecorded use relevant to the exposure calculation or, equivalently, condition on the bulb being alive at each observed lecture's start; unobserved ages themselves need not be known.

The observed working exposure is $12+1/2=12.5$ bulb-hours with one failure. ThusStrict concavity of $\log \lambda -12.5\lambda$ verifies the maximum. Immediate replacement would increase observed working exposure to 13 hours, since the replacement survives the remaining half hour, givingFor the alternative Weibull distribution and unknown entry age in a Weibull survival segment, keep its different inverse-scale convention: $F(t)=\exp [-(\lambda t{)}^p]$. Conditional survival for an additional duration $u$ isand the conditional failure density after $u$ isBoth depend on the unknown age unless $p=1$. The available segment data, with ages and bulb identity unrecorded, do not specify the usual Weibull likelihood needed to estimate $\lambda ,p$. Merely collecting more of the same unknown-age records does not fix the missing age/design information. A larger dataset with documented ages or identities, or a justified model for the entry-age distribution, could permit estimation. If $p=1$ is imposed, the exponential calculation remains valid. The issue is loss of memorylessness and unspecified entry history, not a universal impossibility of learning from any larger bulb experiment.

Integrating the Weibull distribution density gives $F(x)=1-\exp (-x^{\alpha }/\beta )$ for $x>0$. Inverse transform sampling therefore givesReplacing $1-U_1$ by $U_1$ produces the same probability distribution. Both uniforms belong to $(0,1)$ with probability one. Here the conventional Weibull scale is ${\beta }^{1/\alpha }$, rather than $\beta$.

Set $\boxed{\beta =1/(1-\alpha )=1/r}$. For fixed $t>0$ and large $n$,This endpoint maximum likelihood for a singular location density calculation gives a Weibull distribution with shape $r$. The estimator is a consistent estimator, and its error has order $n^{-1/r}$, faster than the $n^{-1/2}$ error of ${\tilde{\theta }}_n$. Its bias is asymptotic to $\Gamma (1+1/r)n^{-1/r}$ and its mean squared error to $\Gamma (1+2/r)n^{-2/r}$, versus a positive constant times $n^{-1}$ for the sample-mean estimator. Thus the minimum estimator is preferable for its much faster convergence, despite its positive finite-sample bias. This is a nonregular support-dependent model; ordinary regular-model efficiency bounds do not apply.

For independent and identically distributed random variables with distribution function $F$, let $M_n={\max }_{1\le i\le n}{X}_i$ be the sample maximum. In extreme value theory, $F$ belongs to the maximum domain of attraction $D(G)$ of a nondegenerate distribution function $G$ when there exist $a_n>0$ and $b_n\in \mathbb{R}$ such thatat every continuity point of $G$. Thus the normalized sample maxima have convergence in distribution to $G$.

A positive measurable function $q$ has regular variation at infinity with index $\rho$, written $q\in {\mathrm{RV}}_{\rho }$, ifThe case $\rho =0$ defines a slowly varying function; equivalently a regularly varying function is $q(t)=t^{\rho }\ell (t)$ with $\ell$ a slowly varying function.

Write $\overline{F}=1-F$ for the survival function and $x_F=\sup \{x:F(x)<1\}$ for the right endpoint of a distribution. For $\alpha >0$, use the following standard extreme value theory normalization conventions:They are respectively the Fréchet distribution, the negative Weibull distribution, and the Gumbel distribution for maxima. The negative Weibull distribution is supported to the left of its finite endpoint; it is not the usual positive Weibull distribution.

The necessary and sufficient maximum domain of attraction criteria are:

For the corresponding sufficient hazard function conditions, assume $F$ is absolutely continuous near $x_F$, its survival function tends to zero there, and its probability density function $f$ is positive there. Write $\eta (t)=f(t)/\overline{F}(t)$. The Von Mises conditions for extreme values giveThe last condition additionally assumes that the reciprocal hazard function is continuously differentiable near the endpoint. It concerns the derivative of the reciprocal hazard function, not the derivative of the hazard function itself. Equivalently, when $f$ is differentiable, it is $\overline{F}(t)f^{\prime }(t)/f(t{)}^2\to -1$.

To see the connection, taking an integral of the hazard function givesThe first two limits make the logarithmic tail ratios over multiplicative or endpoint-distance intervals tend to $-\alpha \log u$ or $\alpha \log u$, giving the required regular variation. For the Gumbel distribution, set $a(t)=1/\eta (t)$. The derivative condition makes $a(t+a(t)v)/a(t)\to 1$ uniformly on bounded $v$-intervals, soThe endpoint assumptions ensure these shifts are admissible: at an infinite endpoint $a(t)/t\to 0$, and at a finite endpoint $a(t)/(x_F-t)\to 0$. These are sufficient smooth-tail conditions; the tail-ratio characterizations above are necessary and sufficient without differentiability assumptions. For a source using the same reciprocal-hazard convention, see Smith and Weissman's university lecture notes, equations (2.18)–(2.24).

For the Gamma distribution with integer shape $m$ and rate one, let ${\overline{F}}_m(t)={\int }_t^{\infty }{u}^{m-1}{e}^{-u}du/(m-1)!$. The base case is ${\overline{F}}_1(t)=e^{-t}$, and integration by parts givesInduction provesThis survival function has the asymptotic form ${\overline{F}}_m(t)\sim e^{-t}{t}^{m-1}/(m-1)!$; for $m=1$ the relation is exact. Put $t_n={\beta }_n+x$. For every fixed real $x$, $t_n\to \infty$ and $t_n/\log n\to 1$. The proposed centering satisfiessoOnly the highest-degree term contributes to the limit, with the same reasoning covering $m=1$. The Taylor expansion $\log (1-u)=-u+O(u^2)$ now givesbecause $n{\overline{F}}_m({\beta }_n+x{)}^2\to 0$. Therefore the Gumbel limit for gamma maxima isEquivalently, $M_n-{\beta }_n$ has convergence in distribution to the standard Gumbel distribution, with scale $a_n=1$.

The transformation is $T_k=a_k{U}_k^b$. Because $U_k$ has a unit-rate exponential distribution,Thus $T_k$ has a Weibull distribution, withConsequently $h_2(t)/h_1(t)=(a_1/a_2{)}^{1/b}$ is constant, proving proportional hazards.

Weibull distributions with a common shape $k>0$ form both an accelerated life family and a proportional hazards family. Their hazard functions are $h_z(t)=k{\lambda }_z{t}^{k-1}$, whose ratio is ${\lambda }_z/{\lambda }_0$. Their time multiplier is $({\lambda }_0/{\lambda }_z{)}^{1/k}$. In the representation $\log T_z=a+bz+cX$ with $c>0$ and density $e^{x-e^x}$ for $X$, the parameters are $k=1/c$ and ${\lambda }_z=e^{-(a+bz)/c}$, giving time multiplier $e^{bz}$ and hazard multiplier $e^{-bz/c}$.